Koopman for stochastic dynamics: error bounds for kernel extended dynamic mode decomposition

TL;DR

This paper establishes rigorous $L^ Infty$-error bounds for kernel extended dynamic mode decomposition (kEDMD) approximations of the Koopman operator in stochastic systems, combining kernel regression and Monte Carlo analysis.

Contribution

It provides the first $L^ Infty$-error bounds for kEDMD in stochastic dynamics, including deterministic and probabilistic error components.

Findings

Derived explicit error bounds for stochastic kEDMD approximations.

Analyzed the impact of data fill distance and sampling on approximation accuracy.

Validated bounds using Langevin-type stochastic differential equations with nonlinear potentials.

Abstract

We prove $L^\infty$-error bounds for kernel extended dynamic mode decomposition (kEDMD) approximants of the Koopman operator for stochastic dynamical systems. To this end, we establish Koopman invariance of suitably chosen reproducing kernel Hilbert spaces and provide an in-depth analysis of the pointwise error in terms of the data points. The latter is split into two parts by showing that kEDMD for stochastic systems involves a kernel regression step leading to a deterministic error in the fill distance as well as Monte Carlo sampling to approximate unknown expected values yielding a probabilistic error in terms of the number of samples. We illustrate the derived bounds by means of Langevin-type stochastic differential equations involving a nonlinear double-well potential.