Singular Dynamic Mode Decompositions

TL;DR

This paper introduces modified dynamic mode decomposition techniques that improve convergence and address limitations in Koopman analysis by using Liouville operators and compactness properties, enabling more reliable data-driven spectral analysis.

Contribution

It proposes two new DMD algorithms based on Liouville operators that ensure convergence and better theoretical understanding, overcoming longstanding issues with Koopman-based methods.

Findings

Operators are shown to be compact for certain Hilbert space pairs.

A reconstruction algorithm is demonstrated without relying on eigenfunctions.

Two DMD algorithms are developed that converge using occupation kernels.

Abstract

This manuscript is aimed at addressing several long standing limitations of dynamic mode decompositions in the application of Koopman analysis. Principle among these limitations are the convergence of associated Dynamic Mode Decomposition algorithms and the existence of Koopman modes. To address these limitations, two major modifications are made, where Koopman operators are removed from the analysis in light of Liouville operators (known as Koopman generators in special cases), and these operators are shown to be compact for certain pairs of Hilbert spaces selected separately as the domain and range of the operator. While eigenfunctions are discarded in the general analysis, a viable reconstruction algorithm is still demonstrated, and the sacrifice of eigenfunctions realizes the theoretical goals of DMD analysis that have yet to be achieved in other contexts. However, in the case where the domain is embedded in the range, an eigenfunction approach is still achievable, where a more typical DMD routine is established, but that leverages a finite rank representation that converges in norm. The manuscript concludes with the description of two Dynamic Mode Decomposition algorithms that converges when a dense collection of occupation kernels, arising from the data, are leveraged in the analysis.