Parametric Interpolation of Dynamic Mode Decomposition for Predicting Nonlinear Systems

TL;DR

The paper introduces piDMD, a parametric reduced-order modeling framework that embeds parameter-affine structures into DMD, enabling accurate predictions across unseen parameters with less data.

Contribution

piDMD is a novel parametric DMD method that learns a single Koopman surrogate model for multiple parameters, improving robustness and efficiency over existing methods.

Findings

piDMD achieves accurate long-term predictions on fluid and electromagnetic benchmarks.

piDMD outperforms existing interpolation-based parametric DMD methods.

piDMD requires fewer training samples and handles multi-dimensional parameter spaces effectively.

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.