Daily Papers

Data assimilation (DA), as an indispensable component within contemporary Numerical Weather Prediction (NWP) systems, plays a crucial role in generating the analysis that significantly impacts forecast performance. Nevertheless, the development of an efficient DA system poses significant challenges, particularly in establishing intricate relationships between the background data and the vast amount of multi-source observation data within limited time windows in operational settings. To address these challenges, researchers design complex pre-processing methods for each observation type, leveraging approximate modeling and the power of super-computing clusters to expedite solutions. The emergence of deep learning (DL) models has been a game-changer, offering unified multi-modal modeling, enhanced nonlinear representation capabilities, and superior parallelization. These advantages have spurred efforts to integrate DL models into various domains of weather modeling. Remarkably, DL models have shown promise in matching, even surpassing, the forecast accuracy of leading operational NWP models worldwide. This success motivates the exploration of DL-based DA frameworks tailored for weather forecasting models. In this study, we introduces FuxiDA, a generalized DL-based DA framework for assimilating satellite observations. By assimilating data from Advanced Geosynchronous Radiation Imager (AGRI) aboard Fengyun-4B, FuXi-DA consistently mitigates analysis errors and significantly improves forecast performance. Furthermore, through a series of single-observation experiments, Fuxi-DA has been validated against established atmospheric physics, demonstrating its consistency and reliability.

Models of many engineering and natural systems are imperfect. The discrepancy between the mathematical representations of a true physical system and its imperfect model is called the model error. These model errors can lead to substantial differences between the numerical solutions of the model and the state of the system, particularly in those involving nonlinear, multi-scale phenomena. Thus, there is increasing interest in reducing model errors, particularly by leveraging the rapidly growing observational data to understand their physics and sources. Here, we introduce a framework named MEDIDA: Model Error Discovery with Interpretability and Data Assimilation. MEDIDA only requires a working numerical solver of the model and a small number of noise-free or noisy sporadic observations of the system. In MEDIDA, first the model error is estimated from differences between the observed states and model-predicted states (the latter are obtained from a number of one-time-step numerical integrations from the previous observed states). If observations are noisy, a data assimilation (DA) technique such as ensemble Kalman filter (EnKF) is employed to provide the analysis state of the system, which is then used to estimate the model error. Finally, an equation-discovery technique, here the relevance vector machine (RVM), a sparsity-promoting Bayesian method, is used to identify an interpretable, parsimonious, and closed-form representation of the model error. Using the chaotic Kuramoto-Sivashinsky (KS) system as the test case, we demonstrate the excellent performance of MEDIDA in discovering different types of structural/parametric model errors, representing different types of missing physics, using noise-free and noisy observations.

Weather forecasting traditionally relies on numerical weather prediction (NWP) systems that integrates global observational systems, data assimilation (DA), and forecasting models. Despite steady improvements in forecast accuracy over recent decades, further advances are increasingly constrained by high computational costs, the underutilization of vast observational datasets, and the challenges of obtaining finer resolution. These limitations, alongside the uneven distribution of observational networks, result in global disparities in forecast accuracy, leaving some regions vulnerable to extreme weather. Recent advances in machine learning present a promising alternative, providing more efficient and accurate forecasts using the same initial conditions as NWP. However, current machine learning models still depend on the initial conditions generated by NWP systems, which require extensive computational resources and expertise. Here we introduce FuXi Weather, a machine learning weather forecasting system that assimilates data from multiple satellites. Operating on a 6-hourly DA and forecast cycle, FuXi Weather generates reliable and accurate 10-day global weather forecasts at a spatial resolution of 0.25^circ. FuXi Weather is the first system to achieve all-grid, all-surface, all-channel, and all-sky DA and forecasting, extending skillful forecast lead times beyond those of the European Centre for Medium-range Weather Forecasts (ECMWF) high-resolution forecasts (HRES) while using significantly fewer observations. FuXi Weather consistently outperforms ECMWF HRES in observation-sparse regions, such as central Africa, demonstrating its potential to improve forecasts where observational infrastructure is limited.

The weather forecasting system is important for science and society, and significant achievements have been made in applying artificial intelligence (AI) to medium-range weather forecasting. However, existing AI-based weather forecasting models rely on analysis or reanalysis products from traditional numerical weather prediction (NWP) systems as initial conditions for making predictions. Initial states are typically generated by traditional data assimilation components, which are computational expensive and time-consuming. Here we present an AI-based data assimilation model, i.e., Adas, for global weather variables. By introducing the confidence matrix, Adas employs gated convolution to handle sparse observations and gated cross-attention for capturing the interactions between the background and observations. Further, we combine Adas with the advanced AI-based forecasting model (i.e., FengWu) to construct the first end-to-end AI-based global weather forecasting system: FengWu-Adas. We demonstrate that Adas can assimilate global observations to produce high-quality analysis, enabling the system operate stably for long term. Moreover, we are the first to apply the methods to real-world scenarios, which is more challenging and has considerable practical application potential. We have also achieved the forecasts based on the analyses generated by AI with a skillful forecast lead time exceeding that of the IFS for the first time.

Earth system models suffer from various structural and parametric errors in their representation of nonlinear, multi-scale processes, leading to uncertainties in their long-term projections. The effects of many of these errors (particularly those due to fast physics) can be quantified in short-term simulations, e.g., as differences between the predicted and observed states (analysis increments). With the increase in the availability of high-quality observations and simulations, learning nudging from these increments to correct model errors has become an active research area. However, most studies focus on using neural networks, which while powerful, are hard to interpret, are data-hungry, and poorly generalize out-of-distribution. Here, we show the capabilities of Model Error Discovery with Interpretability and Data Assimilation (MEDIDA), a general, data-efficient framework that uses sparsity-promoting equation-discovery techniques to learn model errors from analysis increments. Using two-layer quasi-geostrophic turbulence as the test case, MEDIDA is shown to successfully discover various linear and nonlinear structural/parametric errors when full observations are available. Discovery from spatially sparse observations is found to require highly accurate interpolation schemes. While NNs have shown success as interpolators in recent studies, here, they are found inadequate due to their inability to accurately represent small scales, a phenomenon known as spectral bias. We show that a general remedy, adding a random Fourier feature layer to the NN, resolves this issue enabling MEDIDA to successfully discover model errors from sparse observations. These promising results suggest that with further development, MEDIDA could be scaled up to models of the Earth system and real observations.

A data-driven model (DDM) suitable for regional weather forecasting applications is presented. The model extends the Artificial Intelligence Forecasting System by introducing a stretched-grid architecture that dedicates higher resolution over a regional area of interest and maintains a lower resolution elsewhere on the globe. The model is based on graph neural networks, which naturally affords arbitrary multi-resolution grid configurations. The model is applied to short-range weather prediction for the Nordics, producing forecasts at 2.5 km spatial and 6 h temporal resolution. The model is pre-trained on 43 years of global ERA5 data at 31 km resolution and is further refined using 3.3 years of 2.5 km resolution operational analyses from the MetCoOp Ensemble Prediction System (MEPS). The performance of the model is evaluated using surface observations from measurement stations across Norway and is compared to short-range weather forecasts from MEPS. The DDM outperforms both the control run and the ensemble mean of MEPS for 2 m temperature. The model also produces competitive precipitation and wind speed forecasts, but is shown to underestimate extreme events.

Distillation techniques have substantially improved the sampling speed of diffusion models, allowing of the generation within only one step or a few steps. However, these distillation methods require extensive training for each dataset, sampler, and network, which limits their practical applicability. To address this limitation, we propose a straightforward distillation approach, Distilled-ODE solvers (D-ODE solvers), that optimizes the ODE solver rather than training the denoising network. D-ODE solvers are formulated by simply applying a single parameter adjustment to existing ODE solvers. Subsequently, D-ODE solvers with smaller steps are optimized by ODE solvers with larger steps through distillation over a batch of samples. Our comprehensive experiments indicate that D-ODE solvers outperform existing ODE solvers, including DDIM, PNDM, DPM-Solver, DEIS, and EDM, especially when generating samples with fewer steps. Our method incur negligible computational overhead compared to previous distillation techniques, enabling simple and rapid integration with previous samplers. Qualitative analysis further shows that D-ODE solvers enhance image quality while preserving the sampling trajectory of ODE solvers.

Recently, various methods have been proposed to address the inconsistency issue of DDIM inversion to enable image editing, such as EDICT [36] and Null-text inversion [22]. However, the above methods introduce considerable computational overhead. In this paper, we propose a new technique, named bi-directional integration approximation (BDIA), to perform exact diffusion inversion with neglible computational overhead. Suppose we would like to estimate the next diffusion state z_{i-1} at timestep t_i with the historical information (i,z_i) and (i+1,z_{i+1}). We first obtain the estimated Gaussian noise boldsymbol{epsilon}(z_i,i), and then apply the DDIM update procedure twice for approximating the ODE integration over the next time-slot [t_i, t_{i-1}] in the forward manner and the previous time-slot [t_i, t_{t+1}] in the backward manner. The DDIM step for the previous time-slot is used to refine the integration approximation made earlier when computing z_i. A nice property of BDIA-DDIM is that the update expression for z_{i-1} is a linear combination of (z_{i+1}, z_i, boldsymbol{epsilon}(z_i,i)). This allows for exact backward computation of z_{i+1} given (z_i, z_{i-1}), thus leading to exact diffusion inversion. It is demonstrated with experiments that (round-trip) BDIA-DDIM is particularly effective for image editing. Our experiments further show that BDIA-DDIM produces markedly better image sampling qualities than DDIM for text-to-image generation. BDIA can also be applied to improve the performance of other ODE solvers in addition to DDIM. In our work, it is found that applying BDIA to the EDM sampling procedure produces consistently better performance over four pre-trained models.

Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based solutions face important challenges when dealing with spatio-temporal PDEs over long time scales. Specifically, the current theory of NOs does not present a systematic framework to perform data assimilation and efficiently correct the evolution of PDE solutions over time based on sparsely sampled noisy measurements. In this paper, we propose a learning-based state-space approach to compute the solution operators to infinite-dimensional semilinear PDEs. Exploiting the structure of semilinear PDEs and the theory of nonlinear observers in function spaces, we develop a flexible recursive method that allows for both prediction and data assimilation by combining prediction and correction operations. The proposed framework is capable of producing fast and accurate predictions over long time horizons, dealing with irregularly sampled noisy measurements to correct the solution, and benefits from the decoupling between the spatial and temporal dynamics of this class of PDEs. We show through experiments on the Kuramoto-Sivashinsky, Navier-Stokes and Korteweg-de Vries equations that the proposed model is robust to noise and can leverage arbitrary amounts of measurements to correct its prediction over a long time horizon with little computational overhead.

Finding an initial noise vector that produces an input image when fed into the diffusion process (known as inversion) is an important problem in denoising diffusion models (DDMs), with applications for real image editing. The state-of-the-art approach for real image editing with inversion uses denoising diffusion implicit models (DDIMs) to deterministically noise the image to the intermediate state along the path that the denoising would follow given the original conditioning. However, DDIM inversion for real images is unstable as it relies on local linearization assumptions, which result in the propagation of errors, leading to incorrect image reconstruction and loss of content. To alleviate these problems, we propose Exact Diffusion Inversion via Coupled Transformations (EDICT), an inversion method that draws inspiration from affine coupling layers. EDICT enables mathematically exact inversion of real and model-generated images by maintaining two coupled noise vectors which are used to invert each other in an alternating fashion. Using Stable Diffusion, a state-of-the-art latent diffusion model, we demonstrate that EDICT successfully reconstructs real images with high fidelity. On complex image datasets like MS-COCO, EDICT reconstruction significantly outperforms DDIM, improving the mean square error of reconstruction by a factor of two. Using noise vectors inverted from real images, EDICT enables a wide range of image edits--from local and global semantic edits to image stylization--while maintaining fidelity to the original image structure. EDICT requires no model training/finetuning, prompt tuning, or extra data and can be combined with any pretrained DDM. Code is available at https://github.com/salesforce/EDICT.

Data Augmentation (DA), \ie, synthesizing faithful and diverse samples to expand the original training set, is a prevalent and effective strategy to improve various visual recognition tasks. With the powerful image generation ability, diffusion-based DA has shown strong performance gains on different benchmarks. In this paper, we analyze today's diffusion-based DA methods, and argue that they cannot take account of both faithfulness and diversity, which are two critical keys for generating high-quality samples and boosting final classification performance. To this end, we propose a novel Diffusion-based Inversion Interpolation DA method: Diff-II. Specifically, Diff-II consists of three main steps: 1) Category concepts learning: Learning concept embeddings for each category. 2) Inversion interpolation: Calculating the inversion for each image, and conducting spherical interpolation for two randomly sampled inversions from the same category. 3) Two-stage denoising: Using different prompts to generate synthesized images in a coarse-to-fine manner. Extensive experiments on multiple image classification tasks (\eg, few-shot, long-tailed, and out-of-distribution classification) have demonstrated its effectiveness over state-of-the-art diffusion-based DA methods.

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

A popular approach to sample a diffusion-based generative model is to solve an ordinary differential equation (ODE). In existing samplers, the coefficients of the ODE solvers are pre-determined by the ODE formulation, the reverse discrete timesteps, and the employed ODE methods. In this paper, we consider accelerating several popular ODE-based sampling processes (including EDM, DDIM, and DPM-Solver) by optimizing certain coefficients via improved integration approximation (IIA). We propose to minimize, for each time step, a mean squared error (MSE) function with respect to the selected coefficients. The MSE is constructed by applying the original ODE solver for a set of fine-grained timesteps, which in principle provides a more accurate integration approximation in predicting the next diffusion state. The proposed IIA technique does not require any change of a pre-trained model, and only introduces a very small computational overhead for solving a number of quadratic optimization problems. Extensive experiments show that considerably better FID scores can be achieved by using IIA-EDM, IIA-DDIM, and IIA-DPM-Solver than the original counterparts when the neural function evaluation (NFE) is small (i.e., less than 25).

When solving inverse problems, it is increasingly popular to use pre-trained diffusion models as plug-and-play priors. This framework can accommodate different forward models without re-training while preserving the generative capability of diffusion models. Despite their success in many imaging inverse problems, most existing methods rely on privileged information such as derivative, pseudo-inverse, or full knowledge about the forward model. This reliance poses a substantial limitation that restricts their use in a wide range of problems where such information is unavailable, such as in many scientific applications. To address this issue, we propose Ensemble Kalman Diffusion Guidance (EnKG) for diffusion models, a derivative-free approach that can solve inverse problems by only accessing forward model evaluations and a pre-trained diffusion model prior. We study the empirical effectiveness of our method across various inverse problems, including scientific settings such as inferring fluid flows and astronomical objects, which are highly non-linear inverse problems that often only permit black-box access to the forward model.

Democratization of machine learning requires architectures that automatically adapt to new problems. Neural Differential Equations (NDEs) have emerged as a popular modeling framework by removing the need for ML practitioners to choose the number of layers in a recurrent model. While we can control the computational cost by choosing the number of layers in standard architectures, in NDEs the number of neural network evaluations for a forward pass can depend on the number of steps of the adaptive ODE solver. But, can we force the NDE to learn the version with the least steps while not increasing the training cost? Current strategies to overcome slow prediction require high order automatic differentiation, leading to significantly higher training time. We describe a novel regularization method that uses the internal cost heuristics of adaptive differential equation solvers combined with discrete adjoint sensitivities to guide the training process towards learning NDEs that are easier to solve. This approach opens up the blackbox numerical analysis behind the differential equation solver's algorithm and directly uses its local error estimates and stiffness heuristics as cheap and accurate cost estimates. We incorporate our method without any change in the underlying NDE framework and show that our method extends beyond Ordinary Differential Equations to accommodate Neural Stochastic Differential Equations. We demonstrate how our approach can halve the prediction time and, unlike other methods which can increase the training time by an order of magnitude, we demonstrate similar reduction in training times. Together this showcases how the knowledge embedded within state-of-the-art equation solvers can be used to enhance machine learning.

The performance of an algorithm often critically depends on its parameter configuration. While a variety of automated algorithm configuration methods have been proposed to relieve users from the tedious and error-prone task of manually tuning parameters, there is still a lot of untapped potential as the learned configuration is static, i.e., parameter settings remain fixed throughout the run. However, it has been shown that some algorithm parameters are best adjusted dynamically during execution, e.g., to adapt to the current part of the optimization landscape. Thus far, this is most commonly achieved through hand-crafted heuristics. A promising recent alternative is to automatically learn such dynamic parameter adaptation policies from data. In this article, we give the first comprehensive account of this new field of automated dynamic algorithm configuration (DAC), present a series of recent advances, and provide a solid foundation for future research in this field. Specifically, we (i) situate DAC in the broader historical context of AI research; (ii) formalize DAC as a computational problem; (iii) identify the methods used in prior-art to tackle this problem; (iv) conduct empirical case studies for using DAC in evolutionary optimization, AI planning, and machine learning.

During a geosteering operation the well path is intentionally adjusted in response to the new data acquired while drilling. To achieve consistent high-quality decisions, especially when drilling in complex environments, decision support systems can help cope with high volumes of data and interpretation complexities. They can assimilate the real-time measurements into a probabilistic earth model and use the updated model for decision recommendations. Recently, machine learning (ML) techniques have enabled a wide range of methods that redistribute computational cost from on-line to off-line calculations. In this paper, we introduce two ML techniques into the geosteering decision support framework. Firstly, a complex earth model representation is generated using a Generative Adversarial Network (GAN). Secondly, a commercial extra-deep electromagnetic simulator is represented using a Forward Deep Neural Network (FDNN). The numerical experiments demonstrate that the combination of the GAN and the FDNN in an ensemble randomized maximum likelihood data assimilation scheme provides real-time estimates of complex geological uncertainty. This yields reduction of geological uncertainty ahead of the drill-bit from the measurements gathered behind and around the well bore.

