The kernel perspective on dynamic mode decomposition

• URL: http://arxiv.org/abs/2106.00106v3
• Date: Mon, 17 Apr 2023 21:36:55 GMT
• Title: The kernel perspective on dynamic mode decomposition
• Authors: Efrain Gonzalez, Moad Abudia, Michael Jury, Rushikesh Kamalapurkar, Joel A. Rosenfeld
• Abstract summary: This manuscript revisits theoretical assumptions concerning dynamic mode decomposition (DMD) of Koopman operators. Counterexamples that illustrate restrictiveness of the assumptions are provided for each of the assumptions. New framework for DMD requires only densely defined Koopman operators over RKHSs.
• Score: 4.051099980410583
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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