Towards Scalable Koopman Operator Learning: Convergence Rates and A Distributed Learning Algorithm

TL;DR

This paper introduces a distributed algorithm for Koopman operator learning in nonlinear systems, with proven convergence rates, enabling scalable analysis of high-dimensional dynamics.

Contribution

It presents the first distributed Koopman operator learning algorithm with convergence guarantees similar to centralized methods.

Findings

Convergence rate of O(1/T) for constant learning rates.

Convergence rate of O(1/log T) for diminishing learning rates.

Numerical experiments validate theoretical convergence results.

Abstract

We propose an alternating optimization algorithm to the nonconvex Koopman operator learning problem for nonlinear dynamic systems. We show that the proposed algorithm will converge to a critical point with rate $O(1/T)$ and $O(\frac{1}{\log T})$ for the constant and diminishing learning rates, respectively, under some mild conditions. To cope with the high dimensional nonlinear dynamical systems, we present the first-ever distributed Koopman operator learning algorithm. We show that the distributed Koopman operator learning has the same convergence properties as the centralized Koopman operator learning, in the absence of optimal tracker, so long as the basis functions satisfy a set of state-based decomposition conditions. Numerical experiments are provided to complement our theoretical results.