Variance representations and convergence rates for data-driven approximations of Koopman operators

TL;DR

This paper establishes new error bounds and convergence rates for EDMD, a method to approximate Koopman operators, applicable to stochastic and deterministic systems with different sampling strategies, validated through numerical experiments.

Contribution

It introduces novel variance representations and derives the first superlinear convergence rates for ergodic sampling in Koopman operator approximation.

Findings

Exponential convergence for i.i.d. sampling

Superlinear convergence for ergodic sampling

Validated error bounds through numerical simulations

Abstract

We rigorously derive novel error bounds for extended dynamic mode decomposition (EDMD) to approximate the Koopman operator for discrete- and continuous time (stochastic) systems; both for i.i.d. and ergodic sampling under non-restrictive assumptions. We show exponential convergence rates for i.i.d. sampling and provide the first superlinear convergence rates for ergodic sampling of deterministic systems. The proofs are based on novel exact variance representations for the empirical estimators of mass and stiffness matrix. Moreover, we verify the accuracy of the derived error bounds and convergence rates by means of numerical simulations for highly-complex dynamical systems including a nonlinear partial differential equation.