Data-driven deep learning models are transforming global weather forecasting. It is an open question if this success can extend to climate modeling, where the complexity of the data and long inference rollouts pose significant challenges. Here, we present the first conditional generative model that produces accurate and physically consistent global climate ensemble simulations by emulating a coarse version of the United States' primary operational global forecast model, FV3GFS. Our model integrates the dynamics-informed diffusion framework (DYffusion) with the Spherical Fourier Neural Operator (SFNO) architecture, enabling stable 100-year simulations at 6-hourly timesteps while maintaining low computational overhead compared to single-step deterministic baselines. The model achieves near gold-standard performance for climate model emulation, outperforming existing approaches and demonstrating promising ensemble skill. This work represents a significant advance towards efficient, data-driven climate simulations that can enhance our understanding of the climate system and inform adaptation strategies.

Implicit layer deep learning techniques, like Neural Differential Equations, have become an important modeling framework due to their ability to adapt to new problems automatically. Training a neural differential equation is effectively a search over a space of plausible dynamical systems. However, controlling the computational cost for these models is difficult since it relies on the number of steps the adaptive solver takes. Most prior works have used higher-order methods to reduce prediction timings while greatly increasing training time or reducing both training and prediction timings by relying on specific training algorithms, which are harder to use as a drop-in replacement due to strict requirements on automatic differentiation. In this manuscript, we use internal cost heuristics of adaptive differential equation solvers at stochastic time points to guide the training toward learning a dynamical system that is easier to integrate. We "close the black-box" and allow the use of our method with any adjoint technique for gradient calculations of the differential equation solution. We perform experimental studies to compare our method to global regularization to show that we attain similar performance numbers without compromising the flexibility of implementation on ordinary differential equations (ODEs) and stochastic differential equations (SDEs). We develop two sampling strategies to trade off between performance and training time. Our method reduces the number of function evaluations to 0.556-0.733x and accelerates predictions by 1.3-2x.

Diffusion model-based inverse problem solvers have shown impressive performance, but are limited in speed, mostly as they require reverse diffusion sampling starting from noise. Several recent works have tried to alleviate this problem by building a diffusion process, directly bridging the clean and the corrupted for specific inverse problems. In this paper, we first unify these existing works under the name Direct Diffusion Bridges (DDB), showing that while motivated by different theories, the resulting algorithms only differ in the choice of parameters. Then, we highlight a critical limitation of the current DDB framework, namely that it does not ensure data consistency. To address this problem, we propose a modified inference procedure that imposes data consistency without the need for fine-tuning. We term the resulting method data Consistent DDB (CDDB), which outperforms its inconsistent counterpart in terms of both perception and distortion metrics, thereby effectively pushing the Pareto-frontier toward the optimum. Our proposed method achieves state-of-the-art results on both evaluation criteria, showcasing its superiority over existing methods.

Current deep learning models for dynamics forecasting struggle with generalization. They can only forecast in a specific domain and fail when applied to systems with different parameters, external forces, or boundary conditions. We propose a model-based meta-learning method called DyAd which can generalize across heterogeneous domains by partitioning them into different tasks. DyAd has two parts: an encoder which infers the time-invariant hidden features of the task with weak supervision, and a forecaster which learns the shared dynamics of the entire domain. The encoder adapts and controls the forecaster during inference using adaptive instance normalization and adaptive padding. Theoretically, we prove that the generalization error of such procedure is related to the task relatedness in the source domain, as well as the domain differences between source and target. Experimentally, we demonstrate that our model outperforms state-of-the-art approaches on both turbulent flow and real-world ocean data forecasting tasks.

The paper introduces D-CODE, a new framework blending Data Colony Optimization (DCO) algorithms inspired by biological colonies' collective behaviours with Dynamic Efficiency (DE) models for real-time adaptation. DCO utilizes metaheuristic strategies from ant colonies, bee swarms, and fungal networks to efficiently explore complex data landscapes, while DE enables continuous resource recalibration and process adjustments for optimal performance amidst changing conditions. Through a mixed-methods approach involving simulations and case studies, D-CODE outperforms traditional techniques, showing improvements of 3-4% in solution quality, 2-3 times faster convergence rates, and up to 25% higher computational efficiency. The integration of DCO's robust optimization and DE's dynamic responsiveness positions D-CODE as a transformative paradigm for intelligent systems design, with potential applications in operational efficiency, decision support, and computational intelligence, supported by empirical validation and promising outcomes.

Plankton recognition provides novel possibilities to study various environmental aspects and an interesting real-world context to develop domain adaptation (DA) methods. Different imaging instruments cause domain shift between datasets hampering the development of general plankton recognition methods. A promising remedy for this is DA allowing to adapt a model trained on one instrument to other instruments. In this paper, we present a new DA dataset called DAPlankton which consists of phytoplankton images obtained with different instruments. Phytoplankton provides a challenging DA problem due to the fine-grained nature of the task and high class imbalance in real-world datasets. DAPlankton consists of two subsets. DAPlankton_LAB contains images of cultured phytoplankton providing a balanced dataset with minimal label uncertainty. DAPlankton_SEA consists of images collected from the Baltic Sea providing challenging real-world data with large intra-class variance and class imbalance. We further present a benchmark comparison of three widely used DA methods.

Digital Surface Models (DSM) offer a wealth of height information for understanding the Earth's surface as well as monitoring the existence or change in natural and man-made structures. Classical height estimation requires multi-view geospatial imagery or LiDAR point clouds which can be expensive to acquire. Single-view height estimation using neural network based models shows promise however it can struggle with reconstructing high resolution features. The latest advancements in diffusion models for high resolution image synthesis and editing have yet to be utilized for remote sensing imagery, particularly height estimation. Our approach involves training a generative diffusion model to learn the joint distribution of optical and DSM images across both domains as a Markov chain. This is accomplished by minimizing a denoising score matching objective while being conditioned on the source image to generate realistic high resolution 3D surfaces. In this paper we experiment with conditional denoising diffusion probabilistic models (DDPM) for height estimation from a single remotely sensed image and show promising results on the Vaihingen benchmark dataset.

Modern techniques for physical simulations rely on numerical schemes and mesh-refinement methods to address trade-offs between precision and complexity, but these handcrafted solutions are tedious and require high computational power. Data-driven methods based on large-scale machine learning promise high adaptivity by integrating long-range dependencies more directly and efficiently. In this work, we focus on fluid dynamics and address the shortcomings of a large part of the literature, which are based on fixed support for computations and predictions in the form of regular or irregular grids. We propose a novel setup to perform predictions in a continuous spatial and temporal domain while being trained on sparse observations. We formulate the task as a double observation problem and propose a solution with two interlinked dynamical systems defined on, respectively, the sparse positions and the continuous domain, which allows to forecast and interpolate a solution from the initial condition. Our practical implementation involves recurrent GNNs and a spatio-temporal attention observer capable of interpolating the solution at arbitrary locations. Our model not only generalizes to new initial conditions (as standard auto-regressive models do) but also performs evaluation at arbitrary space and time locations. We evaluate on three standard datasets in fluid dynamics and compare to strong baselines, which are outperformed both in classical settings and in the extended new task requiring continuous predictions.

Biophysical modeling, particularly involving partial differential equations (PDEs), offers significant potential for tailoring disease treatment protocols to individual patients. However, the inverse problem-solving aspect of these models presents a substantial challenge, either due to the high computational requirements of model-based approaches or the limited robustness of deep learning (DL) methods. We propose a novel framework that leverages the unique strengths of both approaches in a synergistic manner. Our method incorporates a DL ensemble for initial parameter estimation, facilitating efficient downstream evolutionary sampling initialized with this DL-based prior. We showcase the effectiveness of integrating a rapid deep-learning algorithm with a high-precision evolution strategy in estimating brain tumor cell concentrations from magnetic resonance images. The DL-Prior plays a pivotal role, significantly constraining the effective sampling-parameter space. This reduction results in a fivefold convergence acceleration and a Dice-score of 95%

Global climate models (GCMs) are the main tools for understanding and predicting climate change. However, due to limited numerical resolutions, these models suffer from major structural uncertainties; e.g., they cannot resolve critical processes such as small-scale eddies in atmospheric and oceanic turbulence. Thus, such small-scale processes have to be represented as a function of the resolved scales via closures (parametrization). The accuracy of these closures is particularly important for capturing climate extremes. Traditionally, such closures are based on heuristics and simplifying assumptions about the unresolved physics. Recently, supervised-learned closures, trained offline on high-fidelity data, have been shown to outperform the classical physics-based closures. However, this approach requires a significant amount of high-fidelity training data and can also lead to instabilities. Reinforcement learning is emerging as a potent alternative for developing such closures as it requires only low-order statistics and leads to stable closures. In Scientific Multi-Agent Reinforcement Learning (SMARL) computational elements serve a dual role of discretization points and learning agents. We leverage SMARL and fundamentals of turbulence physics to learn closures for prototypes of atmospheric and oceanic turbulence. The policy is trained using only the enstrophy spectrum, which is nearly invariant and can be estimated from a few high-fidelity samples (these few samples are far from enough for supervised/offline learning). We show that these closures lead to stable low-resolution simulations that, at a fraction of the cost, can reproduce the high-fidelity simulations' statistics, including the tails of the probability density functions. The results demonstrate the high potential of SMARL for closure modeling for GCMs, especially in the regime of scarce data and indirect observations.

Existing customization methods require access to multiple reference examples to align pre-trained diffusion probabilistic models (DPMs) with user-provided concepts. This paper aims to address the challenge of DPM customization when the only available supervision is a differentiable metric defined on the generated contents. Since the sampling procedure of DPMs involves recursive calls to the denoising UNet, na\"ive gradient backpropagation requires storing the intermediate states of all iterations, resulting in extremely high memory consumption. To overcome this issue, we propose a novel method AdjointDPM, which first generates new samples from diffusion models by solving the corresponding probability-flow ODEs. It then uses the adjoint sensitivity method to backpropagate the gradients of the loss to the models' parameters (including conditioning signals, network weights, and initial noises) by solving another augmented ODE. To reduce numerical errors in both the forward generation and gradient backpropagation processes, we further reparameterize the probability-flow ODE and augmented ODE as simple non-stiff ODEs using exponential integration. Finally, we demonstrate the effectiveness of AdjointDPM on three interesting tasks: converting visual effects into identification text embeddings, finetuning DPMs for specific types of stylization, and optimizing initial noise to generate adversarial samples for security auditing.

While diffusion models can successfully generate data and make predictions, they are predominantly designed for static images. We propose an approach for efficiently training diffusion models for probabilistic spatiotemporal forecasting, where generating stable and accurate rollout forecasts remains challenging, Our method, DYffusion, leverages the temporal dynamics in the data, directly coupling it with the diffusion steps in the model. We train a stochastic, time-conditioned interpolator and a forecaster network that mimic the forward and reverse processes of standard diffusion models, respectively. DYffusion naturally facilitates multi-step and long-range forecasting, allowing for highly flexible, continuous-time sampling trajectories and the ability to trade-off performance with accelerated sampling at inference time. In addition, the dynamics-informed diffusion process in DYffusion imposes a strong inductive bias and significantly improves computational efficiency compared to traditional Gaussian noise-based diffusion models. Our approach performs competitively on probabilistic forecasting of complex dynamics in sea surface temperatures, Navier-Stokes flows, and spring mesh systems.

In this paper we present a review of the Kornia differentiable data augmentation (DDA) module for both for spatial (2D) and volumetric (3D) tensors. This module leverages differentiable computer vision solutions from Kornia, with an aim of integrating data augmentation (DA) pipelines and strategies to existing PyTorch components (e.g. autograd for differentiability, optim for optimization). In addition, we provide a benchmark comparing different DA frameworks and a short review for a number of approaches that make use of Kornia DDA.

The 100 MW cryogenic liquid oxygen/hydrogen multi-injector combustor BKD operated by the DLR Institute of Space Propulsion is a research platform that allows the study of thermoacoustic instabilities under realistic conditions, representative of small upper stage rocket engines. We use data from BKD experimental campaigns in which the static chamber pressure and fuel-oxidizer ratio are varied such that the first tangential mode of the combustor is excited under some conditions. We train an autoregressive Bayesian neural network model to forecast the amplitude of the dynamic pressure time series, inputting multiple sensor measurements (injector pressure/ temperature measurements, static chamber pressure, high-frequency dynamic pressure measurements, high-frequency OH* chemiluminescence measurements) and future flow rate control signals. The Bayesian nature of our algorithms allows us to work with a dataset whose size is restricted by the expense of each experimental run, without making overconfident extrapolations. We find that the networks are able to accurately forecast the evolution of the pressure amplitude and anticipate instability events on unseen experimental runs 500 milliseconds in advance. We compare the predictive accuracy of multiple models using different combinations of sensor inputs. We find that the high-frequency dynamic pressure signal is particularly informative. We also use the technique of integrated gradients to interpret the influence of different sensor inputs on the model prediction. The negative log-likelihood of data points in the test dataset indicates that predictive uncertainties are well-characterized by our Bayesian model and simulating a sensor failure event results as expected in a dramatic increase in the epistemic component of the uncertainty.

In the face of escalating climate changes, typhoon intensities and their ensuing damage have surged. Accurate trajectory prediction is crucial for effective damage control. Traditional physics-based models, while comprehensive, are computationally intensive and rely heavily on the expertise of forecasters. Contemporary data-driven methods often rely on reanalysis data, which can be considered to be the closest to the true representation of weather conditions. However, reanalysis data is not produced in real-time and requires time for adjustment because prediction models are calibrated with observational data. This reanalysis data, such as ERA5, falls short in challenging real-world situations. Optimal preparedness necessitates predictions at least 72 hours in advance, beyond the capabilities of standard physics models. In response to these constraints, we present an approach that harnesses real-time Unified Model (UM) data, sidestepping the limitations of reanalysis data. Our model provides predictions at 6-hour intervals for up to 72 hours in advance and outperforms both state-of-the-art data-driven methods and numerical weather prediction models. In line with our efforts to mitigate adversities inflicted by typhoons, we release our preprocessed PHYSICS TRACK dataset, which includes ERA5 reanalysis data, typhoon best-track, and UM forecast data.

Denoising Diffusion Probabilistic Models (DDPMs) can generate high-quality samples such as image and audio samples. However, DDPMs require hundreds to thousands of iterations to produce final samples. Several prior works have successfully accelerated DDPMs through adjusting the variance schedule (e.g., Improved Denoising Diffusion Probabilistic Models) or the denoising equation (e.g., Denoising Diffusion Implicit Models (DDIMs)). However, these acceleration methods cannot maintain the quality of samples and even introduce new noise at a high speedup rate, which limit their practicability. To accelerate the inference process while keeping the sample quality, we provide a fresh perspective that DDPMs should be treated as solving differential equations on manifolds. Under such a perspective, we propose pseudo numerical methods for diffusion models (PNDMs). Specifically, we figure out how to solve differential equations on manifolds and show that DDIMs are simple cases of pseudo numerical methods. We change several classical numerical methods to corresponding pseudo numerical methods and find that the pseudo linear multi-step method is the best in most situations. According to our experiments, by directly using pre-trained models on Cifar10, CelebA and LSUN, PNDMs can generate higher quality synthetic images with only 50 steps compared with 1000-step DDIMs (20x speedup), significantly outperform DDIMs with 250 steps (by around 0.4 in FID) and have good generalization on different variance schedules. Our implementation is available at https://github.com/luping-liu/PNDM.

We present a data-driven approach for forecasting global weather using graph neural networks. The system learns to step forward the current 3D atmospheric state by six hours, and multiple steps are chained together to produce skillful forecasts going out several days into the future. The underlying model is trained on reanalysis data from ERA5 or forecast data from GFS. Test performance on metrics such as Z500 (geopotential height) and T850 (temperature) improves upon previous data-driven approaches and is comparable to operational, full-resolution, physical models from GFS and ECMWF, at least when evaluated on 1-degree scales and when using reanalysis initial conditions. We also show results from connecting this data-driven model to live, operational forecasts from GFS.

Test-time adaptation harnesses test inputs to improve the accuracy of a model trained on source data when tested on shifted target data. Existing methods update the source model by (re-)training on each target domain. While effective, re-training is sensitive to the amount and order of the data and the hyperparameters for optimization. We instead update the target data, by projecting all test inputs toward the source domain with a generative diffusion model. Our diffusion-driven adaptation method, DDA, shares its models for classification and generation across all domains. Both models are trained on the source domain, then fixed during testing. We augment diffusion with image guidance and self-ensembling to automatically decide how much to adapt. Input adaptation by DDA is more robust than prior model adaptation approaches across a variety of corruptions, architectures, and data regimes on the ImageNet-C benchmark. With its input-wise updates, DDA succeeds where model adaptation degrades on too little data in small batches, dependent data in non-uniform order, or mixed data with multiple corruptions.

The optimization of the latents and parameters of diffusion models with respect to some differentiable metric defined on the output of the model is a challenging and complex problem. The sampling for diffusion models is done by solving either the probability flow ODE or diffusion SDE wherein a neural network approximates the score function allowing a numerical ODE/SDE solver to be used. However, naive backpropagation techniques are memory intensive, requiring the storage of all intermediate states, and face additional complexity in handling the injected noise from the diffusion term of the diffusion SDE. We propose a novel family of bespoke ODE solvers to the continuous adjoint equations for diffusion models, which we call AdjointDEIS. We exploit the unique construction of diffusion SDEs to further simplify the formulation of the continuous adjoint equations using exponential integrators. Moreover, we provide convergence order guarantees for our bespoke solvers. Significantly, we show that continuous adjoint equations for diffusion SDEs actually simplify to a simple ODE. Lastly, we demonstrate the effectiveness of AdjointDEIS for guided generation with an adversarial attack in the form of the face morphing problem. Our code will be released on our project page https://zblasingame.github.io/AdjointDEIS/

In this study, we examine the representation learning abilities of Denoising Diffusion Models (DDM) that were originally purposed for image generation. Our philosophy is to deconstruct a DDM, gradually transforming it into a classical Denoising Autoencoder (DAE). This deconstructive procedure allows us to explore how various components of modern DDMs influence self-supervised representation learning. We observe that only a very few modern components are critical for learning good representations, while many others are nonessential. Our study ultimately arrives at an approach that is highly simplified and to a large extent resembles a classical DAE. We hope our study will rekindle interest in a family of classical methods within the realm of modern self-supervised learning.

