Towards Scalable Koopman Operator Learning: Convergence Rates and A Distributed Learning Algorithm
This work addresses scalability issues in Koopman operator learning for high-dimensional nonlinear systems, offering incremental improvements with theoretical guarantees.
The authors tackled the problem of learning Koopman operators for nonlinear dynamic systems by proposing an alternating optimization algorithm with proven convergence rates of O(1/T) and O(1/log T), and introduced the first distributed learning algorithm that maintains these convergence properties under specific conditions.
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.