diff --git "a/GtE0T4oBgHgl3EQfzgJq/content/tmp_files/load_file.txt" "b/GtE0T4oBgHgl3EQfzgJq/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/GtE0T4oBgHgl3EQfzgJq/content/tmp_files/load_file.txt" @@ -0,0 +1,686 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf,len=685 +page_content='Data-driven discovery and extrapolation of parameterized pattern-forming dynamics Zachary G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nicolaou,1 Guanyu Huo,1 Yihui Chen,1 Steven L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton,2 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nathan Kutz1 1Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA 2Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA Pattern-forming systems can exhibit a diverse array of complex behaviors as external parameters are varied, enabling a variety of useful functions in biological and engineered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' First-principle derivations of the underlying transitions can be characterized using bifurcation theory on model sys- tems whose governing equations are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In contrast, data-driven methods for more complicated and realistic systems whose governing evolution dynamics are unknown have only recently been de- veloped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Here, we develop a data-driven approach sparse identification for nonlinear dynamics with control parameters (SINDyCP) to discover dynamics for systems with adjustable control parameters, such as an external driving strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We demonstrate the method on systems of varying complexity, ranging from discrete maps to systems of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To mitigate the impact of measurement noise, we also develop a weak formulation of SINDyCP and assess its performance on noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We demonstrate applications including the discovery of universal pattern-formation equations, and their bifurcation dependencies, directly from data accessible from experiments and the extrapolation of predictions beyond the weakly nonlinear regime near the onset of an instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Data-driven approaches to system identification are undergoing a revolution, spurred by the increasing avail- ability of computational resources, data, and the develop- ment of novel and reliable machine learning algorithms [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The sparse identification of nonlinear dynamics (SINDy) is a particularly simple and flexible mathemat- ical approach that leverages efficient sparse optimization algorithms in the automated discovery of complex sys- tem dynamics and governing equations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In this work, we leverage the SINDy model discovery framework to understand parametric dependencies and underlying bi- furcations in pattern forming systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Specifically, we develop the SINDY with control parameters (SINDyCP) to discover such parameterized dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' It has been thirty years since Cross and Hohenberg’s seminal and authoritative review consolidating an excep- tionally large body of work on pattern formation across a broad range of physical systems [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Universal equations determined by normal forms of canonical bifurcations [6], such as the complex Ginzburg-Landau equation [7], gov- ern the formation of patterns near the onset of instabili- ties across scientific disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Such equations continue to reveal insights into complex systems, including in the study of, for example, synchronization, biophysics, active matter, and quantum dynamics [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Despite the success of pattern-formation theory in modeling complex dynamics, ongoing challenges remain in applying such model equations more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' First- principle derivations and the computation of normal- form parameters in terms of physical driving parameters are tedious, costly, and error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Furthermore, the normal-form approach is only theoretically justified in the weakly-nonlinear regime near the onset of an insta- bility, while interesting and important pattern-forming processes often occur far from the instability threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Recent advances in data-driven system identification are opening new avenues of research to address these chal- lenges, including a paradigm for modeling strongly non- linear regimes beyond the asymptotic approximations re- viewed by Cross and Hohenberg [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The SINDy model discovery framework is particularly well-suited to the modern analysis of bifurcations and normal forms, as it generates interpretable models that have as few terms as possible, balancing model complex- ity and descriptive capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A variety of extensions of the SINDy approach have been developed since its in- troduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' For example, SINDYc enables discovery of systems subject to external control signals [11, 12], while PDEFind [13, 14] enables discovery of spatio-temporal dynamics characterized by partial differential equations (PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' SINDy can also learn to disambiguate between parametric dependency and governing equations [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Model pattern formation equations typically encode the effects of external drive through a number of driving pa- rameters, which characterize the bifurcation leading to the onset of instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Several recent works establish system identification on pattern-forming systems rang- ing from closure models for fluid turbulence [16–18] to biochemical reactions and active active matter systems [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' These approaches show promise, but crucially, they have not demonstrated the ability to extrapolate by detecting pattern-forming instabilities that may de- velop when driving parameter differ the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' While there has been success for discrete maps and or- dinary differential equations (ODEs) [4, 22], combining the PDEFind and SINDYc approaches to discover pa- rameterized spatio-temporal dynamics poses a significant challenge, as we detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The key insight underlying SINDyCP is recognizing the need to introduce distinct libraries of possible de- pendencies for the dependent variables and the control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Our approach is implemented in the open- source PySINDy repository [23, 24], enabling other pow- erful methods to be used in conjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In particu- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='02673v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='PS] 6 Jan 2023 2 Construct library and derivatives from