Koopman operator framework for spectral analysis and identification of infinite-dimensional systems

TL;DR

This paper develops a Koopman operator framework for analyzing and identifying infinite-dimensional nonlinear systems, providing a finite-dimensional approximation method for spectral analysis and PDE identification with proven convergence.

Contribution

It introduces a finite-dimensional projection of the Koopman semigroup for infinite-dimensional systems, generalizing EDMD and enabling linear PDE identification with theoretical guarantees.

Findings

Finite-dimensional approximation of Koopman semigroup for infinite systems

Method for spectral analysis from data in infinite-dimensional context

Linear approach for nonlinear PDE identification with convergence proofs

Abstract

We consider Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals. The properties of the Koopman semigroup are described and a finite-dimensional projection of the semigroup is proposed, which provides a linear finite-dimensional approximation of the underlying infinite-dimensional dynamics. This approximation is used to obtain spectral properties from data, a method which can be seen as a generalization of Extended Dynamic Mode Decomposition for infinite-dimensional systems. Finally, we exploit the proposed framework to identify (a finite-dimensional approximation of) the Lie generator associated with the Koopman semigroup. This approach yields a linear method for nonlinear PDE identification, which is complemented with theoretical convergence results.