A diffusion model, which is formulated to produce an image using thousands of denoising steps, usually suffers from a slow inference speed. Existing acceleration algorithms simplify the sampling by skipping most steps yet exhibit considerable performance degradation. By viewing the generation of diffusion models as a discretized integrating process, we argue that the quality drop is partly caused by applying an inaccurate integral direction to a timestep interval. To rectify this issue, we propose a timestep aligner that helps find a more accurate integral direction for a particular interval at the minimum cost. Specifically, at each denoising step, we replace the original parameterization by conditioning the network on a new timestep, which is obtained by aligning the sampling distribution to the real distribution. Extensive experiments show that our plug-in design can be trained efficiently and boost the inference performance of various state-of-the-art acceleration methods, especially when there are few denoising steps. For example, when using 10 denoising steps on the popular LSUN Bedroom dataset, we improve the FID of DDIM from 9.65 to 6.07, simply by adopting our method for a more appropriate set of timesteps. Code will be made publicly available.

Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation (DLRA) has proven to be a promising approach. DLRA represents the solution as a low-rank tensor factorization and evolves the resulting low-rank factors in time. A central challenge in DLRA is to find time integration schemes that are robust to the arising small singular values. A robust parallel basis update & Galerkin integrator, which simultaneously evolves all low-rank factors, has recently been derived for matrix differential equations. This work extends the parallel low-rank matrix integrator to Tucker tensors and general tree tensor networks, yielding an algorithm in which all bases and connecting tensors are evolved in parallel over a time step. We formulate the algorithm, provide a robust error bound, and demonstrate the efficiency of the new integrators for problems in quantum many-body physics, uncertainty quantification, and radiative transfer.

Despite all recent progress, it is still challenging to edit and manipulate natural images with modern generative models. When using Generative Adversarial Network (GAN), one major hurdle is in the inversion process mapping a real image to its corresponding noise vector in the latent space, since its necessary to be able to reconstruct an image to edit its contents. Likewise for Denoising Diffusion Implicit Models (DDIM), the linearization assumption in each inversion step makes the whole deterministic inversion process unreliable. Existing approaches that have tackled the problem of inversion stability often incur in significant trade-offs in computational efficiency. In this work we propose an Accelerated Iterative Diffusion Inversion method, dubbed AIDI, that significantly improves reconstruction accuracy with minimal additional overhead in space and time complexity. By using a novel blended guidance technique, we show that effective results can be obtained on a large range of image editing tasks without large classifier-free guidance in inversion. Furthermore, when compared with other diffusion inversion based works, our proposed process is shown to be more robust for fast image editing in the 10 and 20 diffusion steps' regimes.

Irregular sampling intervals and missing values in real-world time series data present challenges for conventional methods that assume consistent intervals and complete data. Neural Ordinary Differential Equations (Neural ODEs) offer an alternative approach, utilizing neural networks combined with ODE solvers to learn continuous latent representations through parameterized vector fields. Neural Stochastic Differential Equations (Neural SDEs) extend Neural ODEs by incorporating a diffusion term, although this addition is not trivial, particularly when addressing irregular intervals and missing values. Consequently, careful design of drift and diffusion functions is crucial for maintaining stability and enhancing performance, while incautious choices can result in adverse properties such as the absence of strong solutions, stochastic destabilization, or unstable Euler discretizations, significantly affecting Neural SDEs' performance. In this study, we propose three stable classes of Neural SDEs: Langevin-type SDE, Linear Noise SDE, and Geometric SDE. Then, we rigorously demonstrate their robustness in maintaining excellent performance under distribution shift, while effectively preventing overfitting. To assess the effectiveness of our approach, we conduct extensive experiments on four benchmark datasets for interpolation, forecasting, and classification tasks, and analyze the robustness of our methods with 30 public datasets under different missing rates. Our results demonstrate the efficacy of the proposed method in handling real-world irregular time series data.

Global climate models typically operate at a grid resolution of hundreds of kilometers and fail to resolve atmospheric mesoscale processes, e.g., clouds, precipitation, and gravity waves (GWs). Model representation of these processes and their sources is essential to the global circulation and planetary energy budget, but subgrid scale contributions from these processes are often only approximately represented in models using parameterizations. These parameterizations are subject to approximations and idealizations, which limit their capability and accuracy. The most drastic of these approximations is the "single-column approximation" which completely neglects the horizontal evolution of these processes, resulting in key biases in current climate models. With a focus on atmospheric GWs, we present the first-ever global simulation of atmospheric GW fluxes using machine learning (ML) models trained on the WINDSET dataset to emulate global GW emulation in the atmosphere, as an alternative to traditional single-column parameterizations. Using an Attention U-Net-based architecture trained on globally resolved GW momentum fluxes, we illustrate the importance and effectiveness of global nonlocality, when simulating GWs using data-driven schemes.

Diffusion probabilistic models (DPMs) have exhibited excellent performance for high-fidelity image generation while suffering from inefficient sampling. Recent works accelerate the sampling procedure by proposing fast ODE solvers that leverage the specific ODE form of DPMs. However, they highly rely on specific parameterization during inference (such as noise/data prediction), which might not be the optimal choice. In this work, we propose a novel formulation towards the optimal parameterization during sampling that minimizes the first-order discretization error of the ODE solution. Based on such formulation, we propose DPM-Solver-v3, a new fast ODE solver for DPMs by introducing several coefficients efficiently computed on the pretrained model, which we call empirical model statistics. We further incorporate multistep methods and a predictor-corrector framework, and propose some techniques for improving sample quality at small numbers of function evaluations (NFE) or large guidance scales. Experiments show that DPM-Solver-v3 achieves consistently better or comparable performance in both unconditional and conditional sampling with both pixel-space and latent-space DPMs, especially in 5sim10 NFEs. We achieve FIDs of 12.21 (5 NFE), 2.51 (10 NFE) on unconditional CIFAR10, and MSE of 0.55 (5 NFE, 7.5 guidance scale) on Stable Diffusion, bringing a speed-up of 15\%sim30\% compared to previous state-of-the-art training-free methods. Code is available at https://github.com/thu-ml/DPM-Solver-v3.

Triggered by the realization that AI emulators can rival the performance of traditional numerical weather prediction models running on HPC systems, there is now an increasing number of large AI models that address use cases such as forecasting, downscaling, or nowcasting. While the parallel developments in the AI literature focus on foundation models -- models that can be effectively tuned to address multiple, different use cases -- the developments on the weather and climate side largely focus on single-use cases with particular emphasis on mid-range forecasting. We close this gap by introducing Prithvi WxC, a 2.3 billion parameter foundation model developed using 160 variables from the Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2). Prithvi WxC employs an encoder-decoder-based architecture, incorporating concepts from various recent transformer models to effectively capture both regional and global dependencies in the input data. The model has been designed to accommodate large token counts to model weather phenomena in different topologies at fine resolutions. Furthermore, it is trained with a mixed objective that combines the paradigms of masked reconstruction with forecasting. We test the model on a set of challenging downstream tasks namely: Autoregressive rollout forecasting, Downscaling, Gravity wave flux parameterization, and Extreme events estimation. The pretrained model with 2.3 billion parameters, along with the associated fine-tuning workflows, has been publicly released as an open-source contribution via Hugging Face.

We propose a new differentiable probabilistic model over DAGs (DP-DAG). DP-DAG allows fast and differentiable DAG sampling suited to continuous optimization. To this end, DP-DAG samples a DAG by successively (1) sampling a linear ordering of the node and (2) sampling edges consistent with the sampled linear ordering. We further propose VI-DP-DAG, a new method for DAG learning from observational data which combines DP-DAG with variational inference. Hence,VI-DP-DAG approximates the posterior probability over DAG edges given the observed data. VI-DP-DAG is guaranteed to output a valid DAG at any time during training and does not require any complex augmented Lagrangian optimization scheme in contrast to existing differentiable DAG learning approaches. In our extensive experiments, we compare VI-DP-DAG to other differentiable DAG learning baselines on synthetic and real datasets. VI-DP-DAG significantly improves DAG structure and causal mechanism learning while training faster than competitors.

Calibrating deep learning models to yield uncertainty-aware predictions is crucial as deep neural networks get increasingly deployed in safety-critical applications. While existing post-hoc calibration methods achieve impressive results on in-domain test datasets, they are limited by their inability to yield reliable uncertainty estimates in domain-shift and out-of-domain (OOD) scenarios. We aim to bridge this gap by proposing DAC, an accuracy-preserving as well as Density-Aware Calibration method based on k-nearest-neighbors (KNN). In contrast to existing post-hoc methods, we utilize hidden layers of classifiers as a source for uncertainty-related information and study their importance. We show that DAC is a generic method that can readily be combined with state-of-the-art post-hoc methods. DAC boosts the robustness of calibration performance in domain-shift and OOD, while maintaining excellent in-domain predictive uncertainty estimates. We demonstrate that DAC leads to consistently better calibration across a large number of model architectures, datasets, and metrics. Additionally, we show that DAC improves calibration substantially on recent large-scale neural networks pre-trained on vast amounts of data.

Time series forecasting has been a widely explored task of great importance in many applications. However, it is common that real-world time series data are recorded in a short time period, which results in a big gap between the deep model and the limited and noisy time series. In this work, we propose to address the time series forecasting problem with generative modeling and propose a bidirectional variational auto-encoder (BVAE) equipped with diffusion, denoise, and disentanglement, namely D3VAE. Specifically, a coupled diffusion probabilistic model is proposed to augment the time series data without increasing the aleatoric uncertainty and implement a more tractable inference process with BVAE. To ensure the generated series move toward the true target, we further propose to adapt and integrate the multiscale denoising score matching into the diffusion process for time series forecasting. In addition, to enhance the interpretability and stability of the prediction, we treat the latent variable in a multivariate manner and disentangle them on top of minimizing total correlation. Extensive experiments on synthetic and real-world data show that D3VAE outperforms competitive algorithms with remarkable margins. Our implementation is available at https://github.com/PaddlePaddle/PaddleSpatial/tree/main/research/D3VAE.

Existing ML-based atmospheric models are not suitable for climate prediction, which requires long-term stability and physical consistency. We present ACE (AI2 Climate Emulator), a 200M-parameter, autoregressive machine learning emulator of an existing comprehensive 100-km resolution global atmospheric model. The formulation of ACE allows evaluation of physical laws such as the conservation of mass and moisture. The emulator is stable for 100 years, nearly conserves column moisture without explicit constraints and faithfully reproduces the reference model's climate, outperforming a challenging baseline on over 90% of tracked variables. ACE requires nearly 100x less wall clock time and is 100x more energy efficient than the reference model using typically available resources. Without fine-tuning, ACE can stably generalize to a previously unseen historical sea surface temperature dataset.

We present a unified probabilistic formulation for diffusion-based image editing, where a latent variable is edited in a task-specific manner and generally deviates from the corresponding marginal distribution induced by the original stochastic or ordinary differential equation (SDE or ODE). Instead, it defines a corresponding SDE or ODE for editing. In the formulation, we prove that the Kullback-Leibler divergence between the marginal distributions of the two SDEs gradually decreases while that for the ODEs remains as the time approaches zero, which shows the promise of SDE in image editing. Inspired by it, we provide the SDE counterparts for widely used ODE baselines in various tasks including inpainting and image-to-image translation, where SDE shows a consistent and substantial improvement. Moreover, we propose SDE-Drag -- a simple yet effective method built upon the SDE formulation for point-based content dragging. We build a challenging benchmark (termed DragBench) with open-set natural, art, and AI-generated images for evaluation. A user study on DragBench indicates that SDE-Drag significantly outperforms our ODE baseline, existing diffusion-based methods, and the renowned DragGAN. Our results demonstrate the superiority and versatility of SDE in image editing and push the boundary of diffusion-based editing methods.

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.

Diffusion probabilistic models (DPMs) have achieved impressive success in high-resolution image synthesis, especially in recent large-scale text-to-image generation applications. An essential technique for improving the sample quality of DPMs is guided sampling, which usually needs a large guidance scale to obtain the best sample quality. The commonly-used fast sampler for guided sampling is DDIM, a first-order diffusion ODE solver that generally needs 100 to 250 steps for high-quality samples. Although recent works propose dedicated high-order solvers and achieve a further speedup for sampling without guidance, their effectiveness for guided sampling has not been well-tested before. In this work, we demonstrate that previous high-order fast samplers suffer from instability issues, and they even become slower than DDIM when the guidance scale grows large. To further speed up guided sampling, we propose DPM-Solver++, a high-order solver for the guided sampling of DPMs. DPM-Solver++ solves the diffusion ODE with the data prediction model and adopts thresholding methods to keep the solution matches training data distribution. We further propose a multistep variant of DPM-Solver++ to address the instability issue by reducing the effective step size. Experiments show that DPM-Solver++ can generate high-quality samples within only 15 to 20 steps for guided sampling by pixel-space and latent-space DPMs.

Denoising diffusion bridge models (DDBMs) are a powerful variant of diffusion models for interpolating between two arbitrary paired distributions given as endpoints. Despite their promising performance in tasks like image translation, DDBMs require a computationally intensive sampling process that involves the simulation of a (stochastic) differential equation through hundreds of network evaluations. In this work, we take the first step in fast sampling of DDBMs without extra training, motivated by the well-established recipes in diffusion models. We generalize DDBMs via a class of non-Markovian diffusion bridges defined on the discretized timesteps concerning sampling, which share the same marginal distributions and training objectives, give rise to generative processes ranging from stochastic to deterministic, and result in diffusion bridge implicit models (DBIMs). DBIMs are not only up to 25times faster than the vanilla sampler of DDBMs but also induce a novel, simple, and insightful form of ordinary differential equation (ODE) which inspires high-order numerical solvers. Moreover, DBIMs maintain the generation diversity in a distinguished way, by using a booting noise in the initial sampling step, which enables faithful encoding, reconstruction, and semantic interpolation in image translation tasks. Code is available at https://github.com/thu-ml/DiffusionBridge.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

In offline reinforcement learning (RL), an RL agent learns to solve a task using only a fixed dataset of previously collected data. While offline RL has been successful in learning real-world robot control policies, it typically requires large amounts of expert-quality data to learn effective policies that generalize to out-of-distribution states. Unfortunately, such data is often difficult and expensive to acquire in real-world tasks. Several recent works have leveraged data augmentation (DA) to inexpensively generate additional data, but most DA works apply augmentations in a random fashion and ultimately produce highly suboptimal augmented experience. In this work, we propose Guided Data Augmentation (GuDA), a human-guided DA framework that generates expert-quality augmented data. The key insight behind GuDA is that while it may be difficult to demonstrate the sequence of actions required to produce expert data, a user can often easily characterize when an augmented trajectory segment represents progress toward task completion. Thus, a user can restrict the space of possible augmentations to automatically reject suboptimal augmented data. To extract a policy from GuDA, we use off-the-shelf offline reinforcement learning and behavior cloning algorithms. We evaluate GuDA on a physical robot soccer task as well as simulated D4RL navigation tasks, a simulated autonomous driving task, and a simulated soccer task. Empirically, GuDA enables learning given a small initial dataset of potentially suboptimal experience and outperforms a random DA strategy as well as a model-based DA strategy.

Inverse design refers to the problem of optimizing the input of an objective function in order to enact a target outcome. For many real-world engineering problems, the objective function takes the form of a simulator that predicts how the system state will evolve over time, and the design challenge is to optimize the initial conditions that lead to a target outcome. Recent developments in learned simulation have shown that graph neural networks (GNNs) can be used for accurate, efficient, differentiable estimation of simulator dynamics, and support high-quality design optimization with gradient- or sampling-based optimization procedures. However, optimizing designs from scratch requires many expensive model queries, and these procedures exhibit basic failures on either non-convex or high-dimensional problems.In this work, we show how denoising diffusion models (DDMs) can be used to solve inverse design problems efficiently and propose a particle sampling algorithm for further improving their efficiency. We perform experiments on a number of fluid dynamics design challenges, and find that our approach substantially reduces the number of calls to the simulator compared to standard techniques.

Common image-to-image translation methods rely on joint training over data from both source and target domains. The training process requires concurrent access to both datasets, which hinders data separation and privacy protection; and existing models cannot be easily adapted for translation of new domain pairs. We present Dual Diffusion Implicit Bridges (DDIBs), an image translation method based on diffusion models, that circumvents training on domain pairs. Image translation with DDIBs relies on two diffusion models trained independently on each domain, and is a two-step process: DDIBs first obtain latent encodings for source images with the source diffusion model, and then decode such encodings using the target model to construct target images. Both steps are defined via ordinary differential equations (ODEs), thus the process is cycle consistent only up to discretization errors of the ODE solvers. Theoretically, we interpret DDIBs as concatenation of source to latent, and latent to target Schrodinger Bridges, a form of entropy-regularized optimal transport, to explain the efficacy of the method. Experimentally, we apply DDIBs on synthetic and high-resolution image datasets, to demonstrate their utility in a wide variety of translation tasks and their inherent optimal transport properties.