samples Sparse regression Parameterized equation Trajectories with varying parameters Feature Library, Parameter Library, Time Derivatives, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Schematic of the SINDyCP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Data collected from sample trajectories collected under various driving parame- ters are processed to construct a matrix of time derivatives, a feature library Θfeat of possible governing terms, and a parameter library Θpar of parametric dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sparse regression is applied on the library coefficients ξ to identify a parameterized governing equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' lar, we develop and assess a weak formulation [25–28] of SINDyCP, which shows excellent performance on noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We demonstrate that the method can be easily and effectively employed to discover accurate parameterized models from the kind of data available in typical pat- tern formation experiments and that these parameterized models enable extrapolation beyond the conditions under which they were developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Building the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—Figure 1 illustrates the SINDyCP approach applied to the spatio-temporal evolution of four trajectories of the complex Ginzburg- Landau equation ˙A = A + (1 + ib)∇2A − (1 − ic)|A|2A, (1) which is described by a complex dependent vari- able A(x, t) in two spatial dimensions x = (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Ginzburg-Landau exhibits a stunning variety of pat- terns, depending on the bifurcation parameters b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We generate four trajectories with parameters val- ues (b, c) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='75), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='75), which exhibit differing dynamical phases, corresponding to amplitude turbulence, phase turbulence, stable waves, and frozen spiral glasses, respectively [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Our goal is to discover the partial differential equation for the real and imaginary components A = X + iY parameterized by b and c from time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' As with most SINDy algorithms, we first form a ma- trix of the input data X, whose columns correspond to the dependent variables and whose rows correspond to the sample measurements of the dependent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1, for example, X consists of two columns corresponding to the real and imaginary parts of A and 4NxNyNt rows, where Nx, Ny, and Nt are the number of sample points in the corresponding spatio-temporal di- mensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' again, there are four trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We then de- termine the temporal derivative ˙X for each sample point, either through numerical differentiation or through direct measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In basic SINDy, we define a matrix of library terms Θ = Θ(X) depending on the input data, which includes all possible terms that may be present in the differen- tial equation that describes the temporal derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' These terms may be built from polynomial combina- tions of the dependent variables and their spatial deriva- tives, for example, although more general libraries are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In the SINDYc approach, we augment the li- brary dependence with an external control signal U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=', Θ = Θ(X, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The library terms are typically determined by appending the control variables to the dependent vari- ables and again forming polynomials and derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In the case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1, we can treat the parameters as exter- nal control signals, U = (b, c) and apply SINDYc, but the traditional implementation of this approach will fail for PDEs, as we show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' SINDYc aims to find a sparse linear combination of the library terms determined by the vector of coefficients ξ which minimizes the fit error ξ∗ = argminξ ��� ˙X − Θ(X, U)ξ ��� + λ |ξ|0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (2) Crucially, all SINDy methods employ sparse regression (with appropriate regularization) to determine a sparse set of nonzero coefficients ξ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Such sparsity is expected in physically-relevant dynamics and produces parsimonious and interpretable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A significant challenge arises when applying the tradi- tional SINDYc to control parameters in PDEs with ex- isting implementations such as PySINDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The matrix of library terms Θ is traditionally formed by computing all polynomial combinations of spatial derivatives of the dependent and control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' However, since the con- trol parameters are spatially constant, the spatial deriva- tives will vanish identically, leading to a singular matrix Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To overcome this challenge, we propose construct- ing a more general library through products of a feature 3 library Θfeat(X) and a parameter library Θpar(U), as Θ(X, U) = Θfeat(X) ⊗ Θpar(U), (3) where the product ⊗ here is defined to give the ma- trix consisting of all combinations of products of columns (computed component-wise across the row elements) be- tween the libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' By distinguishing the feature and parameter library dependencies with this SINDyCP ap- proach, we can construct much more targeted and well- conditioned libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Using a feature library consisting of spatial derivatives up to third order and polynomials up to third order along with a linear parameter library, the SINDyCP approach easily discovers Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (1) in Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Details of the numerical integration, an animation illustrating the temporal evolution of the sample trajectories, and additional demonstrations for maps and ODEs are avail- able in the Supplemental Materials [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Beyond weakly nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—SINDyCP enables discovery of nonlinear corrections to weakly nonlin- ear theory directly from data that can be gathered in pattern-formation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To illustrate this result, we implement an in silico experiment of the Belousov- Zhabotinksy chemical reaction system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We numerically integrate the Oregonator model [30], ˙CX = k1CAC2 HCY − k2CHCXCY + k3CACHCX − 2k4C2 X + DX∇2CX, (4a) ˙CY = −k1CAC2 HCY − k2CHCXCY + νk5CBCZ + DY ∇2CY (4b) ˙CZ = 2k3CACHCX − k5CBCZ + DZ∇2CZ, (4c) which describes the evolution of oscillating chemical con- centrations CX, CY , and CZ for given supplied