Factorized Neural Implicit DMD for Parametric Dynamics

TL;DR

This work proposes a physics-coded neural field parameterization of the Koopman operator's spectral decomposition that exposes underlying eigenvalues, modes, and stability of the underlying physical process to enable stable long-term rollouts, interpolation across parameter spaces, and spectral analysis.

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 (10)

• 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.