Synthesizing optimal controllers for dynamical systems often involves solving optimization problems with hard real-time constraints. These constraints determine the class of numerical methods that can be applied: computationally expensive but accurate numerical routines are replaced by fast and inaccurate methods, trading inference time for solution accuracy. This paper provides techniques to improve the quality of optimized control policies given a fixed computational budget. We achieve the above via a hypersolvers approach, which hybridizes a differential equation solver and a neural network. The performance is evaluated in direct and receding-horizon optimal control tasks in both low and high dimensions, where the proposed approach shows consistent Pareto improvements in solution accuracy and control performance.

The Schr\"odinger Bridge (SB) problem offers a powerful framework for combining optimal transport and diffusion models. A promising recent approach to solve the SB problem is the Iterative Markovian Fitting (IMF) procedure, which alternates between Markovian and reciprocal projections of continuous-time stochastic processes. However, the model built by the IMF procedure has a long inference time due to using many steps of numerical solvers for stochastic differential equations. To address this limitation, we propose a novel Discrete-time IMF (D-IMF) procedure in which learning of stochastic processes is replaced by learning just a few transition probabilities in discrete time. Its great advantage is that in practice it can be naturally implemented using the Denoising Diffusion GAN (DD-GAN), an already well-established adversarial generative modeling technique. We show that our D-IMF procedure can provide the same quality of unpaired domain translation as the IMF, using only several generation steps instead of hundreds. We provide the code at https://github.com/Daniil-Selikhanovych/ASBM.

Denoising is intuitively related to projection. Indeed, under the manifold hypothesis, adding random noise is approximately equivalent to orthogonal perturbation. Hence, learning to denoise is approximately learning to project. In this paper, we use this observation to reinterpret denoising diffusion models as approximate gradient descent applied to the Euclidean distance function. We then provide straight-forward convergence analysis of the DDIM sampler under simple assumptions on the projection-error of the denoiser. Finally, we propose a new sampler based on two simple modifications to DDIM using insights from our theoretical results. In as few as 5-10 function evaluations, our sampler achieves state-of-the-art FID scores on pretrained CIFAR-10 and CelebA models and can generate high quality samples on latent diffusion models.

Latent diffusion models are the state-of-the-art for synthetic image generation. To align these models with human preferences, training the models using reinforcement learning on human feedback is crucial. Black et. al 2024 introduced denoising diffusion policy optimisation (DDPO), which accounts for the iterative denoising nature of the generation by modelling it as a Markov chain with a final reward. As the reward is a single value that determines the model's performance on the entire image, the model has to navigate a very sparse reward landscape and so requires a large sample count. In this work, we extend the DDPO by presenting the Pixel-wise Policy Optimisation (PXPO) algorithm, which can take feedback for each pixel, providing a more nuanced reward to the model.

Recent years have witnessed significant progress in developing efficient training and fast sampling approaches for diffusion models. A recent remarkable advancement is the use of stochastic differential equations (SDEs) to describe data perturbation and generative modeling in a unified mathematical framework. In this paper, we reveal several intriguing geometric structures of diffusion models and contribute a simple yet powerful interpretation to their sampling dynamics. Through carefully inspecting a popular variance-exploding SDE and its marginal-preserving ordinary differential equation (ODE) for sampling, we discover that the data distribution and the noise distribution are smoothly connected with an explicit, quasi-linear sampling trajectory, and another implicit denoising trajectory, which even converges faster in terms of visual quality. We also establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm, with which we can characterize the asymptotic behavior of diffusion models and identify the score deviation. These new geometric observations enable us to improve previous sampling algorithms, re-examine latent interpolation, as well as re-explain the working principles of distillation-based fast sampling techniques.

The past few years have witnessed the great success of Diffusion models~(DMs) in generating high-fidelity samples in generative modeling tasks. A major limitation of the DM is its notoriously slow sampling procedure which normally requires hundreds to thousands of time discretization steps of the learned diffusion process to reach the desired accuracy. Our goal is to develop a fast sampling method for DMs with a much less number of steps while retaining high sample quality. To this end, we systematically analyze the sampling procedure in DMs and identify key factors that affect the sample quality, among which the method of discretization is most crucial. By carefully examining the learned diffusion process, we propose Diffusion Exponential Integrator Sampler~(DEIS). It is based on the Exponential Integrator designed for discretizing ordinary differential equations (ODEs) and leverages a semilinear structure of the learned diffusion process to reduce the discretization error. The proposed method can be applied to any DMs and can generate high-fidelity samples in as few as 10 steps. In our experiments, it takes about 3 minutes on one A6000 GPU to generate 50k images from CIFAR10. Moreover, by directly using pre-trained DMs, we achieve the state-of-art sampling performance when the number of score function evaluation~(NFE) is limited, e.g., 4.17 FID with 10 NFEs, 3.37 FID, and 9.74 IS with only 15 NFEs on CIFAR10. Code is available at https://github.com/qsh-zh/deis

Many problems in astrophysics cover multiple orders of magnitude in spatial and temporal scales. While simulating systems that experience rapid changes in these conditions, it is essential to adapt the (time-) step size to capture the behavior of the system during those rapid changes and use a less accurate time step at other, less demanding, moments. We encounter three problems with traditional methods. Firstly, making such changes requires expert knowledge of the astrophysics as well as of the details of the numerical implementation. Secondly, some parameters that determine the time-step size are fixed throughout the simulation, which means that they do not adapt to the rapidly changing conditions of the problem. Lastly, we would like the choice of time-step size to balance accuracy and computation effort. We address these challenges with Reinforcement Learning by training it to select the time-step size dynamically. We use the integration of a system of three equal-mass bodies that move due to their mutual gravity as an example of its application. With our method, the selected integration parameter adapts to the specific requirements of the problem, both in terms of computation time and accuracy while eliminating the expert knowledge needed to set up these simulations. Our method produces results competitive to existing methods and improve the results found with the most commonly-used values of time-step parameter. This method can be applied to other integrators without further retraining. We show that this extrapolation works for variable time-step integrators but does not perform to the desired accuracy for fixed time-step integrators.

Distributed Acoustic Sensing (DAS) is an emerging technology for earthquake monitoring and subsurface imaging. The recorded seismic signals by DAS have several distinct characteristics, such as unknown coupling effects, strong anthropogenic noise, and ultra-dense spatial sampling. These aspects differ from conventional seismic data recorded by seismic networks, making it challenging to utilize DAS at present for seismic monitoring. New data analysis algorithms are needed to extract useful information from DAS data. Previous studies on conventional seismic data demonstrated that deep learning models could achieve performance close to human analysts in picking seismic phases. However, phase picking on DAS data is still a difficult problem due to the lack of manual labels. Further, the differences in mathematical structure between these two data formats, i.e., ultra-dense DAS arrays and sparse seismic networks, make model fine-tuning or transfer learning difficult to implement on DAS data. In this work, we design a new approach using semi-supervised learning to solve the phase-picking task on DAS arrays. We use a pre-trained PhaseNet model as a teacher network to generate noisy labels of P and S arrivals on DAS data and apply the Gaussian mixture model phase association (GaMMA) method to refine these noisy labels to build training datasets. We develop a new deep learning model, PhaseNet-DAS, to process the 2D spatial-temporal data of DAS arrays and train the model on DAS data. The new deep learning model achieves high picking accuracy and good earthquake detection performance. We then apply the model to process continuous data and build earthquake catalogs directly from DAS recording. Our approach using semi-supervised learning provides a way to build effective deep learning models for DAS, which have the potential to improve earthquake monitoring using large-scale fiber networks.

Diffusion Probabilistic Models (DPMs) have achieved considerable success in generation tasks. As sampling from DPMs is equivalent to solving diffusion SDE or ODE which is time-consuming, numerous fast sampling methods built upon improved differential equation solvers are proposed. The majority of such techniques consider solving the diffusion ODE due to its superior efficiency. However, stochastic sampling could offer additional advantages in generating diverse and high-quality data. In this work, we engage in a comprehensive analysis of stochastic sampling from two aspects: variance-controlled diffusion SDE and linear multi-step SDE solver. Based on our analysis, we propose SA-Solver, which is an improved efficient stochastic Adams method for solving diffusion SDE to generate data with high quality. Our experiments show that SA-Solver achieves: 1) improved or comparable performance compared with the existing state-of-the-art sampling methods for few-step sampling; 2) SOTA FID scores on substantial benchmark datasets under a suitable number of function evaluations (NFEs).

We develop a novel, general and computationally efficient framework, called Divide and Conquer Dynamic Programming (DCDP), for localizing change points in time series data with high-dimensional features. DCDP deploys a class of greedy algorithms that are applicable to a broad variety of high-dimensional statistical models and can enjoy almost linear computational complexity. We investigate the performance of DCDP in three commonly studied change point settings in high dimensions: the mean model, the Gaussian graphical model, and the linear regression model. In all three cases, we derive non-asymptotic bounds for the accuracy of the DCDP change point estimators. We demonstrate that the DCDP procedures consistently estimate the change points with sharp, and in some cases, optimal rates while incurring significantly smaller computational costs than the best available algorithms. Our findings are supported by extensive numerical experiments on both synthetic and real data.

Generative processes that involve solving differential equations, such as diffusion models, frequently necessitate balancing speed and quality. ODE-based samplers are fast but plateau in performance while SDE-based samplers deliver higher sample quality at the cost of increased sampling time. We attribute this difference to sampling errors: ODE-samplers involve smaller discretization errors while stochasticity in SDE contracts accumulated errors. Based on these findings, we propose a novel sampling algorithm called Restart in order to better balance discretization errors and contraction. The sampling method alternates between adding substantial noise in additional forward steps and strictly following a backward ODE. Empirically, Restart sampler surpasses previous SDE and ODE samplers in both speed and accuracy. Restart not only outperforms the previous best SDE results, but also accelerates the sampling speed by 10-fold / 2-fold on CIFAR-10 / ImageNet 64 times 64. In addition, it attains significantly better sample quality than ODE samplers within comparable sampling times. Moreover, Restart better balances text-image alignment/visual quality versus diversity than previous samplers in the large-scale text-to-image Stable Diffusion model pre-trained on LAION 512 times 512. Code is available at https://github.com/Newbeeer/diffusion_restart_sampling

Recent advancements in artificial intelligence (AI) for numerical weather prediction (NWP) have significantly transformed atmospheric modeling. AI NWP models outperform traditional physics-based systems, such as the Integrated Forecast System (IFS), across several global metrics while requiring fewer computational resources. However, existing AI NWP models face limitations related to training datasets and timestep choices, often resulting in artifacts that reduce model performance. To address these challenges, we introduce the Community Research Earth Digital Intelligence Twin (CREDIT) framework, developed at NSF NCAR. CREDIT provides a flexible, scalable, and user-friendly platform for training and deploying AI-based atmospheric models on high-performance computing systems. It offers an end-to-end pipeline for data preprocessing, model training, and evaluation, democratizing access to advanced AI NWP capabilities. We demonstrate CREDIT's potential through WXFormer, a novel deterministic vision transformer designed to predict atmospheric states autoregressively, addressing common AI NWP issues like compounding error growth with techniques such as spectral normalization, padding, and multi-step training. Additionally, to illustrate CREDIT's flexibility and state-of-the-art model comparisons, we train the FUXI architecture within this framework. Our findings show that both FUXI and WXFormer, trained on six-hourly ERA5 hybrid sigma-pressure levels, generally outperform IFS HRES in 10-day forecasts, offering potential improvements in efficiency and forecast accuracy. CREDIT's modular design enables researchers to explore various models, datasets, and training configurations, fostering innovation within the scientific community.

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539--542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the Empirical Interpolation Method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.

Most state-of-the-art approaches for weather and climate modeling are based on physics-informed numerical models of the atmosphere. These approaches aim to model the non-linear dynamics and complex interactions between multiple variables, which are challenging to approximate. Additionally, many such numerical models are computationally intensive, especially when modeling the atmospheric phenomenon at a fine-grained spatial and temporal resolution. Recent data-driven approaches based on machine learning instead aim to directly solve a downstream forecasting or projection task by learning a data-driven functional mapping using deep neural networks. However, these networks are trained using curated and homogeneous climate datasets for specific spatiotemporal tasks, and thus lack the generality of numerical models. We develop and demonstrate ClimaX, a flexible and generalizable deep learning model for weather and climate science that can be trained using heterogeneous datasets spanning different variables, spatio-temporal coverage, and physical groundings. ClimaX extends the Transformer architecture with novel encoding and aggregation blocks that allow effective use of available compute while maintaining general utility. ClimaX is pre-trained with a self-supervised learning objective on climate datasets derived from CMIP6. The pre-trained ClimaX can then be fine-tuned to address a breadth of climate and weather tasks, including those that involve atmospheric variables and spatio-temporal scales unseen during pretraining. Compared to existing data-driven baselines, we show that this generality in ClimaX results in superior performance on benchmarks for weather forecasting and climate projections, even when pretrained at lower resolutions and compute budgets.

We introduce Poseidon, a foundation model for learning the solution operators of PDEs. It is based on a multiscale operator transformer, with time-conditioned layer norms that enable continuous-in-time evaluations. A novel training strategy leveraging the semi-group property of time-dependent PDEs to allow for significant scaling-up of the training data is also proposed. Poseidon is pretrained on a diverse, large scale dataset for the governing equations of fluid dynamics. It is then evaluated on a suite of 15 challenging downstream tasks that include a wide variety of PDE types and operators. We show that Poseidon exhibits excellent performance across the board by outperforming baselines significantly, both in terms of sample efficiency and accuracy. Poseidon also generalizes very well to new physics that is not seen during pretraining. Moreover, Poseidon scales with respect to model and data size, both for pretraining and for downstream tasks. Taken together, our results showcase the surprising ability of Poseidon to learn effective representations from a very small set of PDEs during pretraining in order to generalize well to unseen and unrelated PDEs downstream, demonstrating its potential as an effective, general purpose PDE foundation model. Finally, the Poseidon model as well as underlying pretraining and downstream datasets are open sourced, with code being available at https://github.com/camlab-ethz/poseidon and pretrained models and datasets at https://huggingface.co/camlab-ethz.

Diffusion-based generative models use stochastic differential equations (SDEs) and their equivalent ordinary differential equations (ODEs) to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models. We characterize an implicit denoising trajectory and discuss its vital role in forming the coupled sampling trajectory with a strong shape regularity, regardless of the generated content. We also describe a dynamic programming-based scheme to make the time schedule in sampling better fit the underlying trajectory structure. This simple strategy requires minimal modification to any given ODE-based numerical solvers and incurs negligible computational cost, while delivering superior performance in image generation, especially in 5sim 10 function evaluations.

AI emulators for forecasting have emerged as powerful tools that can outperform conventional numerical predictions. The next frontier is to build emulators for long-term climate projections with robust skill across a wide range of spatiotemporal scales, a particularly important goal for the ocean. Our work builds a skillful global emulator of the ocean component of a state-of-the-art climate model. We emulate key ocean variables, sea surface height, horizontal velocities, temperature, and salinity, across their full depth. We use a modified ConvNeXt UNet architecture trained on multidepth levels of ocean data. We show that the ocean emulator - Samudra - which exhibits no drift relative to the truth, can reproduce the depth structure of ocean variables and their interannual variability. Samudra is stable for centuries and 150 times faster than the original ocean model. Samudra struggles to capture the correct magnitude of the forcing trends and simultaneously remains stable, requiring further work.

While 2D diffusion models generate realistic, high-detail images, 3D shape generation methods like Score Distillation Sampling (SDS) built on these 2D diffusion models produce cartoon-like, over-smoothed shapes. To help explain this discrepancy, we show that the image guidance used in Score Distillation can be understood as the velocity field of a 2D denoising generative process, up to the choice of a noise term. In particular, after a change of variables, SDS resembles a high-variance version of Denoising Diffusion Implicit Models (DDIM) with a differently-sampled noise term: SDS introduces noise i.i.d. randomly at each step, while DDIM infers it from the previous noise predictions. This excessive variance can lead to over-smoothing and unrealistic outputs. We show that a better noise approximation can be recovered by inverting DDIM in each SDS update step. This modification makes SDS's generative process for 2D images almost identical to DDIM. In 3D, it removes over-smoothing, preserves higher-frequency detail, and brings the generation quality closer to that of 2D samplers. Experimentally, our method achieves better or similar 3D generation quality compared to other state-of-the-art Score Distillation methods, all without training additional neural networks or multi-view supervision, and providing useful insights into relationship between 2D and 3D asset generation with diffusion models.

Discrete diffusion has achieved state-of-the-art performance, outperforming or approaching autoregressive models on standard benchmarks. In this work, we introduce Discrete Diffusion with Planned Denoising (DDPD), a novel framework that separates the generation process into two models: a planner and a denoiser. At inference time, the planner selects which positions to denoise next by identifying the most corrupted positions in need of denoising, including both initially corrupted and those requiring additional refinement. This plan-and-denoise approach enables more efficient reconstruction during generation by iteratively identifying and denoising corruptions in the optimal order. DDPD outperforms traditional denoiser-only mask diffusion methods, achieving superior results on language modeling benchmarks such as text8, OpenWebText, and token-based generation on ImageNet 256 times 256. Notably, in language modeling, DDPD significantly reduces the performance gap between diffusion-based and autoregressive methods in terms of generative perplexity. Code is available at https://github.com/liusulin/DDPD.

The iterative sampling procedure employed by diffusion models (DMs) often leads to significant inference latency. To address this, we propose Stochastic Consistency Distillation (SCott) to enable accelerated text-to-image generation, where high-quality generations can be achieved with just 1-2 sampling steps, and further improvements can be obtained by adding additional steps. In contrast to vanilla consistency distillation (CD) which distills the ordinary differential equation solvers-based sampling process of a pretrained teacher model into a student, SCott explores the possibility and validates the efficacy of integrating stochastic differential equation (SDE) solvers into CD to fully unleash the potential of the teacher. SCott is augmented with elaborate strategies to control the noise strength and sampling process of the SDE solver. An adversarial loss is further incorporated to strengthen the sample quality with rare sampling steps. Empirically, on the MSCOCO-2017 5K dataset with a Stable Diffusion-V1.5 teacher, SCott achieves an FID (Frechet Inceptio Distance) of 22.1, surpassing that (23.4) of the 1-step InstaFlow (Liu et al., 2023) and matching that of 4-step UFOGen (Xue et al., 2023b). Moreover, SCott can yield more diverse samples than other consistency models for high-resolution image generation (Luo et al., 2023a), with up to 16% improvement in a qualified metric. The code and checkpoints are coming soon.

