A Kernel-Based Approach to Data-Driven Koopman Spectral Analysis

TL;DR

This paper introduces a kernel-based data-driven method for approximating Koopman spectral components in high-dimensional systems, offering improved accuracy and robustness over existing techniques like DMD.

Contribution

The paper presents a novel kernel-based approach that implicitly determines scalar observables for Koopman analysis, overcoming computational challenges of traditional basis function methods.

Findings

More accurate Koopman eigenvalues and modes compared to DMD.

Less sensitive to data distribution in high-dimensional systems.

Effective on the FitzHugh-Nagumo PDE example.

Abstract

A data driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with high dimensional state spaces is presented. This approach approximates the Koopman operator using a set of scalar observables, which are functions defined on state space, that is determined {\em implicitly} by the choice of a kernel. This circumvents the computational issues that arise due to the number of basis functions required to span a "sufficiently rich" subspace of the space of scalar observables in these problems. We illustrate this method on the FitzHugh-Nagumo PDE, a prototypical example of a one-dimensional reaction diffusion system, and compare our results with related methods such as Dynamic Mode Decomposition (DMD) that have the same computational cost as our approach. In this example, the resulting approximations of the leading Koopman eigenvalues, eigenfunctions, and modes are both more accurate and less sensitive to the distribution of the data used in the computation than those produced by DMD.