Factorized Neural Implicit DMD for Parametric Dynamics

Siyuan Chen; Zhecheng Wang; Yixin Chen; Yue Chang; Peter Yichen Chen; Eitan Grinspun; Jonathan Panuelos

cs.LGMar 11, 2026

Factorized Neural Implicit DMD for Parametric Dynamics

Authors: Siyuan Chen, Zhecheng Wang, Yixin Chen, Yue Chang, Peter Yichen Chen, Eitan Grinspun, Jonathan Panuelos

Organizations: Department of Computer Science, University of British Columbia, Vancouver, Canada · Department of Computer Science, University of Toronto, Toronto, Canada

Abstract

A data-driven, model-free approach to modeling the temporal evolution of physical systems mitigates the need for explicit knowledge of the governing equations. Even when physical priors such as partial differential equations are available, such systems often reside in high-dimensional state spaces and exhibit nonlinear dynamics, making traditional numerical solvers computationally expensive and ill-suited for real-time analysis and control. Consider the problem of learning a parametric flow of a dynamical system: with an initial field and a set of physical parameters, we aim to predict the system's evolution over time in a way that supports long-horizon rollouts, generalization to unseen parameters, and spectral analysis. We propose a physics-coded neural field parameterization of the Koopman operator's spectral decomposition. Unlike a physics-constrained neural field, which fits a single solution surface, and neural operators, which directly approximate the solution operator at fixed time horizons, our model learns a factorized flow operator that decouples spatial modes and temporal evolution. This structure exposes underlying eigenvalues, modes, and stability of the underlying physical process to enable stable long-term rollouts, interpolation across parameter spaces, and spectral analysis. We demonstrate the efficacy of our method on a range of dynamics problems, showcasing its ability to accurately predict complex spatiotemporal phenomena while providing insights into the system's dynamic behavior.

Figures & tables

Figure 1: Physics-Coded Neural DMD Pipeline. From left to right: (1) Input dataset at full resolution, (2) Nonlinear optimization using spatial coordinates and physics code, (3) Neural DMD output with continuous eigenvectors $\Phi$ and eigenvalues $\Lambda$ enabling space–time–physics reconstruction.

Figure 2: Burgers’ Equation. Spatio-temporal visualization of the solution $u(x,t)$ for the same test case (horizontal axis: $x$ , vertical axis: $t$ , $u(0,0)$ corresponds to the bottom-left corner of each subplot.

Figure 3: Kármán vortex street. Qualitative comparison of flow-field predictions under varying obstacle positions. KAE and P-DMD exhibit a noticeable double-shadow artifact (see zoom-ins), while our neural-field parameterization yields smoother transitions across physics conditions and remains close to the ground truth (GT). All heatmaps share the same color scale.

Figure 4: Physics code for airfoil parameterization and qualitative rollouts. Top: Illustration of the physics code that maps a 6-dimensional parameter vector to an airfoil geometry, including a radar-style view of the six CST parameters (upper/lower surface controls, trailing-edge thickness, and clockwise rotation). Bottom: Representative examples showing the generated airfoil shapes (left) and the corresponding flow-field predictions over time (right), comparing ground truth (GT) with Ours under different parameter settings in the test set. All heatmaps share the same color scale.

Figure 5: Comparison of basis functions (real part) for the full INR-DMD and ablations. For the deflation baseline, we show the basis learned at each stage; for the non-deflation baseline, we visualize the four lowest-frequency modes. The full pipeline learns clean, symmetric, frequency-separated bases with low inter-mode correlation, whereas removing orthogonality, long-horizon loss, or deflation yields noisier and highly correlated modes, and removing conjugate pairing breaks spectral symmetry.

Figure 6: Short-horizon vs. long-horizon loss weighting sweep. We vary the prediction loss weights from $(\alpha,\beta)=(0,1)$ to $(1,0)$ , where $\alpha$ and $\beta$ are short/long horizon loss weights respectively. The curve reports the resulting rollout error (lower is better) on the Kármán vortex street example.