concen- trations CA, CB, and CH and stoichiometric coefficient ν, which depends on the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We vary the concentration of CB and define a control parame- ter µ ≡ CB − Cc B, where Cc B is the critical value where the Hopf bifurcation occurs (see Supplementary Mate- rials [29] for parameter values and other details in the Oregonator model) to generate six trajectories with µ ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='02 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We use a SINDyCP feature library with polynomial terms up to fifth order and second order spatial deriva- tives and a parameter library with polynomial terms up to second order for the control parameter µ1/2 in con- junction with implicit SINDy [31] to discovers a highly nonlinear parameterized model from time-series measure- ments of CX and CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Figure 2(a) shows the R2 score of the model on test trajectories corresponding to the pa- rameter values that the model was trained on (a value of R2 = 1 means that the fit perfectly predicts the tem- poral derivatives of the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' While the score decreases modestly as µ increases, the model remains very accurate (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Corrections to the weakly nonlinear theory of the Oregonator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (a) R2 score for the parameterized SINDyCP model on test trajectories collected at the param- eter values used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (b) Corrected normal- form parameter values relative to the weakly nonlinear values b0 and c0 as a function of the bifurcation parameter µ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (c) Limit cycles in the homogeneous system exhibiting the highly- nonlinear canard explosion with increasing µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The pattern formation above the canard explosion (upper inset) is quali- tatively different than for smaller driving (lower inset), with more extreme spatio-temporal variation that does not emerge in the weakly nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' on all the testing trajectories, accounting for 99% of the variation in the data in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A nonlinear change of coordinates transforms the dis- covered model into the normal-form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (1) with parameter-dependent values of b(µ) and c(µ) and small quintic corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' These normal-form parameters agree with the analytic values derived [32] from the original model as µ → 0, but here we are able to discover them di- rectly from data without any knowledge of the governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2(b), the pa- rameters vary with µ, representing additional corrections to the weakly nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This variation becomes extreme for µ1/2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='35, which we were able to discover via the implicit version of SINDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In fact, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2(c), the Oregonator model exhibits a canard explo- sion (in which the limit cycle amplitude expands abruptly due to highly nonlinear effects) [30] around µ1/2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='39, where the weakly nonlinear theory breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The SINDyCP model reflects this breakdown and enables the development of higher-order corrections to account for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Weak formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—The weak formulation utilizes in- tegration against compactly supported “test functions” to defined the SINDy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The weak method shows excellent performance for noisy data, owing to its ability to minimize the need for computing numerical deriva- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Rather than forming samples (rows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1) from spatio-temporal points for each trajectory, the weak method constructs the system rows by projecting the data onto weak samples such as wν ik ≡ � Ωk φ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' t)X(ν) i (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' t) dDxdt, (5) 4 where Ωk is a compactly-supported spatio-temporal do- main, φ is the test function, and X(ν) i denotes the νth partial derivative the ith dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' By moving derivatives off of the data and onto the test functions via integration by parts, wν ik = (−1)|ν| � Ωk φ(ν)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' t)Xi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' t) dDxdt, (6) the weak method significantly reduces the impact of mea- surement noise on the SINDy library and improves the fit results [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To maximize the performance for the weak method, we have optimized and fully vectorized numerical inte- gration for the weak formulation in PySINDy, which can be easily combined with the SINDyCP library class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' De- tails about our efficient numerical implementation are available in the Supplemental Material [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Products of weak features do not generally form reasonable samples for a SINDy model, since multiplication and integration do not commute, so on first sight, it is not clear how to combine weak form feature and parameter libraries with SINDyCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' However, when computing the weak samples corresponding to constant functions, such as those that form the parameter library, the integrals simply repre- sent the spatio-temporal volume of the domain Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Our implementation thus rescales the weak features for the temporal derivatives by the same volumetric factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—Using 500 randomly distributed sam- ple domains (measuring 1/10th the spatio-temporal do- main size in each direction), the weak SINDyCP easily identifies the complex Ginzburg-Landau equation using the same data used for the traditional differential form shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Furthermore, it can do so in just a few seconds of run-time on a modern processor in this case (over five times faster than the differential form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To assess the impact of noise, we inject random Gaus- sian noise of varying intensity [34] into the four trajecto- ries used as the training data for the complex Ginzburg- Landau equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We then generate two new sample tra- jectories to use as testing data, with b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5 and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Using the training data, we perform the SINDyCP fits using both the differential for- mulation and the weak formulation and evaluate the R2 score on our test trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Figure 3(a) shows the re- sults for the R2 score on the test trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' While both methods provide good fits for low noise intensity, only the weak method exhibits a robust fit for parameterized equations for large noise intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The SINDyCP fit also requires a sufficient amount of data to identify governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Figure 3(b) shows the performance of SINDyCP on the testing data for fits performed with a varying number of trajectories nt = 2, 3, 4, 5 and of varying length corresponding to a number of time samples Nt = 25, 50, 75, 100, with an in- jected noise intensity of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Unlike the trajectories in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1, the parameters for trajectories were randomly (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Performance of SINDyCP for the fit of the complex Ginzburg-Landau equation with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (a) Model score vs noise intensity using the differential and weak forms of SINDyCP with nt = 4 trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (b) Model score vs num- ber of samples for varying number of randomly generated tra- jectories, varying trajectory length, and noise intensity 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' generated, with (b, c) distributed as Gaussian random variables with means (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0) and standard deviations (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' For too little data, the fit fails to identify the correct model, and the value of 1 − R2 is O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The models improve moderately with an increasing number of samples per trajectory (the product of Nt with the number of spatial grid points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' More importantly, a suf- ficiently large number of trajectories nt is required to achieve a good fit (at least 3 in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The amount of data required will further increase when including a larger number of possible library terms and when identi- fying a larger number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' These requirements should be carefully assessed in order to achieve successful SINDyCP fits for more general pattern forming systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Parameter extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—As a final demonstration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 4), we consider the one-dimensional cubic-quintic Swift-Hohenberg equation ˙u = du − uxxxx − 2uxx − u + eu3 − fu5, (7) with parameters d, e, and f describing the linear, cu- bic, and regularizing quintic terms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This model pattern formation equation has been used to study defect dynamics incorporating corrections beyond the weakly nonlinear approximation and has revealed uni- versal snaking bifurcations leading to the formation of localized states for e > 0 and d < 0 [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The parameters d, e and f are the minimal and natu- ral set to describe the possible dynamics in the Swift- Hohenberg equation derived from normal-form theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' However, in typical pattern formation applications, one does not have direct control over such parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In- stead, experimentally accessible parameters will have a complicated and nonlinear relationship with the normal- form parameters, which requires detailed knowledge and 5 (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Extrapolation of localized states in the cubic-quintic Swift-Hohenberg equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (a) The randomly generated re- lationships between the normal-form parameters (d, e, f) and the experimental parameter ε (bottom panel) gives rise to snaking bifurcations (top panel) near ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Red dotted lines show the values used to train the SINDyCP fit, and dashed colored lines show the coefficients derived from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (b) Localized states extrapolated from numerical simulations of the SINDyCP fit with ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, corresponding to the black dotted line in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' tedious calculations to derive, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=', an expansion and cen- ter manifold transformation around a bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The SINDyCP approach enables an automated discovery of such relationships, which can be used to extrapolate system behavior beyond a set of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To illustrate this idea, we generate random quadratic relationships between an experimental parameter ε and the normal-form parameters (d, e, f), and we create three training trajectories using random values of the param- eter 1 < ε < 3 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To determine the possible dynamics, we numerically continue the solution branches corresponding to the trivial state and localized and pe- riodic states using the AUTO package [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' For all of the training trajectories, ε is sufficiently large that no localized or periodic states are exist, and all trajecto- ries decay to the trivial u = 0 solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We perform the weak SINDyCP fit using these trajectories subject to 1% injected noise with a quadradic parameter library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To test the ability of SINDyCP to extrapolate beyond the parameter regime given in the input data, we sim- ulate the identified model for the experimental param- eter value ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Remarkably, even with limited and noisy training data, the method identifies an accurate relationship between ε and the normal-form parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Thus, simulations of the identified model with random initial conditions converge to localized states [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 4(b)] for ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1 despite the significant extrapolation of the parameter value beyond the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Discussion—The SINDyCP approach represents a simple but powerful generalization of SINDy with con- trol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' By disambiguating the feature and parameter com- ponents of the SINDy libraries, the method enables dis- covery of systems of partial differential equations param- eterized by driving parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Such equations arise nat- urally in the context of pattern formation, where the normal forms of bifurcations lead to parameterized equa- tions near the onset of instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The approach can be easily applied with the data available in typical pattern formation experiments and promises to enable true ex- trapolation beyond the regime that can be theoretically described with weakly nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Combining the SINDyCP approach with autoencoder-assisted discovery of physical coordinates [37–39] will further enable re- searchers to discover nonlinear equations governing com- plex systems directly from data gathered through ex- periments conducted under various driving parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This approach may also help inform universal mecha- nisms leading to the formation of localized states beyond the snaking bifurcations of the Swift-Hohenberg equation [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This work benefited from insightful discussions with Alan Kaptanoglu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Zachary G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nicolaou is a WRF post- doctoral fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We acknowledge support from the Na- tional Science Foundation AI Institute in Dynamic Sys- tems (grant number 2112085).