Recently, data augmentation (DA) has emerged as a method for leveraging domain knowledge to inexpensively generate additional data in reinforcement learning (RL) tasks, often yielding substantial improvements in data efficiency. While prior work has demonstrated the utility of incorporating augmented data directly into model-free RL updates, it is not well-understood when a particular DA strategy will improve data efficiency. In this paper, we seek to identify general aspects of DA responsible for observed learning improvements. Our study focuses on sparse-reward tasks with dynamics-invariant data augmentation functions, serving as an initial step towards a more general understanding of DA and its integration into RL training. Experimentally, we isolate three relevant aspects of DA: state-action coverage, reward density, and the number of augmented transitions generated per update (the augmented replay ratio). From our experiments, we draw two conclusions: (1) increasing state-action coverage often has a much greater impact on data efficiency than increasing reward density, and (2) decreasing the augmented replay ratio substantially improves data efficiency. In fact, certain tasks in our empirical study are solvable only when the replay ratio is sufficiently low.

Active Learning (AL) has gained prominence in integrating data-intensive machine learning (ML) models into domains with limited labeled data. However, its effectiveness diminishes significantly when the labeling budget is low. In this paper, we first empirically observe the performance degradation of existing AL algorithms in the low-budget settings, and then introduce Direct Acquisition Optimization (DAO), a novel AL algorithm that optimizes sample selections based on expected true loss reduction. Specifically, DAO utilizes influence functions to update model parameters and incorporates an additional acquisition strategy to mitigate bias in loss estimation. This approach facilitates a more accurate estimation of the overall error reduction, without extensive computations or reliance on labeled data. Experiments demonstrate DAO's effectiveness in low budget settings, outperforming state-of-the-arts approaches across seven benchmarks.

Reliably reconstructing physical fields from sparse sensor data is a challenge that frequently arises in many scientific domains. In practice, the process generating the data often is not understood to sufficient accuracy. Therefore, there is a growing interest in using the deep neural network route to address the problem. This work presents a novel approach that learns a continuous representation of the physical field using implicit neural representations (INRs). Specifically, after factorizing spatiotemporal variability into spatial and temporal components using the separation of variables technique, the method learns relevant basis functions from sparsely sampled irregular data points to develop a continuous representation of the data. In experimental evaluations, the proposed model outperforms recent INR methods, offering superior reconstruction quality on simulation data from a state-of-the-art climate model and a second dataset that comprises ultra-high resolution satellite-based sea surface temperature fields.

Score-based diffusion models, while achieving remarkable empirical performance, often suffer from low sampling speed, due to extensive function evaluations needed during the sampling phase. Despite a flurry of recent activities towards speeding up diffusion generative modeling in practice, theoretical underpinnings for acceleration techniques remain severely limited. In this paper, we design novel training-free algorithms to accelerate popular deterministic (i.e., DDIM) and stochastic (i.e., DDPM) samplers. Our accelerated deterministic sampler converges at a rate O(1/{T}^2) with T the number of steps, improving upon the O(1/T) rate for the DDIM sampler; and our accelerated stochastic sampler converges at a rate O(1/T), outperforming the rate O(1/T) for the DDPM sampler. The design of our algorithms leverages insights from higher-order approximation, and shares similar intuitions as popular high-order ODE solvers like the DPM-Solver-2. Our theory accommodates ell_2-accurate score estimates, and does not require log-concavity or smoothness on the target distribution.

Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and require large amounts of {\em labeled} training data. We propose the Deep Generative Neural Operator (DGNO), a physics-aware framework that addresses these challenges by leveraging a deep, generative, probabilistic model in combination with a set of lower-dimensional, latent variables that simultaneously encode PDE-inputs and PDE-outputs. This formulation can make use of unlabeled data and significantly improves inverse problem-solving, particularly for discontinuous or discrete-valued input functions. DGNO enforces physics constraints without labeled data by incorporating as virtual observables, weak-form residuals based on compactly supported radial basis functions (CSRBFs). These relax regularity constraints and eliminate higher-order derivatives from the objective function. We also introduce MultiONet, a novel neural operator architecture, which is a more expressive generalization of the popular DeepONet that significantly enhances the approximating power of the proposed model. These innovations make DGNO particularly effective for challenging forward and inverse, PDE-based problems, such as those involving multi-phase media. Numerical experiments demonstrate that DGNO achieves higher accuracy across multiple benchmarks while exhibiting robustness to noise and strong generalization to out-of-distribution cases. Its adaptability, and the ability to handle sparse, noisy data while providing probabilistic estimates, make DGNO a powerful tool for scientific and engineering applications.

Recent inverse problem solvers that leverage generative diffusion priors have garnered significant attention due to their exceptional quality. However, adaptation of the prior is necessary when there exists a discrepancy between the training and testing distributions. In this work, we propose deep diffusion image prior (DDIP), which generalizes the recent adaptation method of SCD by introducing a formal connection to the deep image prior. Under this framework, we propose an efficient adaptation method dubbed D3IP, specified for 3D measurements, which accelerates DDIP by orders of magnitude while achieving superior performance. D3IP enables seamless integration of 3D inverse solvers and thus leads to coherent 3D reconstruction. Moreover, we show that meta-learning techniques can also be applied to yield even better performance. We show that our method is capable of solving diverse 3D reconstructive tasks from the generative prior trained only with phantom images that are vastly different from the training set, opening up new opportunities of applying diffusion inverse solvers even when training with gold standard data is impossible. Code: https://github.com/HJ-harry/DDIP3D

Bayesian filtering approximates the true underlying behavior of a time-varying system by inverting an explicit generative model to convert noisy measurements into state estimates. This process typically requires either storage, inversion, and multiplication of large matrices or Monte Carlo estimation, neither of which are practical in high-dimensional state spaces such as the weight spaces of artificial neural networks. Here, we frame the standard Bayesian filtering problem as optimization over a time-varying objective. Instead of maintaining matrices for the filtering equations or simulating particles, we specify an optimizer that defines the Bayesian filter implicitly. In the linear-Gaussian setting, we show that every Kalman filter has an equivalent formulation using K steps of gradient descent. In the nonlinear setting, our experiments demonstrate that our framework results in filters that are effective, robust, and scalable to high-dimensional systems, comparing well against the standard toolbox of Bayesian filtering solutions. We suggest that it is easier to fine-tune an optimizer than it is to specify the correct filtering equations, making our framework an attractive option for high-dimensional filtering problems.

Adversarial imitation learning (AIL) has stood out as a dominant framework across various imitation learning (IL) applications, with Discriminator Actor Critic (DAC) (Kostrikov et al.,, 2019) demonstrating the effectiveness of off-policy learning algorithms in improving sample efficiency and scalability to higher-dimensional observations. Despite DAC's empirical success, the original AIL objective is on-policy and DAC's ad-hoc application of off-policy training does not guarantee successful imitation (Kostrikov et al., 2019; 2020). Follow-up work such as ValueDICE (Kostrikov et al., 2020) tackles this issue by deriving a fully off-policy AIL objective. Instead in this work, we develop a novel and principled AIL algorithm via the framework of boosting. Like boosting, our new algorithm, AILBoost, maintains an ensemble of properly weighted weak learners (i.e., policies) and trains a discriminator that witnesses the maximum discrepancy between the distributions of the ensemble and the expert policy. We maintain a weighted replay buffer to represent the state-action distribution induced by the ensemble, allowing us to train discriminators using the entire data collected so far. In the weighted replay buffer, the contribution of the data from older policies are properly discounted with the weight computed based on the boosting framework. Empirically, we evaluate our algorithm on both controller state-based and pixel-based environments from the DeepMind Control Suite. AILBoost outperforms DAC on both types of environments, demonstrating the benefit of properly weighting replay buffer data for off-policy training. On state-based environments, DAC outperforms ValueDICE and IQ-Learn (Gary et al., 2021), achieving competitive performance with as little as one expert trajectory.

In recent years, applying deep learning to solve physics problems has attracted much attention. Data-driven deep learning methods produce fast numerical operators that can learn approximate solutions to the whole system of partial differential equations (i.e., surrogate modeling). Although these neural networks may have lower accuracy than traditional numerical methods, they, once trained, are orders of magnitude faster at inference. Hence, one crucial feature is that these operators can generalize to unseen PDE parameters without expensive re-training.In this paper, we construct CFDBench, a benchmark tailored for evaluating the generalization ability of neural operators after training in computational fluid dynamics (CFD) problems. It features four classic CFD problems: lid-driven cavity flow, laminar boundary layer flow in circular tubes, dam flows through the steps, and periodic Karman vortex street. The data contains a total of 302K frames of velocity and pressure fields, involving 739 cases with different operating condition parameters, generated with numerical methods. We evaluate the effectiveness of popular neural operators including feed-forward networks, DeepONet, FNO, U-Net, etc. on CFDBnech by predicting flows with non-periodic boundary conditions, fluid properties, and flow domain shapes that are not seen during training. Appropriate modifications were made to apply popular deep neural networks to CFDBench and enable the accommodation of more changing inputs. Empirical results on CFDBench show many baseline models have errors as high as 300% in some problems, and severe error accumulation when performing autoregressive inference. CFDBench facilitates a more comprehensive comparison between different neural operators for CFD compared to existing benchmarks.

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the L^2 difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

Creating noise from data is easy; creating data from noise is generative modeling. We present a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by slowly removing the noise. Crucially, the reverse-time SDE depends only on the time-dependent gradient field (\aka, score) of the perturbed data distribution. By leveraging advances in score-based generative modeling, we can accurately estimate these scores with neural networks, and use numerical SDE solvers to generate samples. We show that this framework encapsulates previous approaches in score-based generative modeling and diffusion probabilistic modeling, allowing for new sampling procedures and new modeling capabilities. In particular, we introduce a predictor-corrector framework to correct errors in the evolution of the discretized reverse-time SDE. We also derive an equivalent neural ODE that samples from the same distribution as the SDE, but additionally enables exact likelihood computation, and improved sampling efficiency. In addition, we provide a new way to solve inverse problems with score-based models, as demonstrated with experiments on class-conditional generation, image inpainting, and colorization. Combined with multiple architectural improvements, we achieve record-breaking performance for unconditional image generation on CIFAR-10 with an Inception score of 9.89 and FID of 2.20, a competitive likelihood of 2.99 bits/dim, and demonstrate high fidelity generation of 1024 x 1024 images for the first time from a score-based generative model.

Probabilistic forecasting is crucial to decision-making under uncertainty about future weather. The dominant approach is to use an ensemble of forecasts to represent and quantify uncertainty in operational numerical weather prediction. However, generating ensembles is computationally costly. In this paper, we propose to generate ensemble forecasts at scale by leveraging recent advances in generative artificial intelligence. Our approach learns a data-driven probabilistic diffusion model from the 5-member ensemble GEFS reforecast dataset. The model can then be sampled efficiently to produce realistic weather forecasts, conditioned on a few members of the operational GEFS forecasting system. The generated ensembles have similar predictive skill as the full GEFS 31-member ensemble, evaluated against ERA5 reanalysis, and emulate well the statistics of large physics-based ensembles. We also apply the same methodology to developing a diffusion model for generative post-processing: the model directly learns to correct biases present in the emulated forecasting system by leveraging reanalysis data as labels during training. Ensembles from this generative post-processing model show greater reliability and accuracy, particularly in extreme event classification. In general, they are more reliable and forecast the probability of extreme weather more accurately than the GEFS operational ensemble. Our models achieve these results at less than 1/10th of the computational cost incurred by the operational GEFS system.

The growing body of research shows how to replace classical partial differential equation (PDE) integrators with neural networks. The popular strategy is to generate the input-output pairs with a PDE solver, train the neural network in the regression setting, and use the trained model as a cheap surrogate for the solver. The bottleneck in this scheme is the number of expensive queries of a PDE solver needed to generate the dataset. To alleviate the problem, we propose a computationally cheap augmentation strategy based on general covariance and simple random coordinate transformations. Our approach relies on the fact that physical laws are independent of the coordinate choice, so the change in the coordinate system preserves the type of a parametric PDE and only changes PDE's data (e.g., initial conditions, diffusion coefficient). For tried neural networks and partial differential equations, proposed augmentation improves test error by 23% on average. The worst observed result is a 17% increase in test error for multilayer perceptron, and the best case is a 80% decrease for dilated residual network.

This paper describes an efficient algorithm for solving noisy linear inverse problems using pretrained diffusion models. Extending the paradigm of denoising diffusion implicit models (DDIM), we propose constrained diffusion implicit models (CDIM) that modify the diffusion updates to enforce a constraint upon the final output. For noiseless inverse problems, CDIM exactly satisfies the constraints; in the noisy case, we generalize CDIM to satisfy an exact constraint on the residual distribution of the noise. Experiments across a variety of tasks and metrics show strong performance of CDIM, with analogous inference acceleration to unconstrained DDIM: 10 to 50 times faster than previous conditional diffusion methods. We demonstrate the versatility of our approach on many problems including super-resolution, denoising, inpainting, deblurring, and 3D point cloud reconstruction.

We revisit the theoretical properties of Hamiltonian stochastic differential equations (SDES) for Bayesian posterior sampling, and we study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates in the context of data subsampling. Our main result is a novel analysis for the effect of mini-batches through the lens of differential operator splitting, revising previous literature results. The stochastic component of a Hamiltonian SDE is decoupled from the gradient noise, for which we make no normality assumptions. This leads to the identification of a convergence bottleneck: when considering mini-batches, the best achievable error rate is O(eta^2), with eta being the integrator step size. Our theoretical results are supported by an empirical study on a variety of regression and classification tasks for Bayesian neural networks.

Knowledge about the hidden factors that determine particular system dynamics is crucial for both explaining them and pursuing goal-directed interventions. Inferring these factors from time series data without supervision remains an open challenge. Here, we focus on spatiotemporal processes, including wave propagation and weather dynamics, for which we assume that universal causes (e.g. physics) apply throughout space and time. A recently introduced DIstributed SpatioTemporal graph Artificial Neural network Architecture (DISTANA) is used and enhanced to learn such processes, requiring fewer parameters and achieving significantly more accurate predictions compared to temporal convolutional neural networks and other related approaches. We show that DISTANA, when combined with a retrospective latent state inference principle called active tuning, can reliably derive location-respective hidden causal factors. In a current weather prediction benchmark, DISTANA infers our planet's land-sea mask solely by observing temperature dynamics and, meanwhile, uses the self inferred information to improve its own future temperature predictions.

Score-based (denoising diffusion) generative models have recently gained a lot of success in generating realistic and diverse data. These approaches define a forward diffusion process for transforming data to noise and generate data by reversing it (thereby going from noise to data). Unfortunately, current score-based models generate data very slowly due to the sheer number of score network evaluations required by numerical SDE solvers. In this work, we aim to accelerate this process by devising a more efficient SDE solver. Existing approaches rely on the Euler-Maruyama (EM) solver, which uses a fixed step size. We found that naively replacing it with other SDE solvers fares poorly - they either result in low-quality samples or become slower than EM. To get around this issue, we carefully devise an SDE solver with adaptive step sizes tailored to score-based generative models piece by piece. Our solver requires only two score function evaluations, rarely rejects samples, and leads to high-quality samples. Our approach generates data 2 to 10 times faster than EM while achieving better or equal sample quality. For high-resolution images, our method leads to significantly higher quality samples than all other methods tested. Our SDE solver has the benefit of requiring no step size tuning.

Editing real images using a pre-trained text-to-image (T2I) diffusion/flow model often involves inverting the image into its corresponding noise map. However, inversion by itself is typically insufficient for obtaining satisfactory results, and therefore many methods additionally intervene in the sampling process. Such methods achieve improved results but are not seamlessly transferable between model architectures. Here, we introduce FlowEdit, a text-based editing method for pre-trained T2I flow models, which is inversion-free, optimization-free and model agnostic. Our method constructs an ODE that directly maps between the source and target distributions (corresponding to the source and target text prompts) and achieves a lower transport cost than the inversion approach. This leads to state-of-the-art results, as we illustrate with Stable Diffusion 3 and FLUX. Code and examples are available on the project's webpage.

Diffusion models (DMs) demonstrate potent image generation capabilities in various generative modeling tasks. Nevertheless, their primary limitation lies in slow sampling speed, requiring hundreds or thousands of sequential function evaluations through large neural networks to generate high-quality images. Sampling from DMs can be seen alternatively as solving corresponding stochastic differential equations (SDEs) or ordinary differential equations (ODEs). In this work, we formulate the sampling process as an extended reverse-time SDE (ER SDE), unifying prior explorations into ODEs and SDEs. Leveraging the semi-linear structure of ER SDE solutions, we offer exact solutions and arbitrarily high-order approximate solutions for VP SDE and VE SDE, respectively. Based on the solution space of the ER SDE, we yield mathematical insights elucidating the superior performance of ODE solvers over SDE solvers in terms of fast sampling. Additionally, we unveil that VP SDE solvers stand on par with their VE SDE counterparts. Finally, we devise fast and training-free samplers, ER-SDE-Solvers, achieving state-of-the-art performance across all stochastic samplers. Experimental results demonstrate achieving 3.45 FID in 20 function evaluations and 2.24 FID in 50 function evaluations on the ImageNet 64times64 dataset.

