Error bounds for kernel-based approximations of the Koopman operator

TL;DR

This paper provides theoretical error bounds for data-driven kernel-based approximations of the Koopman operator in stochastic systems, with practical insights demonstrated through numerical experiments.

Contribution

It derives new probabilistic bounds and variance expressions for kernel-based Koopman operator approximations in stochastic differential equations.

Findings

Derived an exact variance expression for the kernel cross-covariance operator.

Established probabilistic bounds for finite-data estimation errors.

Provided bounds on the prediction error of observables in RKHS.

Abstract

We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.