The kernel perspective on dynamic mode decomposition
Abstract
This manuscript revisits theoretical assumptions concerning dynamic mode decomposition (DMD) of Koopman operators, including the existence of lattices of eigenfunctions, common eigenfunctions between Koopman operators, and boundedness and compactness of Koopman operators. Counterexamples that illustrate restrictiveness of the assumptions are provided for each of the assumptions. In particular, this manuscript proves that the native reproducing kernel Hilbert space (RKHS) of the Gaussian RBF kernel function only supports bounded Koopman operators if the dynamics are affine. In addition, a new framework for DMD, that requires only densely defined Koopman operators over RKHSs is introduced, and its effectiveness is demonstrated through numerical examples.
Cite
@article{arxiv.2106.00106,
title = {The kernel perspective on dynamic mode decomposition},
author = {Efrain Gonzalez and Moad Abudia and Michael Jury and Rushikesh Kamalapurkar and Joel A. Rosenfeld},
journal= {arXiv preprint arXiv:2106.00106},
year = {2023}
}