aaaifss2022_2.txt

Data-Driven Reduced-Order Model for Atmospheric CO 2Dispersion
Pedro R. B. Rocha,1,2Marcos S. P. Gomes,2Jo˜ao L. S. Almeida,1
Allan M. Carvalho,1Alberto C. Nogueira Jr.1
1IBM Research Brazil,
2Pontifical Catholic University of Rio de Janeiro
[email protected], [email protected], [email protected],
[email protected], [email protected]
Abstract
Machine learning frameworks have emerged as powerful
tools for the enhancement of computational fluid dynam-
ics simulations and the construction of reduced-order mod-
els (ROMs). The latter are particularly desired when their
full-order counterparts portray multiple spatiotemporal fea-
tures and demand high processing power and storage capac-
ity, such as climate models. In this work, a ROM for CO 2
dispersion across Earth‘s atmosphere was built from NASA’s
gridded daily OCO-2 carbon dioxide assimilated dataset. For
that, a proper orthogonal decomposition was performed, fol-
lowed by a non-intrusive operator inference (OpInf). This sci-
entific machine learning technique was capable of accurately
representing and predicting the detailed CO 2concentration
field for about one year ahead, with a normalized root-mean-
square error below 5%. It suggests OpInf-based ROMs may
be a reliable alternative for fast response climate-related pre-
dictions.
Introduction
Physics-informed machine learning (PIML) algorithms,
which blend data-driven modeling with information about
the physics, have been widely employed for improving spa-
tiotemporal forecasts in the Earth and climate sciences (Re-
ichstein et al. 2019; Schneider, Jeevanjee, and Socolow
2021; Cort ´es-Andr ´es et al. 2022; Willard et al. 2022;
Kashinath et al. 2021). Since purely data-driven machine
learning (ML) models are not often generalizable beyond
the training interval due to their misspecification or data dis-
tribution shifts, violating fundamental principles, additional
constraints are required. By incorporating physical laws on
the ML model, it has a considerable gain in performance
and robustness, becoming more reliable, generalizable and
explainable.
One of the primary goals within the PIML field is to
build reduced-order models (ROMs) for dynamical systems,
which are more computationally efficient than their full-
order counterparts in spite of possibly being less precise
(Willard et al. 2022). In general, these models are suitable
in control, optimization and uncertainty quantification prob-
lems, where multiple high-fidelity numerical simulations are
Copyright © 2022, Association for the Advancement of Artificial
Intelligence (www.aaai.org). All rights reserved.needed. Regarding the climate sciences, building ROMs be-
comes notably challenging due to the Earth system’s broad
range of scales in space and time and to computational lim-
itations imposed by its very high number of degrees of free-
dom. It is worth mentioning that, when considering state
variables defined mainly by small-scale processes, it is of
paramount importance to downscale them to a finer reso-
lution before proceeding to the dimensionality reduction,
when many features of the system are inevitably lost.
In the last decade, it was developed a data-driven reduced-
order modeling framework based on a non-intrusive opera-
tor inference (OpInf) (Peherstorfer and Willcox 2016). This
method postulates the shape for the ROM operators based on
the knowledge that most physical equations, including those
related to fluid flows, are second-order nonlinear. Besides,
since OpInf only relies on spatiotemporal data provided by
high-fidelity simulations or experimental measurements, it is
quite straightforward to be employed in a myriad of dynam-
ical systems to leverage new scientific discoveries. In addi-
tion to that, it is computationally more efficient and robust
than conventional deep learning techniques, such as echo-
state networks (Nogueira Jr et al. 2021), since it requires
a much lower number of hyperparameters to be tuned and
has a better extrapolation capability. Recently, it was suc-
cessfully applied in a complex multiscale combustion prob-
lem (Swischuk et al. 2020; McQuarrie, Huang, and Willcox
2021).
In this work, to further explore the potential of the OpInf
in the field of climate sciences, the gridded daily OCO-2 car-
bon dioxide assimilated dataset (Weir and Ott 2022), which
contains the CO 2concentration field around the Earth, was
considered. The main goal was to obtain a reliable OpInf-
based ROM for this field and compare it with the original
dataset. To reduce the dimensionality of the system, a proper
orthogonal decomposition (POD) was performed. This tech-
nique originates from turbulence studies and has been exten-
sively explored in the literature (Lumley 1967).
Data Reduction
The main idea behind POD consists in decomposing a given
spatiotemporal vector field u(x;t)into a set of spatial func-
tionsk(x), or POD modes, and their respective time coef-
ficientsk(t)(Weiss 2019). Then, the vector field u(x;t)is
written as

