Parametric Interpolation of Dynamic Mode Decomposition for Predicting Nonlinear Systems

2026-04-13

AI Summary

Existing parametric dynamic mode decomposition (DMD) methods suffer from unstable predictions and limited accuracy under sparse data or in high-dimensional parameter spaces. This work proposes parametric interpolation DMD (piDMD), which, for the first time, directly embeds parametric affine structure into the DMD regression process to construct a unified Koopman-based reduced-order model. The resulting framework enables efficient prediction of system dynamics at unseen parameter values without retraining. By circumventing post-hoc interpolation of modes, eigenvalues, or operators, piDMD significantly enhances robustness and generalization in multidimensional parameter spaces. Demonstrated on nonlinear systems—including flow past a cylinder, electron beam oscillations, and virtual cathode oscillations—the method achieves high-fidelity long-term predictions with only a few training samples, outperforming current parametric DMD approaches.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Data Mining & Knowledge Management: Recommender Systems

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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.

Problem

• parametric DMD
• nonlinear systems
• sparse training data
• multi-dimensional parameter spaces
• long-horizon prediction

Innovation

• parametric interpolation
• dynamic mode decomposition
• Koopman operator
• reduced-order modeling
• parameter-affine structure