Parametric Interpolation of Dynamic Mode Decomposition for Predicting Nonlinear Systems

Ananda Chakrabarti; Haitham H. Saleh; Indranil Nayak; Balasubramaniam Shanker; Fernando L. Teixeira; Debdipta Goswami

eess.SYApr 13, 2026

Parametric Interpolation of Dynamic Mode Decomposition for Predicting Nonlinear Systems

Authors: Ananda Chakrabarti, Haitham H. Saleh, Indranil Nayak, Balasubramaniam Shanker, Fernando L. Teixeira, Debdipta Goswami

Organizations: Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210, USA · ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212, USA · SLAC National Accelerator Laboratory, Stanford University, Menlo Park, CA 94025, USA · Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA

Abstract

We present parameter-interpolated dynamic mode decomposition (piDMD), a parametric reduced-order modeling framework that embeds known parameter-affine structure directly into the DMD regression step. Unlike existing parametric DMD methods which interpolate modes, eigenvalues, or reduced operators and can be fragile with sparse training data or multi-dimensional parameter spaces, piDMD learns a single parameter-affine Koopman surrogate reduced order model (ROM) across multiple training parameter samples and predicts at unseen parameter values without retraining. We validate piDMD on fluid flow past a cylinder, electron beam oscillations in transverse magnetic fields, and virtual cathode oscillations -- the latter two being simulated using an electromagnetic particle-in-cell (EMPIC) method. Across all benchmarks, piDMD achieves accurate long-horizon predictions and improved robustness over state-of-the-art interpolation-based parametric DMD baselines, with less training samples and with multi-dimensional parameter spaces.

Figures & tables

Figure 1: Time-averaged residual error comparisons for exact DMD, piDMD , stacked parametric DMD and rKOI prediction for the flow past cylinder system over the test $\nu$ values.

Panel of Figure 2 — Figure 2: Horizontal velocity field snapshots for the flow past cylinder system at the final prediction time step for two different test values of $\nu$ . Each row shows the ground truth, piDMD reconstruction, and pointwise absolute error.

Figure 3: Electron beam snapshot at $20.8~\text{ns}$ with $B_{0}=25~\text{mT}$ and $r_{\mathrm{sp}}=2\times 10^{4}$ , with superparticle injection rate of 12 per time step. Each superparticle is colored based on its instantaneous velocity $|\mathbf{v}|$ .

Figure 4: Time-averaged residual errors for exact DMD, piDMD , stacked parametric DMD and rKOI prediction for oscillating electron beam over the test $B_{0}$ values.

Panel of Figure 5 — Figure 5: Electric field snapshots for the oscillating electron beam system at the final prediction time step for three different test values of $B_{0}$ . Each row shows the ground truth, piDMD reconstruction, and pointwise absolute error.

Panel of Figure 6 — Figure 6: Time-averaged residual error distribution for the oscillating electron beam system over the parameter space. Each square corresponds to one two-dimensional parameter $\bm{\theta}=[B_{0}\hskip 9.24994pt\mathcal{N}_{e}]^{\top}$ , colored by the time-averaged residual error obtained over a $1000$ time step prediction horizon. Training $\bm{\theta}$ samples are marked in black with a white $\times$ symbol.

Figure 7: Virtual cathode snapshot at $33.6~\text{ns}$ with $r_{\mathrm{sp}}=1.5\times 10^{6}$ and superparticle injection rate of 100 per time step. Each superparticle is colored based on its instantaneous velocity $|\mathbf{v}|$ .

Figure 8: Time-averaged residual errors for exact DMD, piDMD and stacked parametric DMD for virtual cathode oscillations over the test $\mathcal{N}_{e}$ values.

Panel of Figure 9 — Figure 9: Electric field snapshots for virtual cathode oscillations at the final prediction time step for two different test values of $\mathcal{N}_{e}$ . Each row shows the ground truth, piDMD reconstruction, and pointwise absolute error.

Reduced-Order Modeling

Reduced-order modeling builds compact approximations of complex systems to make simulation, prediction, and control more efficient.

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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.

Abstract

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