Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics

Alex Guo; Michael D. Graham

cs.LGJul 21, 2025

Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics

Authors: Alex Guo, Michael D. Graham

Organizations: Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison WI 53706, USA

Abstract

While data-driven techniques are powerful tools for reduced-order modeling of systems with chaotic dynamics, great potential remains for leveraging known physics (i.e. a full-order model (FOM)) to improve predictive capability. We develop a hybrid reduced order model (ROM), informed by both data and FOM, for evolving spatiotemporal chaotic dynamics on an invariant manifold whose coordinates are found using an autoencoder. This approach projects the vector field of the FOM onto the invariant manifold; then, this physics-derived vector field is either corrected using dynamic data, or used as a Bayesian prior that is updated with data. In both cases, the neural ordinary differential equation approach is used. We consider simulated data from the Kuramoto-Sivashinsky and complex Ginzburg-Landau equations. Relative to the data-only approach, for scenarios of abundant data, scarce data, and even an incorrect FOM (i.e. erroneous parameter values), the hybrid approach yields substantially improved time-series predictions.

Figures & tables

Figure 1: Data-driven manifold dynamics (DManD) is a data-driven ROM that reduces dimension of the state $u$ with an autoencoder (a) and evolves trajectories with a vector field on the latent space $h$ learned by a neural ODE (b). (c) Physics-informed DManD (PI-DManD) is a hybrid ROM that uses Manifold Galerkin to project the vector field on the full order model (FOM) $f_{\mathrm{FOM}}$ onto that of the latent space, learns a neural network approximation of the latent space vector field as $g_{mn}$ , and either uses a corrective neural ODE $g_{c}$ to fix numerical and FOM approximation errors (PI-DManD-c) or uses $g_{mn}$ as a Bayesian-motivated prior for training a DManD model (PI-DManD-B).

Network	Function	Layers	Activations
Encoder	$\mathcal{E}$	$d_{u}$ :4 $d_{u}$ :2 $d_{u}$ : $d_{h}$	ReLU:ReLU:Linear
Decoder	$\mathcal{D}$	$d_{h}$ :2 $d_{u}$ :4 $d_{u}$ : $d_{u}$	ReLU:ReLU:Linear
NODE (DManD)	$g_{d}$	$d_{h}$ :256:256: $d_{h}$	GeLU:GeLU:Linear
Neural network (MG-N)	$g_{mn}$	$d_{h}$ :256:256: $d_{h}$	GeLU:GeLU:Linear
NODE (PI-DManD-c)	$g_{c}$	$d_{h}$ :256:256: $d_{h}$	GeLU:GeLU:Linear
NODE (PI-DManD-B)	$g_{B}$	$d_{h}$ :256:256: $d_{h}$	GeLU:GeLU:Linear

Table 1: Neural network architectures for the autoencoder, DManD, MG-N, and the two PI-DManD methods.

Figure 2: KSE, $L=22$ (for both data and FOM), $N=400,000$ , $\tau=0.25$ . (a) Ensemble-averaged short-time tracking error for all methods. (b) Representative trajectories of ground truth and model predictions (contours at $u=\pm 1$ ). (c) Difference plots between ground truth and model predictions (contours at difference values of $\pm 1$ ). Representative trajectories and difference plots for other models look similar to those of PI-DManD-c. (d) NODE training convergence for purely data-driven and hybrid methods (note the log-log scale) .

Figure 3: KSE,

L=22

(for both data and FOM). Short-time tracking error at one Lyapunov time for scarce data in terms of (a) less data with

N\leq 400,000

and

\tau=0.25

(i.e. shorter time series), (b) widely spaced data with

\tau>0.25

and

N=400,000

, and (c) increasing

\tau

and decreasing

N

proportionally. Here and elsewhere, error bars show standard error, and all subplots follow the legend in (a). Note the different vertical axis limits for each subplot. Missing PI-DManD-c bars, denoted with asterisk, in (b) and (c) have errors of about 1 (i.e. much larger than the case shown) and are omitted to improve clarity in the vertical axis.

Figure 4: KSE, $N=400,000$ , $\tau=0.25$ . Effects of using an erroneous parameter ( $L=20$ ) in the FOM to model data generated with $L=22$ . (a) Ground truth trajectory (left) and example trajectory with erroneous parameter (right). (b) Ensemble-averaged short-time tracking error. (c) NODE training convergence for purely data-driven and hybrid methods.

