Error bounds for kernel-based approximations of the Koopman operator

By Friedrich M. Philipp, Manuel Schaller, Karl Worthmann, Sebastian Peitz, Feliks Nuske. Applied and Computational Harmonic Analysis, 2023

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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.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/dac261b84f52d77f78e42b6ab26fd2f7e11e1a2e