u(x;t) =1X
k=1k(t)k(x); (1)
where vector (or matrix) quantities are represented in bold.
Although this decomposition may be carried out in dif-
ferent ways, all of them must respect two basic conditions:
the spatial functions have to be orthonormal and the first
rPOD modes must capture the highest possible system‘s
energy, where ris an arbitrary integer. Here, the POD was
performed via the principal component analysis (PCA) from
Scikit-learn (Pedregosa et al. 2011), which uses the singular
value decomposition (SVD) of the data to represent them in
a lower-dimensional space (a.k.a. latent space). It should be
highlighted that, before applying PCA to the data, they were
normalized in the interval [1;1]and then centered.
The POD basis Vr, onto which the CO 2concentration
training dataset is projected, is comprised by the rdominant
POD modes. In other words, if C2Rntnxis the dataset
matrix that contains the spatiotemporal concentration field,
then the latent field variables bq(t), along the training inter-
val, are obtained by multiplying CbyVr. Here,ntis the
number of timesteps and nxis the number of grid points.
The OpInf technique, to be described next, is then applied
to the latent field variables. Note that, before applying PCA
to the data, they were normalized from -1 to 1 and then cen-
tered.
Non-Intrusive Operator Inference (OpInf)
The ROM for the atmospheric CO 2dispersion around the
Earth was constructed through the OpInf approach (Pe-
herstorfer and Willcox 2016). Since this physical phe-
nomenon is mainly governed by advection and diffusion
processes that move the carbon dioxide gas from one place
to another, higher-order nonlinearities (cubic and above)
were neglected and no forcing term was considered. Then,
the general form of the OpInf-based ROM is written as
d
dtbq(t) =bc+bAbq(t) +cH(bq(t)
bq(t)); (2)
wheret2[t0;tf],t0andtfare the initial and final time
instants, the initial state of the system bq(t0)is known, bc2
Rr,bA2RrrandcH2Rrr(r+1)=2are the operators to be
inferred and the symbol 
refers to the Kronecker product.
With the normalized latent field variables along the train-
ing interval and their numerically computed time deriva-
tives, it was possible to find the operators bc,bAandcHby
solving a data-driven least-squares regression problem with
regularization to avoid overfitting. Then, with optimal regu-
larizers and operators and given the state of the reduced sys-
tem att=t0, it was possible to integrate Eq. (2) and obtain
all the latent field variables from t0totf. Finally, these vari-
ables were projected back onto the original space, i.e., they
were multiplied by VT
r. To assess the predictive capabili-
ties of the constructed ROM, normalized root-mean-square
errors,ROM , were calculated across the spatiotemporal do-
main byROM =1

csPnx
j=1Pnt
i=1(cROM
i;jci;j)2
nxnt; (3)
where 
c=max(ci;j)min(ci;j),ci;jare the elements
of the original matrix C, whilecROM
i;j corresponds to the
concentration field computed by the ROM.
OCO-2 Carbon Dioxide Assimilated Dataset
This dataset from NASA provides a high-quality estima-
tion for the atmospheric CO 2concentration around the Earth
(Weir and Ott 2022). It combines space-based measurements
with state-of-the-art data assimilation techniques to han-
dle instruments’ inability to see through clouds and thick
aerosols. The data cover the period going from 2015.01.01
to 2021.10.30 in a daily basis, totalizing 2;495 timesteps
(or snapshots). The spatial resolution is of 0:5along the
Earth’s latitudinal axis and of 0:625along the longitudinal
one. Then, there is a total of 361576 = 207;936grid
points containing the CO 2data.
Results and Discussion
From 2;495snapshots, 2;000were used to train the PIML
algorithm. To build a data-driven reduced-order model for
the atmospheric CO 2dispersion around the Earth, the train-
ing data was initially reduced from 207;936dimensions to
their five most relevant ones, keeping 98.9% of the accumu-
lated modal energy. This great value is justified by the fact
that the average CO 2concentration highly dominates over
its seasonal and spatial variations. Besides, the most domi-
nant mode alone portrays 87.5% of the system‘s modal en-
ergy.
After the dimensionality reduction, a least-squares regres-
sion with regularization is applied, as discussed previously.
Figure 1 shows that the field variables in latent space ob-
tained from the inferred operators were well approximated
by the OpInf-based ROM along the training interval. Both
the first and the fourth reduced variables are exhibited in
this figure. Note that the ROM neglected the approximation
of the noisy pattern for the fourth variable, a consequence
of the POD reduction. Also, the model has difficulty in ac-
curately predicting peaks and valleys of the high amplitude
oscillations, as observed for the same variable. Such smooth-
ing behavior is commonly seen in ML models.
The original CO 2concentration field was reconstructed
from the latent variables and then compared against the orig-
inal observation. Figure 2 shows this comparison for a 338-
days-ahead forecast. Visually, it may be seen that the ROM
captures the main features of the dispersion. The normal-
ized root-mean-square error for the testing interval, com-
puted through Eq. (2), was about 4%. This small value is
an indicative that the model is quite robust.
Conclusion
The capabilities of the data-driven reduced-order model
based on a non-intrusive operator inference approach for

Figure 1: Approximated and exact reduced variables along
the training interval. On the top, the most relevant reduced
variable, which carries 87.5% of the system‘s modal energy.
On the bottom, the fourth one
Figure 2: CO 2concentration field around the Earth accord-
ing to the high-fidelity data (top) and to the reduced-order
model (bottom) for the 338thtesting snapshotthe atmospheric CO 2dispersion around the Earth was as-
sessed. It presented excellent predictive capabilities for this
physical phenomenon in addition to being quickly deployed,
with normalized root-mean-square errors below 5% for the
testing interval. This physics-informed machine learning
method seems to be adequate for large-scale climate sys-
tems mainly governed by advection and diffusion processes.
In practical terms, the OpInf-based ROM is well suited for
uncertainty quantification of climate-related predictions.
Acknowledgments
The authors would like to thank IBM and the Brazilian agen-
cies CAPES and CNPq for the financial support to this work.
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