Variant	# params	error $\downarrow$
Ours	316,510	$8.093\times 10^{-7}$
w/o Orthogonality	316,510	$1.156\times 10^{-6}$
w/o Long Horizon	316,510	$5.697\times 10^{-4}$
w/o Deflation	354,172	$1.819\times 10^{-4}$
w/o Conj. Pairing	316,510	$4.089\times 10^{-6}$

Table 1: Ablation study of key design choices on Kármán vortex street.

\downarrow

: lower is better.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Experiments	$\phi$ MLP	$\lambda$ MLP	$\alpha$	$\beta$	# Modes
Burgers’ Equation	$64\times 3$ (Sine)	$64\times 3$ (ELU)	$0.90$	$0.10$	$2$
Double Shear	$128\times 3$ (Sine)	$64\times 3$ (ELU)	$0.05$	$0.95$	$6$
Kármán Vortex Street	$128\times 3$ (Sine)	$64\times 3$ (ELU)	$0.10$	$0.90$	$5$
Airfoil	$128\times 3$ (Sine)	$64\times 3$ (ELU)	$0.05$	$0.95$	$5$

Appendix

Table 3: Network architectures and loss configurations across experiments.

Figure 7: Architecture ablation on network width and depth. We evaluate the impact of the number of neurons per layer (64, 128, 256) and network depth (3 vs. 5 layers) on the training loss. While increasing model capacity generally reduces the loss, the improvement saturates beyond 128 neurons. Deeper networks do not consistently outperform shallower ones under the same width. In our experiments, we choose the 3-layer configuration with 128 neurons as our default setting.

Figure 8: Burgers’ Equation across testing set viscosities. Spatiotemporal solutions $u(x,t)$ are shown as heatmaps. Each row corresponds to a different viscosity value from the testing set (increasing from top to bottom, as indicated on the left). Columns compare the ground truth (GT) with predictions from Ours , FNO, KAE, P-DMD, KNO, and RKN. All panels use the same color scale.

Figure 9: Double Shear Layer Over Time. Snapshots of the solution field at four time instants (left to right, increasing time as indicated by the arrow) with shear separation $=0.21$ . Rows show the ground truth (GT) and predictions from Ours , FNO, KAE, P-DMD, KNO, PDE-T, and RKN. All panels share the same color scale.

Figure 10: Kármán Vortex Street Over Time. Snapshots of the solution field at five time instants (left to right, increasing time as indicated by the arrow) with the cylinder placed at $x=0.435$ . Rows show the ground truth (GT) and predictions from Ours, FNO, KAE, and P-DMD. All panels share the same color scale.

Dynamical Systems

Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.

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Parameterized Temporal Extrapolation of PDEs via Disentangled Latent Dynamics Manifold Fusion

Zhangyong Liang, Jihong Zhu, Huanhuan Gao

National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, PR China · School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, 710072, China · School of Mechanical and Aerospace Engineering, Jilin University, Changchun, 130025, China

Latent Dynamics Modeling PDE Solving PDE Surrogate Modeling Physics-Informed ML

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Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable. The problem becomes even more severe when the model must also predict beyond the training time range. Existing methods usually cannot handle parameter generalization and temporal extrapolation at the same time. Standard parameterized models treat time as just another input and therefore fail to capture intrinsic dynamics, while recent continuous-time latent methods often rely on expensive test-time auto-decoding for each instance, which is inefficient and can disrupt continuity across the parameterized solution space. To address this, we propose Disentangled Latent Dynamics Manifold Fusion (DLDMF), a physics-informed framework that explicitly separates space, time, and parameters. Instead of unstable auto-decoding, DLDMF maps PDE parameters directly to a continuous latent embedding through a feed-forward network. This embedding initializes and conditions a latent state whose evolution is governed by a parameter-conditioned Neural ODE. We further introduce a dynamic manifold fusion mechanism that uses a shared decoder to combine spatial coordinates, parameter embeddings, and time-evolving latent states to reconstruct the corresponding spatiotemporal solution. By modeling prediction as latent dynamic evolution rather than static coordinate fitting, DLDMF reduces interference between parameter variation and temporal evolution while preserving a smooth and coherent solution manifold. As a result, it performs well on unseen parameter settings and in long-term temporal extrapolation. Experiments on several benchmark problems show that DLDMF consistently outperforms state-of-the-art baselines in accuracy, parameter generalization, and extrapolation robustness.

Abstract

Dynamical Systems

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