The Kernel Perspective on Dynamic Mode Decomposition
Abstract: The purpose of the new DMD algorithm developed in this paper is to show that DMD methods very similar to KDMD emerge naturally out of a finite rank representation of the Koopman operator. It should be noted that the developed algorithm, while derived in a different way than traditional KDMD, involves computations that are nearly identical to KDMD, and as such, is not expected to offer any performance benefits over KDMD. Moreover, the algorithmic development of the present method does not invoke feature space representations and infinite matrices as in Williams et al., rather this method uses directly the properties of Koopman (or composition) operators and kernel functions. By doing so, this makes the theoretical dependencies of kernel based DMD methods transparent as densely defined operators over infinite dimensional kernel spaces. In order to present this new kernel perspective of Koopman analysis, the manuscript first introduces reproducing kernel Hilbert spaces (RKHSs) and examines the properties of Koopman operators over said spaces. Additionally, the examination of these properties led to the proof that the Koopman operator over the Gaussian RBF's native space is only bounded when it corresponds to discrete dynamics that are affine.
Submission Length: Long submission (more than 12 pages of main content)
Previous TMLR Submission Url: https://openreview.net/forum?id=4csshMM3HB
Changes Since Last Submission: We added an introduction that better captures the motivation behind the study. Additionally, we added a more formal metric for comparing the snapshots that were obtained from the new DMD algorithm to those from the original data set. We also added a second numerical experiment. A comparison to previous DMD methods was added to the paper. Pseudocode for the new DMD algorithm was added in order to add clarity to the paper. Furthermore, a more extensive explanation of our method has been added to the manuscript.
Assigned Action Editor: ~William_T_Redman1
Submission Number: 2417
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