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arXiv · 2301.08637

Error bounds for kernel-based approximations of the Koopman operator

Friedrich Philipp·Manuel Schaller·Karl Worthmann·Sebastian Peitz·Feliks Nüske
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Abstract

We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.

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Friedrich Philipp, Manuel Schaller, Karl Worthmann, Sebastian Peitz, Feliks Nüske. 2023-01-20. Error bounds for kernel-based approximations of the Koopman operator. https://arxiv.org/abs/2301.08637

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