Error Bounds for Finite-Dimensional Stochastic Koopman Approximations

Abstract

Koopman operator theory provides a popular framework for lifting a nonlinear dynamical system to a linear system in a higher dimensional space using a dictionary of observable functions. When a finite-dimensional dictionary is selected, it is important to determine whether the corresponding observable subspace is invariant under the Koopman operator. If this invariance property does not hold, then an approximation error is introduced in the lifted linear representation. In the stochastic setting, process noise gives rise to an additional source of error. Although the stochastic Koopman operator accounts for process noise through conditional expectations, a residual error remains in the latent space. In this paper, we study these error sources and derive analytical and data-driven upper bounds for them. We first consider the case of exact invariance and then extend the analysis to approximately invariant observable subspaces. The resulting combined one-step bound explicitly captures both process noise and non-invariance effects. Simulation results illustrate that the proposed bounds upper-bound the observed one-step prediction error.