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Data-driven science and engineering: Machine learning, dynamical systems, and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (Cambridge University Press, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Udrescu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Tegmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' AI Feynman: A physics- inspired method for symbolic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 6, eaay2631 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Karniadakis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kevrekidis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Perdikaris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Wang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Physics-informed machine learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nature Reviews Physics 3, 422-440 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Proctor, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 113, 3932-3937 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Cross and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Hohenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Pattern formation outside of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 65, 851 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kuznetsov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kuznetsov, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kuznetsov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Elements of applied bifurcation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' New York: Springer, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [7] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Aranson and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kramer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The world of the com- plex Ginzburg-Landau equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 74, 99 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [8] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nicolaou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Riecke, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Motter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Chimera states in continuous media: Existence and distinctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 119, 244101 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [9] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Heinonen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Abraham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S�lomka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Burns, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S´aenz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Dunkel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Emergent universal statistics in nonequilibrium systems with dynamical scale selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01627 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Proctor, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sparse iden- tification of nonlinear dynamics with control (SINDYc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' IFAC-PapersOnLine 49, 710-715 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kaiser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, Sparse identifi- 6 cation of nonlinear dynamics for model predictive control in the low-data limit, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Royal Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A 474, 20180335 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [12] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Fasel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kaiser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sindy with control: A tutorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In 2021 60th IEEE Conference on Decision and Control (CDC), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 16-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Rudy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Proctor, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Data-driven discovery of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 3, e1602614 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [14] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Schaeffer, Learning partial differential equations via data discovery and sparse optimization, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Royal Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A 473, 20160446 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Rudy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Alla, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz, Data- driven identification of parametric partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 18, 643-660 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [16] Schmelzer, Martin, Richard P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Dwight, and Paola Cin- nella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' ”Discovery of algebraic Reynolds-stress models us- ing sparse symbolic regression.” Flow, Turbulence and Combustion 104, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2 (2020): 579-603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Zanna and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Bolton, Data-driven equation discov- ery of ocean mesoscale closures, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 47, e2020GL088376 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Beetham, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Fox, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Capecelatro, Sparse iden- tification of multiphase turbulence closures for coupled fluid–particle flows, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 914, A11 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [19] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Wu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Garikipati, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Huan, A per- spective on regression and Bayesian approaches for sys- tem identification of pattern formation dynamics, Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 10, 188-194 