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
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 ν values.
Figure 2: Horizontal velocity field snapshots for the flow past cylinder system at the final prediction time step for two different test values of ν . Each row shows the ground truth, piDMD reconstruction, and pointwise absolute error.
Figure 3: Electron beam snapshot at 20.8ns with B0=25mT and rsp=2×104 , with superparticle injection rate of 12 per time step. Each superparticle is colored based on its instantaneous velocity ∣v∣ .
Figure 4: Time-averaged residual errors for exact DMD, piDMD , stacked parametric DMD and rKOI prediction for oscillating electron beam over the test B0 values.
Figure 5: Electric field snapshots for the oscillating electron beam system at the final prediction time step for three different test values of B0 . Each row shows the ground truth, piDMD reconstruction, and pointwise absolute error.
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 θ=[B0Ne]⊤ , colored by the time-averaged residual error obtained over a 1000 time step prediction horizon. Training θ samples are marked in black with a white × symbol.
Figure 7: Virtual cathode snapshot at 33.6ns with rsp=1.5×106 and superparticle injection rate of 100 per time step. Each superparticle is colored based on its instantaneous velocity ∣v∣ .
Figure 8: Time-averaged residual errors for exact DMD, piDMD and stacked parametric DMD for virtual cathode oscillations over the test Ne values.
Figure 9: Electric field snapshots for virtual cathode oscillations at the final prediction time step for two different test values of Ne . Each row shows the ground truth, piDMD reconstruction, and pointwise absolute error.
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.
Siyuan Chen, Zhecheng Wang, Yixin Chen +4
Department of Computer Science, University of British Columbia, Vancouver, Canada · Department of Computer Science, University of Toronto, Toronto, Canada
Dynamic mode decomposition (DMD) is a data-driven method that computes the best linear approximation of the underlying dynamical system and decomposes the dynamics into a superposition of characteristic spatiotemporal patterns. Originally introduced by the fluid dynamics community, DMD and its extensions have found widespread use in many other research areas such as molecular dynamics, climate science, engineering, finance, and neuroscience. Applications include dimensionality reduction, forecasting, system identification, control, and spectral clustering. In order to apply DMD to partial differential equations, the spatial domain is typically first discretized using finite difference or finite element techniques, thus implicitly rendering the problem finite-dimensional. We extend projected and exact DMD to infinite-dimensional systems. Rather than estimating matrices from vector-valued observations, our DMD variants learn finite-rank operators from functional data such as observables, densities, or wavefunctions. We show that conventional DMD algorithms can be regarded as special cases of their functional DMD counterparts. All results will be illustrated with the aid of guiding examples. We focus in particular on Koopman, Perron-Frobenius, and Koopman-von Neumann operators associated with graphons, ordinary differential equations, and stochastic differential equations.
Stefan Klus, Eirini Ioannou
School of Mathematical & Computer Sciences, Heriot–Watt University, Edinburgh, UK · Maxwell Institute for Mathematical Sciences, University of Edinburgh and Heriot–Watt University, Edinburgh, UK
Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this problem, we evaluate Koopman-based upsampling techniques implemented with dynamic mode decomposition (DMD), extended DMD (EDMD), and optimized DMD. These methods learn finite-dimensional approximations of Koopman evolution on selected observables and are used to interpolate and denoise snapshots inside the observed time window before derivative estimation and sparse regression. The empirical benchmark comprises two ODE systems, Lorenz-63 and Van der Pol, and three periodic PDE systems, Burgers, Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP), and linear advection-diffusion, over sparse and noisy sampling regimes. Polynomial EDMD gives the strongest ODE results, especially in coefficient accuracy. The PDE results are system-dependent: low-rank DMD-assisted reconstructions improve Burgers and advection-diffusion discovery, while the raw baseline (without upsampling) remains competitive for the Fisher-KPP data. A comparison against linear and smoothing-spline interpolation techniques shows that the selected Koopman-based preprocessors provide overall performance gains over these non-dynamical alternatives. We also demonstrate that DMD-assisted upsampling can stabilize Pareto-based non-oracle support-size selection. Overall, Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step that can reduce derivative-estimation error when its observable representation and low-rank structure are appropriate for the data.
Pongpisit Thanasutives, Yoshinobu Kawahara
Center for Advanced Intelligence Project (AIP), RIKEN, Osaka, Japan · 2Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan