Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator

TL;DR

This paper introduces a kernel-based method to approximate principal eigenfunctions of the Koopman operator, enabling analysis of nonlinear systems' dynamics without explicit operator computation.

Contribution

The paper presents a novel kernel method for approximating Koopman eigenfunctions, avoiding spectral pollution and spurious eigenvalues, and decomposing eigenfunctions into linear and nonlinear parts.

Findings

Effective approximation of Koopman eigenfunctions demonstrated

Method avoids spectral pollution and spurious eigenvalues

Numerical examples validate the approach

Abstract

The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct computations. In this paper, we introduce a kernel-based method to construct the principal eigenfunctions of the Koopman operator without explicitly computing the operator itself. These principal eigenfunctions are associated with the equilibrium dynamics, and their eigenvalues match those of the linearization of the nonlinear system at the equilibrium point. We exploit the structure of the principal eigenfunctions by decomposing them into linear and nonlinear components. The linear part corresponds to the left eigenvector of the system's linearization at the equilibrium, while the nonlinear part is obtained by solving a partial differential equation (PDE) using kernel methods. Our approach avoids common issues such as spectral pollution and spurious eigenvalues, which can arise in previous methods. We demonstrate the effectiveness of our algorithm through numerical examples.