Path-Integral Formula for Computing Koopman Eigenfunctions

TL;DR

This paper introduces a novel path-integral method for computing Koopman eigenfunctions, including a neural network framework for high-dimensional systems, with applications in stability analysis and manifold computation.

Contribution

It presents a new path-integral formula for Koopman eigenfunctions and a neural network approach for high-dimensional systems, advancing stability and manifold analysis.

Findings

Successful computation of principal eigenfunctions in simulations

Application to stability and manifold analysis

Neural network framework for high-dimensional systems

Abstract

The paper is about the computation of the principal spectrum of the Koopman operator (i.e., eigenvalues and eigenfunctions). The principal eigenfunctions of the Koopman operator are the ones with the corresponding eigenvalues equal to the eigenvalues of the linearization of the nonlinear system at an equilibrium point. The main contribution of this paper is to provide a novel approach for computing the principal eigenfunctions using a path-integral formula. Furthermore, we provide conditions based on the stability property of the dynamical system and the eigenvalues of the linearization towards computing the principal eigenfunction using the path-integral formula. Further, we provide a Deep Neural Network framework that utilizes our proposed path-integral approach for eigenfunction computation in high-dimension systems. Finally, we present simulation results for the computation of principal eigenfunction and demonstrate their application for determining the stable and unstable manifolds and constructing the Lyapunov function.