Numerical methods to evaluate Koopman matrix from system equations

TL;DR

This paper introduces a novel method for evaluating the Koopman matrix directly from system equations, especially for stochastic differential equations, bypassing the need for data sets and improving convergence over traditional EDMD methods.

Contribution

A new approach to compute the Koopman matrix from system equations without data sets, using combinatorics, resolvent approximation, and extrapolations, applicable to stochastic systems.

Findings

The proposed method performs well on noisy van der Pol systems.

It converges faster than the EDMD in certain cases.

The method provides reasonable results even with noisy data.

Abstract

The Koopman operator is beneficial for analyzing nonlinear and stochastic dynamics; it is linear but infinite-dimensional, and it governs the evolution of observables. The extended dynamic mode decomposition (EDMD) is one of the famous methods in the Koopman operator approach. The EDMD employs a data set of snapshot pairs and a specific dictionary to evaluate an approximation for the Koopman operator, i.e., the Koopman matrix. In this study, we focus on stochastic differential equations, and a method to obtain the Koopman matrix is proposed. The proposed method does not need any data set, which employs the original system equations to evaluate some of the targeted elements of the Koopman matrix. The proposed method comprises combinatorics, an approximation of the resolvent, and extrapolations. Comparisons with the EDMD are performed for a noisy van der Pol system. The proposed method yields reasonable results even in cases wherein the EDMD exhibits a slow convergence behavior.