Nonparametric Sparse Online Learning of the Koopman Operator
This work addresses a mis-specified scenario in Koopman operator learning for dynamical systems, offering an incremental improvement with online sparse methods.
The paper tackles the problem of learning the Koopman operator for nonlinear dynamical systems when the chosen function space may be mis-specified, by relating it to conditional mean embeddings and proposing an online sparse learning algorithm with complexity control, achieving effectiveness confirmed through numerical examples.
The Koopman operator provides a powerful framework for representing the dynamics of general nonlinear dynamical systems. Data-driven techniques to learn the Koopman operator typically assume that the chosen function space is closed under system dynamics. In this paper, we study the Koopman operator via its action on the reproducing kernel Hilbert space (RKHS), and explore the mis-specified scenario where the dynamics may escape the chosen function space. We relate the Koopman operator to the conditional mean embeddings (CME) operator and then present an operator stochastic approximation algorithm to learn the Koopman operator iteratively with control over the complexity of the representation. We provide both asymptotic and finite-time last-iterate guarantees of the online sparse learning algorithm with trajectory-based sampling with an analysis that is substantially more involved than that for finite-dimensional stochastic approximation. Numerical examples confirm the effectiveness of the proposed algorithm.