	Exact physics, abundant data	Exact physics, scarce data	Erroneous physics, abundant data
% reduction in relative trajectory error at $t=\tau_{L}$	46%–51% (Figs. 2 (a), 8 (a))	None (shorter time series, finer sampling; Figs. 3 (a), 10 (a)) 22%–62% (longer time series, coarser sampling; Figs. 3 (b), 10 (b)) 6%–29% (fixed time series length, coarser sampling; Figs. 3 (c), 10 (c))	43%–64% (Figs. 4 (b), 11 (b), 11 (c))

Table 2: Summary of the improvement or reduction in

\epsilon(t=\tau_{L})

(the relative ensemble averaged short-time tracking error at one Lyapunov time) for the hybrid PI-DManD-B method over the solely data-driven DManD method; specifically, the quantities correspond to

(\epsilon_{\text{DManD}}-\epsilon_{\text{PI-DManD-B}})/\epsilon_{\text{DManD}}\times 100\%

. We varied the accuracy of the known physics (i.e. the full order model), the length of the time series (i.e. number of time units covered), and the sampling rate of the data.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

	DManD	MG-N	MG	PI-DManD-c	PI-DManD-B
Train time (hr)	2.2	8.3	n/a	13.1	12.0
Test time (s)	8.3	8.2	102	11.4	7.7

Appendix

Table 3: Comparing computation time for methods applied to KSE. First row is time (in hours) for training. Second row is computation time (in seconds) for testing, where an initial condition is integrated for 100 Lyapunov times. Calculations were run on an Intel Xeon E5-2640 CPU.

Figure 5: KSE, $L=22$ (for both data and FOM), $N=400,000$ , $\tau=0.25$ . (a) Comparison of joint pdfs. (b) Earth movers distance (EMD) quantifying the difference between pdfs of the ground truth and model predictions, where the dashed red line is the baseline EMD calculated with the pdf of another ground truth trajectory. Joint pdfs for models not shown look similar to joint pdfs of models with similar EMD values.

Figure 6: KSE, $N=400,000$ , $\tau=0.25$ . Effects of using an erroneous parameter ( $L=20$ ) in the FOM when making model predictions on the data generated with $L=22$ . (a) Comparison of joint pdfs. (b) Earth movers distance (EMD) quantifying the difference between pdfs of the ground truth and model predictions, where the dashed red line is the baseline EMD calculated with the pdf of another ground truth trajectory. Joint pdfs for models not shown look similar to joint pdfs of models with similar EMD values.

Figure 7: Singular value spectra of the covariance of the latent space

z

for IRMAE-WD applied to CGLE. Red dashed line located at 40th singular value index.

Figure 8: CGLE, $\beta=-2$ (for both data and FOM), $N=400,000$ , $\tau=0.05$ . (a) Ensemble-averaged short-time tracking error for all methods. (b) Absolute value of representative trajectories of ground truth and model predictions (contours at $|A|=0.4,0.8$ ). (c) Difference plots between ground truth and model predictions (contours at difference values of 0.4, 0.8). Representative trajectories and difference plots for other models look similar to those of PI-DManD-c. (d) NODE training convergence for purely data-driven and hybrid methods (note the log-log scale).

	DManD	MG-N	MG	PI-DManD-c	PI-DManD-B
Train time (hr)	2.2	9.3	n/a	13.0	12.3
Test time (s)	4.9	5.0	218.0	6.8	4.5

Appendix

Table 4: Comparing computation time for methods applied to CGLE. First row is time (in hours) for training. Second row is computation time (in seconds) for testing, where an initial condition is integrated for 100 Lyapunov times. Calculations were run on an Intel Xeon E5-2640 CPU.

Figure 9: CGLE, $\beta=-2$ (for both data and FOM), $N=400,000$ , $\tau=0.05$ . (a) Comparison of joint pdfs. (b) Earth movers distance (EMD) quantifying the difference between pdfs of the ground truth and model predictions, where the dashed red line is the baseline EMD calculated with the pdf of another ground truth trajectory. Joint pdfs for models not shown look similar to joint pdfs of models with similar EMD values.

Figure 10: CGLE,

\beta=-2

(for both data and FOM). Short-time tracking error at one Lyapunov time for scarce data in terms of (a) less data with

N\leq 400,000

and

\tau=0.05

(i.e. shorter time series), (b) widely spaced data with

\tau>0.05

and

N=400,000

, and (c) increasing

\tau

and decreasing

N

proportionally. Error bars are standard errors, and all subplots follow the legend in (a). Note the different vertical axis limits for each subplot. Missing PI-DManD-c bars, denoted with asterisk, in (b) and (c) have errors of about 1 and are omitted to improve clarity in vertical axis.

Figure 11: CGLE, $N=400,000$ , $\tau=0.05$ . Effects of using an erroneous parameter ( $\beta=-1.8$ or $\beta=-1$ ) in the FOM to model data generated with $\beta=-2$ . (a) Ground truth trajectory (top-left) and example trajectories for the two erroneous $\beta$ values (top-right and bottom); note the different colorbar axis limits. Ensemble-averaged short-time tracking error when using $\beta=-1.8$ (b) or $\beta=-1$ (c) in the FOM.

Figure 12: CGLE, N=400,000, $\tau=0.05$ . Effects of using an erroneous parameter ( $\beta=-1.8$ ) in the FOM to model the data generated with $\beta=-2$ . (a) Comparison of joint pdfs for $\beta=-1.8$ . (b) Earth movers distance (EMD) quantifying the difference between pdfs of the ground truth and model predictions for $\beta=-1.8$ . Dashed red line is the baseline EMD calculated with the pdf of another ground truth trajectory. Joint pdfs for models not shown look similar to joint pdfs of models with similar EMD values.

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