Improved error bounds for Koopman operator and reconstructed trajectories approximations with kernel-based methods

TL;DR

This paper introduces a new error bound for Koopman operator approximation using kernel methods, along with a lifting technique for trajectory reconstruction, supported by theoretical analysis and numerical experiments.

Contribution

It presents a novel $O(N^{-1/2})$ error bound for Koopman operator approximation and introduces a lifting back operator for trajectory reconstruction.

Findings

Error bound of $O(N^{-1/2})$ for Koopman approximation

Successful nonlinear system approximation with exponential decay faster than $-1/2$

Numerical results confirm theoretical error bounds

Abstract

In this article, we propose a new error bound for Koopman operator approximation using Kernel Extended Dynamic Mode Decomposition. The new estimate is $O(N^{-1/2})$, with a constant related to the probability of success of the bound, given by Hoeffding's inequality, similar to other methodologies, such as Philipp et al. Furthermore, we propose a \textit{lifting back} operator to obtain trajectories generated by embedding the initial state and iterating a linear system in a higher dimension. This naturally yields an $O(N^{-1/2})$ error bound for mean trajectories. Finally, we show numerical results including an example of nonlinear system, exhibiting successful approximation with exponential decay faster than $-1/2$, as suggested by the theoretical results.