Estimating Koopman operators for nonlinear dynamical systems: a nonparametric approach

TL;DR

This paper introduces a nonparametric kernel-based method to estimate Koopman operators for nonlinear dynamical systems, leveraging RKHS to achieve finite-dimensional approximations and improve upon existing techniques.

Contribution

It establishes a novel link between kernel methods and Koopman operators, enabling their estimation in a data-driven, finite-dimensional setting.

Findings

Kernel methods effectively estimate Koopman operators.

The approach outperforms standard procedures in simulations.

Finite-dimensional RKHS captures nonlinear dynamics accurately.

Abstract

The Koopman operator is a mathematical tool that allows for a linear description of non-linear systems, but working in infinite dimensional spaces. Dynamic Mode Decomposition and Extended Dynamic Mode Decomposition are amongst the most popular finite dimensional approximation. In this paper we capture their core essence as a dual version of the same framework, incorporating them into the Kernel framework. To do so, we leverage the RKHS as a suitable space for learning the Koopman dynamics, thanks to its intrinsic finite-dimensional nature, shaped by data. We finally establish a strong link between kernel methods and Koopman operators, leading to the estimation of the latter through Kernel functions. We provide also simulations for comparison with standard procedures.