Current diffusion-based image restoration methods feed degraded input images as conditions into the noise estimation network. However, interpreting this diffusion process is challenging since it essentially generates the target image from the noise. To establish a unified and more interpretable model for image generation and restoration, we propose residual denoising diffusion models (RDDM). In contrast to existing diffusion models (e.g., DDPM or DDIM) that focus solely on noise estimation, our RDDM predicts residuals to represent directional diffusion from the target domain to the input domain, while concurrently estimating noise to account for random perturbations in the diffusion process. The introduction of residuals allows us to redefine the forward diffusion process, wherein the target image progressively diffuses into a purely noisy image or a noise-carrying input image, thus unifying image generation and restoration. We demonstrate that our sampling process is consistent with that of DDPM and DDIM through coefficient transformation, and propose a partially path-independent generation process to better understand the reverse process. Notably, with native support for conditional inputs, our RDDM enables a generic UNet, trained with only an ell _1 loss and a batch size of 1, to compete with state-of-the-art image restoration methods. We provide code and pre-trained models to encourage further exploration, application, and development of our innovative framework (https://github.com/nachifur/RDDM).

Reducing traffic accidents is an important public safety challenge, therefore, accident analysis and prediction has been a topic of much research over the past few decades. Using small-scale datasets with limited coverage, being dependent on extensive set of data, and being not applicable for real-time purposes are the important shortcomings of the existing studies. To address these challenges, we propose a new solution for real-time traffic accident prediction using easy-to-obtain, but sparse data. Our solution relies on a deep-neural-network model (which we have named DAP, for Deep Accident Prediction); which utilizes a variety of data attributes such as traffic events, weather data, points-of-interest, and time. DAP incorporates multiple components including a recurrent (for time-sensitive data), a fully connected (for time-insensitive data), and a trainable embedding component (to capture spatial heterogeneity). To fill the data gap, we have - through a comprehensive process of data collection, integration, and augmentation - created a large-scale publicly available database of accident information named US-Accidents. By employing the US-Accidents dataset and through an extensive set of experiments across several large cities, we have evaluated our proposal against several baselines. Our analysis and results show significant improvements to predict rare accident events. Further, we have shown the impact of traffic information, time, and points-of-interest data for real-time accident prediction.

Text-guided diffusion models have significantly advanced image editing, enabling high-quality and diverse modifications driven by text prompts. However, effective editing requires inverting the source image into a latent space, a process often hindered by prediction errors inherent in DDIM inversion. These errors accumulate during the diffusion process, resulting in inferior content preservation and edit fidelity, especially with conditional inputs. We address these challenges by investigating the primary contributors to error accumulation in DDIM inversion and identify the singularity problem in traditional noise schedules as a key issue. To resolve this, we introduce the Logistic Schedule, a novel noise schedule designed to eliminate singularities, improve inversion stability, and provide a better noise space for image editing. This schedule reduces noise prediction errors, enabling more faithful editing that preserves the original content of the source image. Our approach requires no additional retraining and is compatible with various existing editing methods. Experiments across eight editing tasks demonstrate the Logistic Schedule's superior performance in content preservation and edit fidelity compared to traditional noise schedules, highlighting its adaptability and effectiveness.

Generative models transform random noise into images; their inversion aims to transform images back to structured noise for recovery and editing. This paper addresses two key tasks: (i) inversion and (ii) editing of a real image using stochastic equivalents of rectified flow models (such as Flux). Although Diffusion Models (DMs) have recently dominated the field of generative modeling for images, their inversion presents faithfulness and editability challenges due to nonlinearities in drift and diffusion. Existing state-of-the-art DM inversion approaches rely on training of additional parameters or test-time optimization of latent variables; both are expensive in practice. Rectified Flows (RFs) offer a promising alternative to diffusion models, yet their inversion has been underexplored. We propose RF inversion using dynamic optimal control derived via a linear quadratic regulator. We prove that the resulting vector field is equivalent to a rectified stochastic differential equation. Additionally, we extend our framework to design a stochastic sampler for Flux. Our inversion method allows for state-of-the-art performance in zero-shot inversion and editing, outperforming prior works in stroke-to-image synthesis and semantic image editing, with large-scale human evaluations confirming user preference.

Recent studies on inverse problems have proposed posterior samplers that leverage the pre-trained diffusion models as powerful priors. These attempts have paved the way for using diffusion models in a wide range of inverse problems. However, the existing methods entail computationally demanding iterative sampling procedures and optimize a separate solution for each measurement, which leads to limited scalability and lack of generalization capability across unseen samples. To address these limitations, we propose a novel approach, Diffusion prior-based Amortized Variational Inference (DAVI) that solves inverse problems with a diffusion prior from an amortized variational inference perspective. Specifically, instead of separate measurement-wise optimization, our amortized inference learns a function that directly maps measurements to the implicit posterior distributions of corresponding clean data, enabling a single-step posterior sampling even for unseen measurements. Extensive experiments on image restoration tasks, e.g., Gaussian deblur, 4times super-resolution, and box inpainting with two benchmark datasets, demonstrate our approach's superior performance over strong baselines. Code is available at https://github.com/mlvlab/DAVI.

We present an algorithm for minimizing an objective with hard-to-compute gradients by using a related, easier-to-access function as a proxy. Our algorithm is based on approximate proximal point iterations on the proxy combined with relatively few stochastic gradients from the objective. When the difference between the objective and the proxy is delta-smooth, our algorithm guarantees convergence at a rate matching stochastic gradient descent on a delta-smooth objective, which can lead to substantially better sample efficiency. Our algorithm has many potential applications in machine learning, and provides a principled means of leveraging synthetic data, physics simulators, mixed public and private data, and more.

An essential and challenging problem in causal inference is causal effect estimation from observational data. The problem becomes more difficult with the presence of unobserved confounding variables. The front-door adjustment is a practical approach for dealing with unobserved confounding variables. However, the restriction for the standard front-door adjustment is difficult to satisfy in practice. In this paper, we relax some of the restrictions by proposing the concept of conditional front-door (CFD) adjustment and develop the theorem that guarantees the causal effect identifiability of CFD adjustment. Furthermore, as it is often impossible for a CFD variable to be given in practice, it is desirable to learn it from data. By leveraging the ability of deep generative models, we propose CFDiVAE to learn the representation of the CFD adjustment variable directly from data with the identifiable Variational AutoEncoder and formally prove the model identifiability. Extensive experiments on synthetic datasets validate the effectiveness of CFDiVAE and its superiority over existing methods. The experiments also show that the performance of CFDiVAE is less sensitive to the causal strength of unobserved confounding variables. We further apply CFDiVAE to a real-world dataset to demonstrate its potential application.

Diffusion models have recently achieved success in solving Bayesian inverse problems with learned data priors. Current methods build on top of the diffusion sampling process, where each denoising step makes small modifications to samples from the previous step. However, this process struggles to correct errors from earlier sampling steps, leading to worse performance in complicated nonlinear inverse problems, such as phase retrieval. To address this challenge, we propose a new method called Decoupled Annealing Posterior Sampling (DAPS) that relies on a novel noise annealing process. Specifically, we decouple consecutive steps in a diffusion sampling trajectory, allowing them to vary considerably from one another while ensuring their time-marginals anneal to the true posterior as we reduce noise levels. This approach enables the exploration of a larger solution space, improving the success rate for accurate reconstructions. We demonstrate that DAPS significantly improves sample quality and stability across multiple image restoration tasks, particularly in complicated nonlinear inverse problems. For example, we achieve a PSNR of 30.72dB on the FFHQ 256 dataset for phase retrieval, which is an improvement of 9.12dB compared to existing methods.

Predictive uncertainty estimation is essential for safe deployment of Deep Neural Networks in real-world autonomous systems. However, disentangling the different types and sources of uncertainty is non trivial for most datasets, especially since there is no ground truth for uncertainty. In addition, while adverse weather conditions of varying intensities can disrupt neural network predictions, they are usually under-represented in both training and test sets in public datasets.We attempt to mitigate these setbacks and introduce the MUAD dataset (Multiple Uncertainties for Autonomous Driving), consisting of 10,413 realistic synthetic images with diverse adverse weather conditions (night, fog, rain, snow), out-of-distribution objects, and annotations for semantic segmentation, depth estimation, object, and instance detection. MUAD allows to better assess the impact of different sources of uncertainty on model performance. We conduct a thorough experimental study of this impact on several baseline Deep Neural Networks across multiple tasks, and release our dataset to allow researchers to benchmark their algorithm methodically in adverse conditions. More visualizations and the download link for MUAD are available at https://muad-dataset.github.io/.

We introduce ODEFormer, the first transformer able to infer multidimensional ordinary differential equation (ODE) systems in symbolic form from the observation of a single solution trajectory. We perform extensive evaluations on two datasets: (i) the existing "Strogatz" dataset featuring two-dimensional systems; (ii) ODEBench, a collection of one- to four-dimensional systems that we carefully curated from the literature to provide a more holistic benchmark. ODEFormer consistently outperforms existing methods while displaying substantially improved robustness to noisy and irregularly sampled observations, as well as faster inference. We release our code, model and benchmark dataset publicly.

Solving time-dependent parametric partial differential equations (PDEs) is challenging, as models must adapt to variations in parameters such as coefficients, forcing terms, and boundary conditions. Data-driven neural solvers either train on data sampled from the PDE parameters distribution in the hope that the model generalizes to new instances or rely on gradient-based adaptation and meta-learning to implicitly encode the dynamics from observations. This often comes with increased inference complexity. Inspired by the in-context learning capabilities of large language models (LLMs), we introduce Zebra, a novel generative auto-regressive transformer designed to solve parametric PDEs without requiring gradient adaptation at inference. By leveraging in-context information during both pre-training and inference, Zebra dynamically adapts to new tasks by conditioning on input sequences that incorporate context trajectories or preceding states. This approach enables Zebra to flexibly handle arbitrarily sized context inputs and supports uncertainty quantification through the sampling of multiple solution trajectories. We evaluate Zebra across a variety of challenging PDE scenarios, demonstrating its adaptability, robustness, and superior performance compared to existing approaches.

Sampling from diffusion probabilistic models (DPMs) can be viewed as a piecewise distribution transformation, which generally requires hundreds or thousands of steps of the inverse diffusion trajectory to get a high-quality image. Recent progress in designing fast samplers for DPMs achieves a trade-off between sampling speed and sample quality by knowledge distillation or adjusting the variance schedule or the denoising equation. However, it can't be optimal in both aspects and often suffer from mode mixture in short steps. To tackle this problem, we innovatively regard inverse diffusion as an optimal transport (OT) problem between latents at different stages and propose the DPM-OT, a unified learning framework for fast DPMs with a direct expressway represented by OT map, which can generate high-quality samples within around 10 function evaluations. By calculating the semi-discrete optimal transport map between the data latents and the white noise, we obtain an expressway from the prior distribution to the data distribution, while significantly alleviating the problem of mode mixture. In addition, we give the error bound of the proposed method, which theoretically guarantees the stability of the algorithm. Extensive experiments validate the effectiveness and advantages of DPM-OT in terms of speed and quality (FID and mode mixture), thus representing an efficient solution for generative modeling. Source codes are available at https://github.com/cognaclee/DPM-OT

Score-based diffusion models learn to reverse a stochastic differential equation that maps data to noise. However, for complex tasks, numerical error can compound and result in highly unnatural samples. Previous work mitigates this drift with thresholding, which projects to the natural data domain (such as pixel space for images) after each diffusion step, but this leads to a mismatch between the training and generative processes. To incorporate data constraints in a principled manner, we present Reflected Diffusion Models, which instead reverse a reflected stochastic differential equation evolving on the support of the data. Our approach learns the perturbed score function through a generalized score matching loss and extends key components of standard diffusion models including diffusion guidance, likelihood-based training, and ODE sampling. We also bridge the theoretical gap with thresholding: such schemes are just discretizations of reflected SDEs. On standard image benchmarks, our method is competitive with or surpasses the state of the art without architectural modifications and, for classifier-free guidance, our approach enables fast exact sampling with ODEs and produces more faithful samples under high guidance weight.

Floods wreak havoc throughout the world, causing billions of dollars in damages, and uprooting communities, ecosystems and economies. The NASA Impact Flood Detection competition tasked participants with predicting flooded pixels after training with synthetic aperture radar (SAR) images in a supervised setting. We propose a semi-supervised learning pseudo-labeling scheme that derives confidence estimates from U-Net ensembles, progressively improving accuracy. Concretely, we use a cyclical approach involving multiple stages (1) training an ensemble model of multiple U-Net architectures with the provided high confidence hand-labeled data and, generated pseudo labels or low confidence labels on the entire unlabeled test dataset, and then, (2) filter out quality generated labels and, (3) combine the generated labels with the previously available high confidence hand-labeled dataset. This assimilated dataset is used for the next round of training ensemble models and the cyclical process is repeated until the performance improvement plateaus. We post process our results with Conditional Random Fields. Our approach sets a new state-of-the-art on the Sentinel-1 dataset with 0.7654 IoU, an impressive improvement over the 0.60 IoU baseline. Our method, which we release with all the code and models, can also be used as an open science benchmark for the Sentinel-1 dataset.

In centralized settings, it is well known that stochastic gradient descent (SGD) avoids saddle points and converges to local minima in nonconvex problems. However, similar guarantees are lacking for distributed first-order algorithms. The paper studies distributed stochastic gradient descent (D-SGD)--a simple network-based implementation of SGD. Conditions under which D-SGD avoids saddle points and converges to local minima are studied. First, we consider the problem of computing critical points. Assuming loss functions are nonconvex and possibly nonsmooth, it is shown that, for each fixed initialization, D-SGD converges to critical points of the loss with probability one. Next, we consider the problem of avoiding saddle points. In this case, we again assume that loss functions may be nonconvex and nonsmooth, but are smooth in a neighborhood of a saddle point. It is shown that, for any fixed initialization, D-SGD avoids such saddle points with probability one. Results are proved by studying the underlying (distributed) gradient flow, using the ordinary differential equation (ODE) method of stochastic approximation, and extending classical techniques from dynamical systems theory such as stable manifolds. Results are proved in the general context of subspace-constrained optimization, of which D-SGD is a special case.

Dynamic Algorithm Configuration (DAC) aims to dynamically control a target algorithm's hyperparameters in order to improve its performance. Several theoretical and empirical results have demonstrated the benefits of dynamically controlling hyperparameters in domains like evolutionary computation, AI Planning or deep learning. Replicating these results, as well as studying new methods for DAC, however, is difficult since existing benchmarks are often specialized and incompatible with the same interfaces. To facilitate benchmarking and thus research on DAC, we propose DACBench, a benchmark library that seeks to collect and standardize existing DAC benchmarks from different AI domains, as well as provide a template for new ones. For the design of DACBench, we focused on important desiderata, such as (i) flexibility, (ii) reproducibility, (iii) extensibility and (iv) automatic documentation and visualization. To show the potential, broad applicability and challenges of DAC, we explore how a set of six initial benchmarks compare in several dimensions of difficulty.

Approximating nonlinear differential equations using a neural network provides a robust and efficient tool for various scientific computing tasks, including real-time predictions, inverse problems, optimal controls, and surrogate modeling. Previous works have focused on embedding dynamical systems into networks through two approaches: learning a single solution operator (i.e., the mapping from input parametrized functions to solutions) or learning the governing system of equations (i.e., the constitutive model relative to the state variables). Both of these approaches yield different representations for the same underlying data or function. Additionally, observing that families of differential equations often share key characteristics, we seek one network representation across a wide range of equations. Our method, called Predicting Operators and Symbolic Expressions (PROSE), learns maps from multimodal inputs to multimodal outputs, capable of generating both numerical predictions and mathematical equations. By using a transformer structure and a feature fusion approach, our network can simultaneously embed sets of solution operators for various parametric differential equations using a single trained network. Detailed experiments demonstrate that the network benefits from its multimodal nature, resulting in improved prediction accuracy and better generalization. The network is shown to be able to handle noise in the data and errors in the symbolic representation, including noisy numerical values, model misspecification, and erroneous addition or deletion of terms. PROSE provides a new neural network framework for differential equations which allows for more flexibility and generality in learning operators and governing equations from data.

Numerical simulations in climate, chemistry, or astrophysics are computationally too expensive for uncertainty quantification or parameter-exploration at high-resolution. Reduced-order or surrogate models are multiple orders of magnitude faster, but traditional surrogates are inflexible or inaccurate and pure machine learning (ML)-based surrogates too data-hungry. We propose a hybrid, flexible surrogate model that exploits known physics for simulating large-scale dynamics and limits learning to the hard-to-model term, which is called parametrization or closure and captures the effect of fine- onto large-scale dynamics. Leveraging neural operators, we are the first to learn grid-independent, non-local, and flexible parametrizations. Our multiscale neural operator is motivated by a rich literature in multiscale modeling, has quasilinear runtime complexity, is more accurate or flexible than state-of-the-art parametrizations and demonstrated on the chaotic equation multiscale Lorenz96.

In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the d-dimensional advection-diffusion equation, our proposal is a two-level algorithm that merges the solutions obtained from the discretization of the equation over highly anisotropic submeshes to compute an initial approximation of the fine solution. The algorithm then iteratively refines the initial guess by leveraging the structure of the residual. Performing costly calculations on anisotropic submeshes enable us to reduce the dimensionality of the problem by one, and the merging process, which involves the computation of solutions over disjoint domains, allows for parallel implementation.

