The Kernel Perspective on Dynamic Mode Decomposition
Abstract: This manuscript takes a critical look at the interactions between Koopman theory and reproducing kernel Hilbert spaces with an eye towards giving a tighter theoretical foundation for
Koopman based dynamic mode decomposition (DMD), a data driven method for modeling
a nonlinear dynamical system from snapshots. In particular, this paper explores the various
necessary conditions imposed on the dynamics when a Koopman operator is bounded or
compact over a reproducing kernel Hilbert space.
Ultimately, it is determined that for many RKHSs, the imposition of compactness or boundedness on a Koopman operator forces the dynamics to be affine.
However, a numerical method is still recovered in more general cases through the consideration of the Koopman operator as a closed and densely defined operator, which requires
a closer examination of the connection between the Koopman operator and a RKHS. By
abandoning the feature representation of RKHSs, the tools of function theory are brought to
bear, and a simpler algorithm is obtained for DMD than what was introduced in Williams
et al (2016). This algorithm is also generalized to utilize vector valued RKHSs.
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 links to our code, links to videos regarding the work that we have discussed, author names, and funding sources.
Video: https://youtube.com/playlist?list=PLldiDnQu2phuB64ccOYWxeSBZxq0Gg47g
Code: https://github.com/scc-lab/publications-code/tree/master/2024-TMLR-KernelPerspective
Assigned Action Editor: ~William_T_Redman1
Submission Number: 2417
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