# Error bounds for kernel-based approximations of the Koopman operator

F. Philipp, M. Schaller, K. Worthmann, S. Peitz, F. Nüske, ArXiv:2301.08637 (2023).

Preprint | English
Author
Philipp, Friedrich; Schaller, Manuel; Worthmann, Karl; Peitz, SebastianLibreCat ; Nüske, Feliks
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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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Journal Title
arXiv:2301.08637
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### Cite this

Philipp F, Schaller M, Worthmann K, Peitz S, Nüske F. Error bounds for kernel-based approximations of the Koopman operator. arXiv:230108637. Published online 2023.
Philipp, F., Schaller, M., Worthmann, K., Peitz, S., & Nüske, F. (2023). Error bounds for kernel-based approximations of the Koopman operator. In arXiv:2301.08637.
@article{Philipp_Schaller_Worthmann_Peitz_Nüske_2023, title={Error bounds for kernel-based approximations of the Koopman operator}, journal={arXiv:2301.08637}, author={Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl and Peitz, Sebastian and Nüske, Feliks}, year={2023} }
Philipp, Friedrich, Manuel Schaller, Karl Worthmann, Sebastian Peitz, and Feliks Nüske. “Error Bounds for Kernel-Based Approximations of the Koopman Operator.” ArXiv:2301.08637, 2023.
F. Philipp, M. Schaller, K. Worthmann, S. Peitz, and F. Nüske, “Error bounds for kernel-based approximations of the Koopman operator,” arXiv:2301.08637. 2023.
Philipp, Friedrich, et al. “Error Bounds for Kernel-Based Approximations of the Koopman Operator.” ArXiv:2301.08637, 2023.
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arXiv 2301.08637