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [20] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Romeo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Hastewell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Mietke, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Dunkel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Learning developmental mode dynamics from single-cell trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Elife 10 e68679 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Supekar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Song, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Hastewell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Mietke, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Dunkel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Learning hydrodynamic equations for active matter from particle simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='06568 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Schaeffer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Tran, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Ward, Learning dynam- ical systems and bifurcation via group sparsity, arXiv preprint arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01558 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kaptanoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' PySINDy: A comprehensive Python package for robust sparse system identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' of Open Source Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 7, 3994 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [24] The pysindy repository is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' com/dynamicslab/pysindy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [25] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Reinbold, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Gurevich, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Grigoriev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Using noisy or incomplete data to discover models of spa- tiotemporal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' E 101, 010203 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Reinbold, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kageorge, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Schatz, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Grigoriev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Robust learning from noisy, incom- plete, high-dimensional experimental data via physically constrained symbolic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 12, 3219 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Messenger and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Bortz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Weak SINDy for partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' of Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 443, 110525 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Messenger and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Bortz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Learning mean-field equations from particle data using WSINDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Physica D, 133406 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [29] See Supplemental Material for details about numerical integration, additional demonstrations, the oregonator model, and the weak form implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Mazzotti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Morbidelli, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Serravalle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Bifurca- tion analysis of the Oregonator model in the 3-D space bromate/malonic acid/stoichiometric coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J Phys Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 99, 4501 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [31] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Mangan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Proctor, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz, Inferring biological networks by sparse iden- tification of nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Multi-Scale Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Ipsen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Hynne, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sørensen, Amplitude equa- tions for reaction–diffusion systems with a Hopf bifurca- tion and slow real modes, Physica D 136, 66 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [33] It is not possible to remove all numerical derivatives in the weak formulation, but the maximum order of deriva- tives can generally be reduced to at most half the original order for the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [34] Noise intensity here refers to the pointwise standard devi- ation on the spatio-temporal grid employed in the simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' True white noise has a Dirac delta variance, and intensity should thus scale with grid spacing and time step to 1/2 power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [35] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Burke and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Knobloch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Homoclinic snaking: structure and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Chaos 17, 037102 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [36] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Doedel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Champneys, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Dercole, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Fair- grieve, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kuznetsov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Oldeman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Paffenroth, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sandstede, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Wang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' AUTO- 07P: Continuation and bifurcation software for ordinary differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [37] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Champion, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Lusch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Data-driven discovery of coordinates and governing equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 116, 22445-22451 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [38] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Raghupathi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Chandratreya, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Du, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Lipson, Automated discovery of fundamental variables hidden in experimental data, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2, 433-442 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Bakarji, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Champion, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Kutz, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brun- ton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Discovering governing equations from partial mea- surements with deep delay autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='05136 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [40] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Chen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Upadhyaya, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Vitelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nonlinear con- duction via solitons in a topological mechanical insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 111, 13004-13009 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [41] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nicolaou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Case, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Wee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Driscoll, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Motter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Heterogeneity-stabilized homogeneous states in driven media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 12, 4486 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1 Supplementary Material for “Data-driven discovery and extrapolation of parameterized pattern-forming dynamics” Zachary G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nicolaou, Guanyu Huo,Yihui Chen, Steven L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nathan Kutz S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' NUMERICAL INTEGRATION For the complex Ginzburg-Landau equation, we use a pseudospectral integration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We take a periodic domain of size of size L = 32π in each direction and discretize using Nx = Ny = 128 grid points in each spatial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Derivatives are calculated using fast Fourier transforms, and the discretized system is integrated with a 4(5) Runge-Kutta-Fehlberg method (which is also used for the other equations, with relative and absolute error tolerances of 10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To produce states in the dynamical phases of interest, we take random initial conditions A0 = � nm αnmeiknm·x + ϵeik2 2·x, where αnm are complex random Gaussian amplitudes with mean zero and variance σ2/(1 + n2 + m2), knm = 2π(nˆx + mˆy)/L, the sum ranges over −2 ≤ n, m ≤ 2, and ϵ is the scale of an initial plane wave perturbation with wavevector k2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The mode amplitudes are determined by σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01 for the four trajectories used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The system is allowed to approach an attractor for the first 90 time units, then the trajectory is formed by the next 10 time units, in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We also provide an animation showing the phase and amplitude for longer runs of 100 time units (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A similar pseudospectral approach was used for the Oregonator and Swift-Hohenberg examples, but, in the Swift-Hohenberg case, with Nx = 256 discretization points, a domain of size L = 64π, an integration time of 5 time units, and random initial condition given by the real part of u0 = �20 n=−20 αneiknx with kn = 2πn/L and αn complex random Gaussian amplitudes with mean zero and variance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0/(1 + � |n|)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Snapshot of the animation showing the phase φ and amplitude r of the trajectories, where A = reiφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 0 r/2 3π/2 2rl L/2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='2 0 L/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='9 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6 r L L/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0 L/2 L L/2 0 L/2 7-7 L/2 0 L/2 7 x x2 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' DEMONSTRATIONS Demonstrations of SINDyCP in discrete maps, ODEs and PDEs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The left panels illustrate the logistic map, xn+1 = rxn(1 − xn), (S1) which is a discrete-time system with a single dependent variable xn and a single parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This equation is the model for a universal period-doubling route to chaos as the parameter r increases past 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='56995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We perform the SINDyCP fit using four sample trajectories of 1000 iterations, corresponding to parameter values r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='7, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='9 (red dotted lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We employ a library consisting of polynomials up to third order in the dependent variable xn and linear functions of the control parameter r, and the SINDyCP approach correctly identifies the parameterized equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The middle panels illustrate the Lorenz system, ˙x = σ(y − x), ˙y = x(ρ − z) − y, ˙z = xy − βz, (S2) which consists of three ordinary differential equations in three dependent variables x, y, and z and three parameters σ, ρ and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This equation exhibits the iconic butterfly-shaped Lorenz attractor for certain parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We perform the SINDyCP fit using five sample trajectories that have converged to their attractors, corresponding to the randomly selected parameter values σ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='9, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='3, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, ρ = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='2, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='3, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, and β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We use feature and parameter libraries consisting of polynomials up to fourth order in the dependent variables (x, y, z) and linear functions in the parameters (σ, ρ, β), and the SINDyCP approach again correctly identifies the parameterized equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Finally, the right panels illustrate the CGLE described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' SINDyCP ft Input data Model Logistic map Lorenz system Complex Ginzburg-Landau equation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Demonstrations of the SINDyCP approach for three models (top row) of nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Several trajectories produced from different parameter values (middle row) are supplied as input, and the SINDyCP fit (bottom row) correctly identifies the governing equations in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 3 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' OREGONATOR MODEL AND NORMAL FORM TRANSFORMATION We mainly follow the analyses of the Oregonator model in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [30,32], with realistic parameter values shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The fixed point (CX, CY , CZ) = (C0 X, C0 Y , C0 Z) undergoes a Hopf bifurcation as µ increases from zero, leading to oscillatory chemical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' For small µ, the weakly nonlinear