Geometric Considerations of a Good Dictionary for Koopman Analysis of Dynamical Systems: Cardinality, 'Primary Eigenfunction,' and Efficient Representation

TL;DR

This paper explores the geometric properties of eigenfunctions in Koopman analysis, introduces the concept of primary eigenfunctions, and proposes an optimal construction for efficient data-driven representation of dynamical systems.

Contribution

It introduces the notion of primary eigenfunctions and presents oKEEDMD, an optimal method for efficient Koopman eigenfunction approximation from data.

Findings

Primary eigenfunctions share level sets, aiding in understanding geometric multiplicity.

oKEEDMD provides a least squares optimal eigenfunction approximation.

Geometric insights improve the efficiency of Koopman-based reduced order models.

Abstract

Representation of a dynamical system in terms of simplifying modes is a central premise of reduced order modelling and a primary concern of the increasingly popular DMD (dynamic mode decomposition) empirical interpretation of Koopman operator analysis of complex systems. In the spirit of optimal approximation and reduced order modelling the goal of DMD methods and variants are to describe the dynamical evolution as a linear evolution in an appropriately transformed lower rank space, as best as possible. That Koopman eigenfunctions follow a linear PDE that is solvable by the method of characteristics yields several interesting relationships between geometric and algebraic properties. Corresponding to freedom to arbitrarily define functions on a data surface, for each eigenvalue, there are infinitely many eigenfunctions emanating along characteristics. We focus on contrasting cardinality and equivalence. In particular, we introduce an equivalence class, "primary eigenfunctions," consisting of those eigenfunctions with identical sets of level sets, that helps contrast algebraic multiplicity from other geometric aspects. Popularly, Koopman methods and notably dynamic mode decomposition (DMD) and variants, allow data-driven study of how measurable functions evolve along orbits. As far as we know, there has not been an in depth study regarding the underlying geometry as related to an efficient representation. We present a construction that leads to functions on the data surface whose corresponding eigenfunctions are efficient in a least squares sense. We call this construction optimal Koopman eigenfunction DMD, (oKEEDMD), and we highlight with examples.