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.