Diffusion models suffer from slow sample generation at inference time. Therefore, developing a principled framework for fast deterministic/stochastic sampling for a broader class of diffusion models is a promising direction. We propose two complementary frameworks for accelerating sample generation in pre-trained models: Conjugate Integrators and Splitting Integrators. Conjugate integrators generalize DDIM, mapping the reverse diffusion dynamics to a more amenable space for sampling. In contrast, splitting-based integrators, commonly used in molecular dynamics, reduce the numerical simulation error by cleverly alternating between numerical updates involving the data and auxiliary variables. After extensively studying these methods empirically and theoretically, we present a hybrid method that leads to the best-reported performance for diffusion models in augmented spaces. Applied to Phase Space Langevin Diffusion [Pandey & Mandt, 2023] on CIFAR-10, our deterministic and stochastic samplers achieve FID scores of 2.11 and 2.36 in only 100 network function evaluations (NFE) as compared to 2.57 and 2.63 for the best-performing baselines, respectively. Our code and model checkpoints will be made publicly available at https://github.com/mandt-lab/PSLD.

Diffusion-based imitation learning improves Behavioral Cloning (BC) on multi-modal decision-making, but comes at the cost of significantly slower inference due to the recursion in the diffusion process. It urges us to design efficient policy generators while keeping the ability to generate diverse actions. To address this challenge, we propose AdaFlow, an imitation learning framework based on flow-based generative modeling. AdaFlow represents the policy with state-conditioned ordinary differential equations (ODEs), which are known as probability flows. We reveal an intriguing connection between the conditional variance of their training loss and the discretization error of the ODEs. With this insight, we propose a variance-adaptive ODE solver that can adjust its step size in the inference stage, making AdaFlow an adaptive decision-maker, offering rapid inference without sacrificing diversity. Interestingly, it automatically reduces to a one-step generator when the action distribution is uni-modal. Our comprehensive empirical evaluation shows that AdaFlow achieves high performance with fast inference speed.

Full-complexity Earth system models (ESMs) are computationally very expensive, limiting their use in exploring the climate outcomes of multiple emission pathways. More efficient emulators that approximate ESMs can directly map emissions onto climate outcomes, and benchmarks are being used to evaluate their accuracy on standardized tasks and datasets. We investigate a popular benchmark in data-driven climate emulation, ClimateBench, on which deep learning-based emulators are currently achieving the best performance. We implement a linear regression-based emulator, akin to pattern scaling, and find that it outperforms the incumbent 100M-parameter deep learning foundation model, ClimaX, on 3 out of 4 regionally-resolved surface-level climate variables. While emulating surface temperature is expected to be predominantly linear, this result is surprising for emulating precipitation. We identify that this outcome is a result of high levels of internal variability in the benchmark targets. To address internal variability, we update the benchmark targets with ensemble averages from the MPI-ESM1.2-LR model that contain 50 instead of 3 climate simulations per emission pathway. Using the new targets, we show that linear pattern scaling continues to be more accurate on temperature, but can be outperformed by a deep learning-based model for emulating precipitation. We publish our code, data, and an interactive tutorial at github.com/blutjens/climate-emulator.

The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code.

Recent successful generative models are trained by fitting a neural network to an a-priori defined tractable probability density path taking noise to training examples. In this paper we investigate the space of Gaussian probability paths, which includes diffusion paths as an instance, and look for an optimal member in some useful sense. In particular, minimizing the Kinetic Energy (KE) of a path is known to make particles' trajectories simple, hence easier to sample, and empirically improve performance in terms of likelihood of unseen data and sample generation quality. We investigate Kinetic Optimal (KO) Gaussian paths and offer the following observations: (i) We show the KE takes a simplified form on the space of Gaussian paths, where the data is incorporated only through a single, one dimensional scalar function, called the data separation function. (ii) We characterize the KO solutions with a one dimensional ODE. (iii) We approximate data-dependent KO paths by approximating the data separation function and minimizing the KE. (iv) We prove that the data separation function converges to 1 in the general case of arbitrary normalized dataset consisting of n samples in d dimension as n/drightarrow 0. A consequence of this result is that the Conditional Optimal Transport (Cond-OT) path becomes kinetic optimal as n/drightarrow 0. We further support this theory with empirical experiments on ImageNet.

Deep reinforcement learning (DRL) has shown significant promise for uncovering sophisticated control policies that interact in environments with complicated dynamics, such as stabilizing the magnetohydrodynamics of a tokamak fusion reactor or minimizing the drag force exerted on an object in a fluid flow. However, these algorithms require an abundance of training examples and may become prohibitively expensive for many applications. In addition, the reliance on deep neural networks often results in an uninterpretable, black-box policy that may be too computationally expensive to use with certain embedded systems. Recent advances in sparse dictionary learning, such as the sparse identification of nonlinear dynamics (SINDy), have shown promise for creating efficient and interpretable data-driven models in the low-data regime. In this work we introduce SINDy-RL, a unifying framework for combining SINDy and DRL to create efficient, interpretable, and trustworthy representations of the dynamics model, reward function, and control policy. We demonstrate the effectiveness of our approaches on benchmark control environments and challenging fluids problems. SINDy-RL achieves comparable performance to state-of-the-art DRL algorithms using significantly fewer interactions in the environment and results in an interpretable control policy orders of magnitude smaller than a deep neural network policy.

Approximating Stochastic Gradient Descent (SGD) as a Stochastic Differential Equation (SDE) has allowed researchers to enjoy the benefits of studying a continuous optimization trajectory while carefully preserving the stochasticity of SGD. Analogous study of adaptive gradient methods, such as RMSprop and Adam, has been challenging because there were no rigorously proven SDE approximations for these methods. This paper derives the SDE approximations for RMSprop and Adam, giving theoretical guarantees of their correctness as well as experimental validation of their applicability to common large-scaling vision and language settings. A key practical result is the derivation of a square root scaling rule to adjust the optimization hyperparameters of RMSprop and Adam when changing batch size, and its empirical validation in deep learning settings.

This paper rigorously shows how over-parameterization changes the convergence behaviors of gradient descent (GD) for the matrix sensing problem, where the goal is to recover an unknown low-rank ground-truth matrix from near-isotropic linear measurements. First, we consider the symmetric setting with the symmetric parameterization where M^* in R^{n times n} is a positive semi-definite unknown matrix of rank r ll n, and one uses a symmetric parameterization XX^top to learn M^*. Here X in R^{n times k} with k > r is the factor matrix. We give a novel Omega (1/T^2) lower bound of randomly initialized GD for the over-parameterized case (k >r) where T is the number of iterations. This is in stark contrast to the exact-parameterization scenario (k=r) where the convergence rate is exp (-Omega (T)). Next, we study asymmetric setting where M^* in R^{n_1 times n_2} is the unknown matrix of rank r ll min{n_1,n_2}, and one uses an asymmetric parameterization FG^top to learn M^* where F in R^{n_1 times k} and G in R^{n_2 times k}. Building on prior work, we give a global exact convergence result of randomly initialized GD for the exact-parameterization case (k=r) with an exp (-Omega(T)) rate. Furthermore, we give the first global exact convergence result for the over-parameterization case (k>r) with an exp(-Omega(alpha^2 T)) rate where alpha is the initialization scale. This linear convergence result in the over-parameterization case is especially significant because one can apply the asymmetric parameterization to the symmetric setting to speed up from Omega (1/T^2) to linear convergence. On the other hand, we propose a novel method that only modifies one step of GD and obtains a convergence rate independent of alpha, recovering the rate in the exact-parameterization case.

Inferring unbiased treatment effects has received widespread attention in the machine learning community. In recent years, our community has proposed numerous solutions in standard settings, high-dimensional treatment settings, and even longitudinal settings. While very diverse, the solution has mostly relied on neural networks for inference and simultaneous correction of assignment bias. New approaches typically build on top of previous approaches by proposing new (or refined) architectures and learning algorithms. However, the end result -- a neural-network-based inference machine -- remains unchallenged. In this paper, we introduce a different type of solution in the longitudinal setting: a closed-form ordinary differential equation (ODE). While we still rely on continuous optimization to learn an ODE, the resulting inference machine is no longer a neural network. Doing so yields several advantages such as interpretability, irregular sampling, and a different set of identification assumptions. Above all, we consider the introduction of a completely new type of solution to be our most important contribution as it may spark entirely new innovations in treatment effects in general. We facilitate this by formulating our contribution as a framework that can transform any ODE discovery method into a treatment effects method.

Omnidirectional images (ODIs) are commonly used in real-world visual tasks, and high-resolution ODIs help improve the performance of related visual tasks. Most existing super-resolution methods for ODIs use end-to-end learning strategies, resulting in inferior realness of generated images and a lack of effective out-of-domain generalization capabilities in training methods. Image generation methods represented by diffusion model provide strong priors for visual tasks and have been proven to be effectively applied to image restoration tasks. Leveraging the image priors of the Stable Diffusion (SD) model, we achieve omnidirectional image super-resolution with both fidelity and realness, dubbed as OmniSSR. Firstly, we transform the equirectangular projection (ERP) images into tangent projection (TP) images, whose distribution approximates the planar image domain. Then, we use SD to iteratively sample initial high-resolution results. At each denoising iteration, we further correct and update the initial results using the proposed Octadecaplex Tangent Information Interaction (OTII) and Gradient Decomposition (GD) technique to ensure better consistency. Finally, the TP images are transformed back to obtain the final high-resolution results. Our method is zero-shot, requiring no training or fine-tuning. Experiments of our method on two benchmark datasets demonstrate the effectiveness of our proposed method.

Data-driven modeling based on machine learning (ML) is showing enormous potential for weather forecasting. Rapid progress has been made with impressive results for some applications. The uptake of ML methods could be a game-changer for the incremental progress in traditional numerical weather prediction (NWP) known as the 'quiet revolution' of weather forecasting. The computational cost of running a forecast with standard NWP systems greatly hinders the improvements that can be made from increasing model resolution and ensemble sizes. An emerging new generation of ML models, developed using high-quality reanalysis datasets like ERA5 for training, allow forecasts that require much lower computational costs and that are highly-competitive in terms of accuracy. Here, we compare for the first time ML-generated forecasts with standard NWP-based forecasts in an operational-like context, initialized from the same initial conditions. Focusing on deterministic forecasts, we apply common forecast verification tools to assess to what extent a data-driven forecast produced with one of the recently developed ML models (PanguWeather) matches the quality and attributes of a forecast from one of the leading global NWP systems (the ECMWF IFS). The results are very promising, with comparable skill for both global metrics and extreme events, when verified against both the operational analysis and synoptic observations. Increasing forecast smoothness and bias drift with forecast lead time are identified as current drawbacks of ML-based forecasts. A new NWP paradigm is emerging relying on inference from ML models and state-of-the-art analysis and reanalysis datasets for forecast initialization and model training.

Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality.

Diffusion models are a class of flexible generative models trained with an approximation to the log-likelihood objective. However, most use cases of diffusion models are not concerned with likelihoods, but instead with downstream objectives such as human-perceived image quality or drug effectiveness. In this paper, we investigate reinforcement learning methods for directly optimizing diffusion models for such objectives. We describe how posing denoising as a multi-step decision-making problem enables a class of policy gradient algorithms, which we refer to as denoising diffusion policy optimization (DDPO), that are more effective than alternative reward-weighted likelihood approaches. Empirically, DDPO is able to adapt text-to-image diffusion models to objectives that are difficult to express via prompting, such as image compressibility, and those derived from human feedback, such as aesthetic quality. Finally, we show that DDPO can improve prompt-image alignment using feedback from a vision-language model without the need for additional data collection or human annotation.

Recently, a series of diffusion-aware distillation algorithms have emerged to alleviate the computational overhead associated with the multi-step inference process of Diffusion Models (DMs). Current distillation techniques often dichotomize into two distinct aspects: i) ODE Trajectory Preservation; and ii) ODE Trajectory Reformulation. However, these approaches suffer from severe performance degradation or domain shifts. To address these limitations, we propose Hyper-SD, a novel framework that synergistically amalgamates the advantages of ODE Trajectory Preservation and Reformulation, while maintaining near-lossless performance during step compression. Firstly, we introduce Trajectory Segmented Consistency Distillation to progressively perform consistent distillation within pre-defined time-step segments, which facilitates the preservation of the original ODE trajectory from a higher-order perspective. Secondly, we incorporate human feedback learning to boost the performance of the model in a low-step regime and mitigate the performance loss incurred by the distillation process. Thirdly, we integrate score distillation to further improve the low-step generation capability of the model and offer the first attempt to leverage a unified LoRA to support the inference process at all steps. Extensive experiments and user studies demonstrate that Hyper-SD achieves SOTA performance from 1 to 8 inference steps for both SDXL and SD1.5. For example, Hyper-SDXL surpasses SDXL-Lightning by +0.68 in CLIP Score and +0.51 in Aes Score in the 1-step inference.

Denoising Diffusion Models (DDMs) have become a popular tool for generating high-quality samples from complex data distributions. These models are able to capture sophisticated patterns and structures in the data, and can generate samples that are highly diverse and representative of the underlying distribution. However, one of the main limitations of diffusion models is the complexity of sample generation, since a large number of inference timesteps is required to faithfully capture the data distribution. In this paper, we present MMD-DDM, a novel method for fast sampling of diffusion models. Our approach is based on the idea of using the Maximum Mean Discrepancy (MMD) to finetune the learned distribution with a given budget of timesteps. This allows the finetuned model to significantly improve the speed-quality trade-off, by substantially increasing fidelity in inference regimes with few steps or, equivalently, by reducing the required number of steps to reach a target fidelity, thus paving the way for a more practical adoption of diffusion models in a wide range of applications. We evaluate our approach on unconditional image generation with extensive experiments across the CIFAR-10, CelebA, ImageNet and LSUN-Church datasets. Our findings show that the proposed method is able to produce high-quality samples in a fraction of the time required by widely-used diffusion models, and outperforms state-of-the-art techniques for accelerated sampling. Code is available at: https://github.com/diegovalsesia/MMD-DDM.

Diffusion models have exhibited excellent performance in various domains. The probability flow ordinary differential equation (ODE) of diffusion models (i.e., diffusion ODEs) is a particular case of continuous normalizing flows (CNFs), which enables deterministic inference and exact likelihood evaluation. However, the likelihood estimation results by diffusion ODEs are still far from those of the state-of-the-art likelihood-based generative models. In this work, we propose several improved techniques for maximum likelihood estimation for diffusion ODEs, including both training and evaluation perspectives. For training, we propose velocity parameterization and explore variance reduction techniques for faster convergence. We also derive an error-bounded high-order flow matching objective for finetuning, which improves the ODE likelihood and smooths its trajectory. For evaluation, we propose a novel training-free truncated-normal dequantization to fill the training-evaluation gap commonly existing in diffusion ODEs. Building upon these techniques, we achieve state-of-the-art likelihood estimation results on image datasets (2.56 on CIFAR-10, 3.43/3.69 on ImageNet-32) without variational dequantization or data augmentation.

Partial Differential Equations are foundational in modeling science and natural systems such as fluid dynamics and weather forecasting. The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers by proposing a scalable and efficient alternative. To enhance the efficiency and accuracy of LE-PDE, we incorporate the Mamba model, an advanced machine learning model known for its predictive efficiency and robustness in handling complex dynamic systems with a progressive learning strategy. The LE-PDE was tested on several benchmark problems. The method demonstrated a marked reduction in computational time compared to traditional solvers and standalone deep learning models while maintaining high accuracy in predicting system behavior over time. Our method doubles the inference speed compared to the LE-PDE while retaining the same level of parameter efficiency, making it well-suited for scenarios requiring long-term predictions.

Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied data distribution are expected to adhere to specific governing equations. We present a framework that unifies generative modeling and partial differential equation fulfillment by introducing a first-principle-based loss term that enforces generated samples to fulfill the underlying physical constraints. Our approach reduces the residual error by up to two orders of magnitude compared to previous work in a fluid flow case study and outperforms task-specific frameworks in relevant metrics for structural topology optimization. We also present numerical evidence that our extended training objective acts as a natural regularization mechanism against overfitting. Our framework is simple to implement and versatile in its applicability for imposing equality and inequality constraints as well as auxiliary optimization objectives.

In this study, a novel machine learning approach was used to classify three types of synoptic weather events in Taiwan area from 2001 to 2010. We used reanalysis data with three machine learning algorithms to recognize weather systems and evaluated their performance. Overall, the classifiers successfully identified 52-83% of weather events (hit rate), which is higher than the performance of traditional objective methods. The results showed that the machine learning approach gave low false alarm rate in general, while the support vector machine (SVM) with more principal components of reanalysis data had higher hit rate on all tested weather events. The sensitivity tests of grid data resolution indicated that the differences between the high- and low-resolution datasets are limited, which implied that the proposed method can achieve reasonable performance in weather forecasting with minimal resources. By identifying daily weather systems in historical reanalysis data, this method can be used to study long-term weather changes, to monitor climatological-scale variations, and to provide a better estimate of climate projections. Furthermore, this method can also serve as an alternative to model output statistics and potentially be used for synoptic weather forecasting.

Diffusion models (DMs) represent state-of-the-art generative models for continuous inputs. DMs work by constructing a Stochastic Differential Equation (SDE) in the input space (ie, position space), and using a neural network to reverse it. In this work, we introduce a novel generative modeling framework grounded in phase space dynamics, where a phase space is defined as {an augmented space encompassing both position and velocity.} Leveraging insights from Stochastic Optimal Control, we construct a path measure in the phase space that enables efficient sampling. {In contrast to DMs, our framework demonstrates the capability to generate realistic data points at an early stage of dynamics propagation.} This early prediction sets the stage for efficient data generation by leveraging additional velocity information along the trajectory. On standard image generation benchmarks, our model yields favorable performance over baselines in the regime of small Number of Function Evaluations (NFEs). Furthermore, our approach rivals the performance of diffusion models equipped with efficient sampling techniques, underscoring its potential as a new tool generative modeling.

