Finite-data error bounds for Koopman-based prediction and control

TL;DR

This paper establishes probabilistic finite-data error bounds for Koopman-based methods in predicting and controlling dynamical systems, including stochastic and nonlinear cases, with demonstrated advantages over existing techniques.

Contribution

It provides the first finite-data error analysis for Koopman methods in stochastic and control settings, including new bounds and a bilinear surrogate approach that avoids the curse of dimensionality.

Findings

Derived probabilistic error bounds depending on data size.

Extended analysis to stochastic nonlinear control systems.

Demonstrated the superiority of the bilinear approach over state-of-the-art methods.

Abstract

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein-Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.