theory follows from a perturbative expansion of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Take x ≡ (CX, CY , CZ) − (C0 X, C0 Y , C0 Z) and express Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (4)-(6) as ˙x = F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Define the multilinear operators of partial derivatives Fxn(ei1, · · · , ein) = ∂nF/∂xi1 · · · ∂xin with ei the ith component unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Then the Taylor expansion for the system is ˙x = (∂F/∂µ) µ + Fx1(x) + (∂F/∂µ)x1 (x)µ + 1 2Fx2(x, x) + 1 6Fx3(x, x, x) + D · ∇2x + · · · , (S3) where D is a diagonal matrix with elements DX, DY and DZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We develop a transformation x = y+h(y, µ) perturbatively, where y ≡ Aeiω0tu + ¯Ae−iω0t¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Here u is one of the critical eigenvectors of the Jacobian matrix Fx1 with eigenvalue λ = iω0 (with zero real part for µ = 0) and overbars represent complex conjugates, and we also define the corresponding left eigenvector at u⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The near-identity transformation function h(y, µ) is selected so as to eliminate the non-resonant terms in the evolution equation of A, which can be accomplished under general conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This results in an amplitude equation ˙A = µσA + g|A|2A + d∇2A, where σ = u⊥ · (∂F/∂µ)x1 (u) − u⊥ · Fx2 � u, (Fx1)−1 (∂F/∂µ) � , (S4) g = 1 2u⊥ · Fx3 (u, u, ¯u) − u⊥ · Fx2 � u, [Fx1]−1 [Fx2 (u, ¯u)] � − 1 2u⊥ · Fx2 � ¯u, � Fx1 − � λ − ¯λ � I �−1 [Fx2 (u, u)] � , (S5) d = u⊥ · D · u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S6) By rescaling the amplitude by a factor of µ1/2, time by a factor of 1/µ, and space by a factor of 1/µ1/2 and employing additional rescalings to unitize the real components and eliminate the mean rotation, we can arrive at the CGLE in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (2), where b ≡ Im(d)/Re(d) = b0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='173 and c ≡ −Im(g)/Re(g) = c0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' As expected, these parameter values correspond to the amplitude turbulence regime of the CGLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' For our numerical simulations, we use a spatial domain of length L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='4/µ1/2 cm and an integration time of T = 200/µ s, where we scaled by µ to ensure the trajectories have corresponding scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We strobe the time in steps of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='94804 s, which corresponds to the critical frequency of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We then interpolate the time series in steps of T/1000 to generate the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The first 200 time steps are discarded as the trajectories relax to their attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The next 400 time steps are used to train the SINDyCP model, while the remaining 400 steps are used as test trajectories to evaluate the R2 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We finally employ the normal form transformation described above for the SINDyCP model to evaluate the parameterized b(µ) and c(µ) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2(b) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Consistently, the normal form parameters very closely approximate the analytic results b(0) ≈ b0 and c(0) ≈ c0, but significant variations emerge for larger µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' k1 k2 k3 k4 k5 DX DY DZ CH CA CB/(1 − µ) ν 2 106 10 2 × 103 1 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='787 1 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Parameter values for the Oregonator model, in cgs units (suppressed for brevity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 4 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' WEAK FORMULATION IMPLEMENTATION We refer the reader to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [25-28] for the theory of the weak formulation of SINDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Here, we only briefly describe our efficient numerical integration method for the weak formulation used in pysindy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We suppose that the spatial grid is one-dimensional, for the moment, and the values of the coordinates on the grid points are xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The weak form requires us to calculate the integral of interpolated data f(x) weighted by the dth derivatives of test function φ(x), I(d) ≡ � xN x0 f(x)φ(d)(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S7) We choose to use test functions φ(x) = (x2 − 1)p in our implementation, and thus their dth derivatives are φ(d)(x) = ∂ ∂xd (x2 − 1)p = p � k=0 � p k � (−1)k (2(p − k))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (2(p − k) − d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='x2(p−k)−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S8) We are provided with some feature values ui at the grid points, and we consider the value of a library function f applied to that feature, fi ≡ f(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We linearly interpolate the function as f(x) = fi + x−xi xi+1−xi (fi+1 − fi) where xi ≤ x ≤ xi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Expanding the interpolation, and integrating the xφ(d)(x) terms by parts, I(d) = N−1 � i=0 � xi+1 xi � fi + x − xi xi+1 − xi (fi+1 − fi) � φ(d)(x)dx = N−1 � i=0 fixi+1 − fi+1xi xi+1 − xi � Φ(d)(xi+1) − Φ(d)(xi) � + fi+1 − fi xi+1 − xi � Φ(d−1)(xi+1) − Φ(d−1)(xi) � , (S9) where Φ(d)(x) are the antiderivatives of φ(d) [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Φ(d)(x) = φ(d−1)(x) for d > 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' By relabelling the dummy summation variables, we can recast Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S9) as a dot product between the input data fj and a weight wj I(d) = N−1 � j=0 wj · fj, (S10) with wj ≡ xj+1 � Φ(d)(xj+1) − Φ(d)(xj) � xj+1 − xj − xj−1 � Φ(d)(xj) − Φ(d)(xj−1) � xj − xj−1 + Φ(d−1)(xj) − Φ(d−1)(xj−1) xj − xj−1 − Φ(d−1)(xj+1) − Φ(d−1)(xj) xj+1 − xj , (S11) where 0 < j < N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' At the left and right sides of the domain (for j = 0 and j = N − 1), we must adjust the weights to correct for boundary effects, w0 ≡ x1 � Φ(d)(x1) − Φ(d)(x0) � x1 − x0 − Φ(d−1)(x1) − Φ(d−1)(x0) x1 − x0 , (S12) wN−1 ≡ −xN−2 � Φ(d)(xN−1) − Φ(d)(xN−2) � xN−1 − xN−2 + Φ(d−1)(xN−1) − Φ(d−1)(xN−2) xN−1 − xN−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S13) 5 Expressing the integrals along each dimension as dot products [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S10)] enables efficient vectorization with BLAS operations, and the integration weights [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S11)-(S13)] only need to be evaluated a single time when the library is first initialized (in a vectorized fashion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We further vectorize the code by forming tensor products over all integration dimensions to calculate multidimensional integrals using a single tensor dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'}