On Convergent Dynamic Mode Decomposition and its Equivalence with Occupation Kernel Regression

TL;DR

This paper introduces a norm-convergent dynamic mode decomposition method for deterministic systems, linking it to occupation kernel regression through finite-rank Liouville operator representations.

Contribution

It develops a new finite-rank approximation of the Liouville operator that ensures norm convergence and establishes its equivalence with occupation kernel regression.

Findings

Finite-rank Liouville operator approximation achieved.

Convergence of the DMD method demonstrated.

Connection between DMD and occupation kernel regression established.

Abstract

This paper presents a new technique for norm-convergent dynamic mode decomposition of deterministic systems. The developed method utilizes recent results on singular dynamic mode decomposition where it is shown that by appropriate selection of domain and range Hilbert spaces, the Liouville operator (also known as the Koopman generator) can be made to be compact. In this paper, it is shown that by selecting appropriate collections of finite basis functions in the domain and the range, a novel finite-rank representation of the Liouville operator may be obtained. It is also shown that the model resulting from dynamic mode decomposition of the finite-rank representation is closely related to regularized regression using the so-called occupation kernels as basis functions.