Data-driven approximation of Koopman operators and generators: Convergence rates and error bounds

TL;DR

This paper introduces a unified Monte Carlo-based framework for approximating transfer operators of dynamical systems, providing convergence proofs, explicit rates, and noise robustness, applicable to EDMD, gEDMD, and more.

Contribution

It offers a general approximation framework with proven convergence, explicit error bounds, and noise handling, extending and refining existing methods like EDMD and gEDMD.

Findings

Proves convergence of operator and spectrum approximations

Derives explicit convergence rates under broad conditions

Demonstrates robustness to observational noise

Abstract

Global information about dynamical systems can be extracted by analysing associated infinite-dimensional transfer operators, such as Perron-Frobenius and Koopman operators as well as their infinitesimal generators. In practice, these operators typically need to be approximated from data. Popular approximation methods are extended dynamic mode decomposition (EDMD) and generator extended mode decomposition (gEDMD). We propose a unified framework that leverages Monte Carlo sampling to approximate the operator of interest on a finite-dimensional space spanned by a set of basis functions. Our framework contains EDMD and gEDMD as special cases, but can also be used to approximate more general operators. Our key contributions are proofs of the convergence of the approximating operator and its spectrum under non-restrictive conditions. Moreover, we derive explicit convergence rates and account for the presence of noise in the observations. Whilst all these results are broadly applicable, they also refine previous analyses of EDMD and gEDMD. We verify the analytical results with the aid of several numerical experiments.