Earth Observation (EO) data are increasingly used in policy analysis by enabling granular estimation of treatment effects. However, a challenge in EO-based causal inference lies in balancing the trade-off between capturing fine-grained individual heterogeneity and broader contextual information. This paper introduces Multi-scale Concatenation, a family of composable procedures that transform arbitrary single-scale CATE estimation algorithms into multi-scale algorithms. We benchmark the performance of Multi-scale Concatenation on a CATE estimation pipeline combining Vision Transformer (ViT) models fine-tuned on satellite images to encode images of different scales with Causal Forests to obtain the final CATE estimate. We first perform simulation studies, showing how a multi-scale approach captures multi-level dynamics that single-scale ViT models fail to capture. We then apply the multi-scale method to two randomized controlled trials (RCTs) conducted in Peru and Uganda using Landsat satellite imagery. In the RCT analysis, the Rank Average Treatment Effect Ratio (RATE Ratio) measure is employed to assess performance without ground truth individual treatment effects. Results indicate that Multi-scale Concatenation improves the performance of deep learning models in EO-based CATE estimation without the complexity of designing new multi-scale architectures for a specific use case.

We deal with the combinatorial problem of learning directed acyclic graph (DAG) structure from observational data adhering to a linear structural equation model (SEM). Leveraging advances in differentiable, nonconvex characterizations of acyclicity, recent efforts have advocated a continuous constrained optimization paradigm to efficiently explore the space of DAGs. Most existing methods employ lasso-type score functions to guide this search, which (i) require expensive penalty parameter retuning when the unknown SEM noise variances change across problem instances; and (ii) implicitly rely on limiting homoscedasticity assumptions. In this work, we propose a new convex score function for sparsity-aware learning of linear DAGs, which incorporates concomitant estimation of scale and thus effectively decouples the sparsity parameter from the exogenous noise levels. Regularization via a smooth, nonconvex acyclicity penalty term yields CoLiDE (Concomitant Linear DAG Estimation), a regression-based criterion amenable to efficient gradient computation and closed-form estimation of noise variances in heteroscedastic scenarios. Our algorithm outperforms state-of-the-art methods without incurring added complexity, especially when the DAGs are larger and the noise level profile is heterogeneous. We also find CoLiDE exhibits enhanced stability manifested via reduced standard deviations in several domain-specific metrics, underscoring the robustness of our novel linear DAG estimator.

D-Adaptation is an approach to automatically setting the learning rate which asymptotically achieves the optimal rate of convergence for minimizing convex Lipschitz functions, with no back-tracking or line searches, and no additional function value or gradient evaluations per step. Our approach is the first hyper-parameter free method for this class without additional multiplicative log factors in the convergence rate. We present extensive experiments for SGD and Adam variants of our method, where the method automatically matches hand-tuned learning rates across more than a dozen diverse machine learning problems, including large-scale vision and language problems. An open-source implementation is available.

Acquisition of large datasets for three-dimensional (3D) partial differential equations are usually very expensive. Physics-informed neural operator (PINO) eliminates the high costs associated with generation of training datasets, and shows great potential in a variety of partial differential equations. In this work, we employ physics-informed neural operator, encoding the large-eddy simulation (LES) equations directly into the neural operator for simulating three-dimensional incompressible turbulent flows. We develop the LESnets (Large-Eddy Simulation nets) by adding large-eddy simulation equations to two different data-driven models, including Fourier neural operator (FNO) and implicit Fourier neural operator (IFNO) without using label data. Notably, by leveraging only PDE constraints to learn the spatio-temporal dynamics problem, LESnets retains the computational efficiency of data-driven approaches while obviating the necessity for data. Meanwhile, using large-eddy simulation equations as PDE constraints makes it possible to efficiently predict complex turbulence at coarse grids. We investigate the performance of the LESnets with two standard three-dimensional turbulent flows: decaying homogeneous isotropic turbulence and temporally evolving turbulent mixing layer. In the numerical experiments, the LESnets model shows a similar or even better accuracy as compared to traditional large-eddy simulation and data-driven models of FNO and IFNO. Moreover, the well-trained LESnets is significantly faster than traditional LES, and has a similar efficiency as the data-driven FNO and IFNO models. Thus, physics-informed neural operators have a strong potential for 3D nonlinear engineering applications.

Modern climate projections lack adequate spatial and temporal resolution due to computational constraints. A consequence is inaccurate and imprecise predictions of critical processes such as storms. Hybrid methods that combine physics with machine learning (ML) have introduced a new generation of higher fidelity climate simulators that can sidestep Moore's Law by outsourcing compute-hungry, short, high-resolution simulations to ML emulators. However, this hybrid ML-physics simulation approach requires domain-specific treatment and has been inaccessible to ML experts because of lack of training data and relevant, easy-to-use workflows. We present ClimSim, the largest-ever dataset designed for hybrid ML-physics research. It comprises multi-scale climate simulations, developed by a consortium of climate scientists and ML researchers. It consists of 5.7 billion pairs of multivariate input and output vectors that isolate the influence of locally-nested, high-resolution, high-fidelity physics on a host climate simulator's macro-scale physical state. The dataset is global in coverage, spans multiple years at high sampling frequency, and is designed such that resulting emulators are compatible with downstream coupling into operational climate simulators. We implement a range of deterministic and stochastic regression baselines to highlight the ML challenges and their scoring. The data (https://huggingface.co/datasets/LEAP/ClimSim_high-res, https://huggingface.co/datasets/LEAP/ClimSim_low-res, and https://huggingface.co/datasets/LEAP/ClimSim_low-res_aqua-planet) and code (https://leap-stc.github.io/ClimSim) are released openly to support the development of hybrid ML-physics and high-fidelity climate simulations for the benefit of science and society.

The availability of reliable, high-resolution climate and weather data is important to inform long-term decisions on climate adaptation and mitigation and to guide rapid responses to extreme events. Forecasting models are limited by computational costs and, therefore, often generate coarse-resolution predictions. Statistical downscaling, including super-resolution methods from deep learning, can provide an efficient method of upsampling low-resolution data. However, despite achieving visually compelling results in some cases, such models frequently violate conservation laws when predicting physical variables. In order to conserve physical quantities, here we introduce methods that guarantee statistical constraints are satisfied by a deep learning downscaling model, while also improving their performance according to traditional metrics. We compare different constraining approaches and demonstrate their applicability across different neural architectures as well as a variety of climate and weather data sets. Besides enabling faster and more accurate climate predictions through downscaling, we also show that our novel methodologies can improve super-resolution for satellite data and natural images data sets.

Diffusion inversion is the problem of taking an image and a text prompt that describes it and finding a noise latent that would generate the image. Most current inversion techniques operate by approximately solving an implicit equation and may converge slowly or yield poor reconstructed images. Here, we formulate the problem as finding the roots of an implicit equation and design a method to solve it efficiently. Our solution is based on Newton-Raphson (NR), a well-known technique in numerical analysis. A naive application of NR may be computationally infeasible and tends to converge to incorrect solutions. We describe an efficient regularized formulation that converges quickly to a solution that provides high-quality reconstructions. We also identify a source of inconsistency stemming from prompt conditioning during the inversion process, which significantly degrades the inversion quality. To address this, we introduce a prompt-aware adjustment of the encoding, effectively correcting this issue. Our solution, Regularized Newton-Raphson Inversion, inverts an image within 0.5 sec for latent consistency models, opening the door for interactive image editing. We further demonstrate improved results in image interpolation and generation of rare objects.

The quantitative prediction of the intensity of rainfall events (light or heavy) has remained a challenge in Numerical Weather Prediction (NWP) models. For the first time the mean coefficient of diffusional growth rates are calculated using an Eulerian-Lagrangian particle-based small-scale model on in situ airborne measurement data of Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX) during monsoon over Indian sub-continent. The results show that diffusional growth rates varies in the range of 0.00025 - 0.0015(cm/s). The generic problem of the overestimation of light rain in NWP models might be related with the choice of cm in the model. It is also shown from DNS experiment using Eulerian-Lagrangian particle-based small-scale model that the relative dispersion is constrained with average values in the range of ~ 0.2 - 0.37 (~ 0.1- 0.26) in less humid (more humid) conditions. This is in agreement with in situ airborne observation (dispersion ~ 0.36) and previous study over Indian sub-continent. The linear relationship between relative dispersion and cloud droplet number concentration (NC) is obtained from this study using CAIPEEX observation over Indian subcontinent. The dispersion based autoconversion-scheme for Indian region must be useful for the Indian summer monsoon precipitation calculation in the general circulation model. The present study also provide valuable guidance for the parameterization of effective radius, important for radiation scheme.

Denoising diffusion (score-based) generative models have recently achieved significant accomplishments in generating realistic and diverse data. These approaches define a forward diffusion process for transforming data into noise and a backward denoising process for sampling data from noise. Unfortunately, the generation process of current denoising diffusion models is notoriously slow due to the lengthy iterative noise estimations, which rely on cumbersome neural networks. It prevents the diffusion models from being widely deployed, especially on edge devices. Previous works accelerate the generation process of diffusion model (DM) via finding shorter yet effective sampling trajectories. However, they overlook the cost of noise estimation with a heavy network in every iteration. In this work, we accelerate generation from the perspective of compressing the noise estimation network. Due to the difficulty of retraining DMs, we exclude mainstream training-aware compression paradigms and introduce post-training quantization (PTQ) into DM acceleration. However, the output distributions of noise estimation networks change with time-step, making previous PTQ methods fail in DMs since they are designed for single-time step scenarios. To devise a DM-specific PTQ method, we explore PTQ on DM in three aspects: quantized operations, calibration dataset, and calibration metric. We summarize and use several observations derived from all-inclusive investigations to formulate our method, which especially targets the unique multi-time-step structure of DMs. Experimentally, our method can directly quantize full-precision DMs into 8-bit models while maintaining or even improving their performance in a training-free manner. Importantly, our method can serve as a plug-and-play module on other fast-sampling methods, e.g., DDIM. The code is available at https://github.com/42Shawn/PTQ4DM .

Various data augmentation techniques have been recently proposed in image-based deep reinforcement learning (DRL). Although they empirically demonstrate the effectiveness of data augmentation for improving sample efficiency or generalization, which technique should be preferred is not always clear. To tackle this question, we analyze existing methods to better understand them and to uncover how they are connected. Notably, by expressing the variance of the Q-targets and that of the empirical actor/critic losses of these methods, we can analyze the effects of their different components and compare them. We furthermore formulate an explanation about how these methods may be affected by choosing different data augmentation transformations in calculating the target Q-values. This analysis suggests recommendations on how to exploit data augmentation in a more principled way. In addition, we include a regularization term called tangent prop, previously proposed in computer vision, but whose adaptation to DRL is novel to the best of our knowledge. We evaluate our proposition and validate our analysis in several domains. Compared to different relevant baselines, we demonstrate that it achieves state-of-the-art performance in most environments and shows higher sample efficiency and better generalization ability in some complex environments.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

Denoising diffusion models (DDMs) have recently attracted increasing attention by showing impressive synthesis quality. DDMs are built on a diffusion process that pushes data to the noise distribution and the models learn to denoise. In this paper, we establish the interpretation of DDMs in terms of image restoration (IR). Integrating IR literature allows us to use an alternative objective and diverse forward processes, not confining to the diffusion process. By imposing prior knowledge on the loss function grounded on MAP-based estimation, we eliminate the need for the expensive sampling of DDMs. Also, we propose a multi-scale training, which improves the performance compared to the diffusion process, by taking advantage of the flexibility of the forward process. Experimental results demonstrate that our model improves the quality and efficiency of both training and inference. Furthermore, we show the applicability of our model to inverse problems. We believe that our framework paves the way for designing a new type of flexible general generative model.

The development of foundation models has revolutionized our ability to interpret the Earth's surface using satellite observational data. Traditional models have been siloed, tailored to specific sensors or data types like optical, radar, and hyperspectral, each with its own unique characteristics. This specialization hinders the potential for a holistic analysis that could benefit from the combined strengths of these diverse data sources. Our novel approach introduces the Dynamic One-For-All (DOFA) model, leveraging the concept of neural plasticity in brain science to integrate various data modalities into a single framework adaptively. This dynamic hypernetwork, adjusting to different wavelengths, enables a single versatile Transformer jointly trained on data from five sensors to excel across 12 distinct Earth observation tasks, including sensors never seen during pretraining. DOFA's innovative design offers a promising leap towards more accurate, efficient, and unified Earth observation analysis, showcasing remarkable adaptability and performance in harnessing the potential of multimodal Earth observation data.

Image deblurring aims to remove undesired blurs from an image captured in a dynamic scene. Much research has been dedicated to improving deblurring performance through model architectural designs. However, there is little work on data augmentation for image deblurring. Since continuous motion causes blurred artifacts during image exposure, we aspire to develop a groundbreaking blur augmentation method to generate diverse blurred images by simulating motion trajectories in a continuous space. This paper proposes Implicit Diffusion-based reBLurring AUgmentation (ID-Blau), utilizing a sharp image paired with a controllable blur condition map to produce a corresponding blurred image. We parameterize the blur patterns of a blurred image with their orientations and magnitudes as a pixel-wise blur condition map to simulate motion trajectories and implicitly represent them in a continuous space. By sampling diverse blur conditions, ID-Blau can generate various blurred images unseen in the training set. Experimental results demonstrate that ID-Blau can produce realistic blurred images for training and thus significantly improve performance for state-of-the-art deblurring models.

Over the past few years, due to the rapid development of machine learning (ML) models for weather forecasting, state-of-the-art ML models have shown superior performance compared to the European Centre for Medium-Range Weather Forecasts (ECMWF)'s high-resolution forecast (HRES) in 10-day forecasts at a spatial resolution of 0.25 degree. However, the challenge remains to perform comparably to the ECMWF ensemble mean (EM) in 15-day forecasts. Previous studies have demonstrated the importance of mitigating the accumulation of forecast errors for effective long-term forecasts. Despite numerous efforts to reduce accumulation errors, including autoregressive multi-time step loss, using a single model is found to be insufficient to achieve optimal performance in both short and long lead times. Therefore, we present FuXi, a cascaded ML weather forecasting system that provides 15-day global forecasts with a temporal resolution of 6 hours and a spatial resolution of 0.25 degree. FuXi is developed using 39 years of the ECMWF ERA5 reanalysis dataset. The performance evaluation, based on latitude-weighted root mean square error (RMSE) and anomaly correlation coefficient (ACC), demonstrates that FuXi has comparable forecast performance to ECMWF EM in 15-day forecasts, making FuXi the first ML-based weather forecasting system to accomplish this achievement.

Learning from off-policy data is essential for sample-efficient reinforcement learning. In the present work, we build on the insight that the advantage function can be understood as the causal effect of an action on the return, and show that this allows us to decompose the return of a trajectory into parts caused by the agent's actions (skill) and parts outside of the agent's control (luck). Furthermore, this decomposition enables us to naturally extend Direct Advantage Estimation (DAE) to off-policy settings (Off-policy DAE). The resulting method can learn from off-policy trajectories without relying on importance sampling techniques or truncating off-policy actions. We draw connections between Off-policy DAE and previous methods to demonstrate how it can speed up learning and when the proposed off-policy corrections are important. Finally, we use the MinAtar environments to illustrate how ignoring off-policy corrections can lead to suboptimal policy optimization performance.

Neural Controlled Differential Equations (NCDEs) are a state-of-the-art tool for supervised learning with irregularly sampled time series (Kidger, 2020). However, no theoretical analysis of their performance has been provided yet, and it remains unclear in particular how the irregularity of the time series affects their predictions. By merging the rich theory of controlled differential equations (CDE) and Lipschitz-based measures of the complexity of deep neural nets, we take a first step towards the theoretical understanding of NCDE. Our first result is a generalization bound for this class of predictors that depends on the regularity of the time series data. In a second time, we leverage the continuity of the flow of CDEs to provide a detailed analysis of both the sampling-induced bias and the approximation bias. Regarding this last result, we show how classical approximation results on neural nets may transfer to NCDEs. Our theoretical results are validated through a series of experiments.

A faster than real time forecast system for solar wind and interplanetary magnetic field transients that is driven by hourly updated solar magnetograms is proposed to provide a continuous nowcast of the solar corona (<0.1AU) and 24-hours forecast of the solar wind at 1 AU by solving a full 3-D MHD model. This new model has been inspired by the concept of relativity of simultaneity used in the theory of special relativity. It is based on time transformation between two coordinate systems: the solar rest frame and a boosted system in which the current observations of the solar magnetic field and tomorrow's measurement of the solar wind at 1 AU are simultaneous. In this paper we derive the modified governing equations for both hydrodynamics (HD) and magnetohydrodynamics (MHD) and present a new numerical algorithm that only modifies the conserved quantities but preserves the original HD/MHD numerical flux. The proposed method enables an efficient numerical implementation, and thus a significantly longer forecast time than the traditional method.

Chaos and unpredictability are traditionally synonymous, yet large-scale machine learning methods recently have demonstrated a surprising ability to forecast chaotic systems well beyond typical predictability horizons. However, recent works disagree on whether specialized methods grounded in dynamical systems theory, such as reservoir computers or neural ordinary differential equations, outperform general-purpose large-scale learning methods such as transformers or recurrent neural networks. These prior studies perform comparisons on few individually-chosen chaotic systems, thereby precluding robust quantification of how statistical modeling choices and dynamical invariants of different chaotic systems jointly determine empirical predictability. Here, we perform the largest to-date comparative study of forecasting methods on the classical problem of forecasting chaos: we benchmark 24 state-of-the-art forecasting methods on a crowdsourced database of 135 low-dimensional systems with 17 forecast metrics. We find that large-scale, domain-agnostic forecasting methods consistently produce predictions that remain accurate up to two dozen Lyapunov times, thereby accessing a new long-horizon forecasting regime well beyond classical methods. We find that, in this regime, accuracy decorrelates with classical invariant measures of predictability like the Lyapunov exponent. However, in data-limited settings outside the long-horizon regime, we find that physics-based hybrid methods retain a comparative advantage due to their strong inductive biases.