Consistent spectral approximation of Koopman operators using resolvent compactification

TL;DR

This paper introduces a spectrally accurate, data-driven method for approximating Koopman operators in measure-preserving systems by compactifying the resolvent, enabling analysis of both periodic and mixing dynamics.

Contribution

It develops a novel spectral approximation technique using resolvent compactification and kernel operators, with proven convergence for large data samples.

Findings

Converges for large data in measure-preserving systems

Applicable to systems with pure point and continuous spectra

Demonstrated on tori and Lorenz 63 system

Abstract

Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on $L^2$ for measure-preserving, continuous-time systems via a ``compactification'' of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.