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Nonparametric Sparse Online Learning of the Koopman Operator

arXiv 2501.16489 ↗First posted 27 Jan 2025stat.ML cs.LG cs.SYcited by 0

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Abstract (arXiv)

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.

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References · 20

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8 more
  1. 2013Sparsity-promoting dynamic mode decompositioncited by 30
  2. 2017Online dynamic mode decomposition for time-varying systemscited by 22
  3. 2023Koopman Kernel Regressioncited by 22
  4. 2020Nonparametric approximation of conditional expectation operatorscited by 21
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  7. 2024Adversarial dynamical systems characterize when data-driven learning succeeds or failscited by 10
  8. 2024Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stabilitycited by 4

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