Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework

TL;DR

This paper introduces wavelet-based observables for Koopman analysis, deriving their properties and integrating them into an extended DMD framework for improved numerical approximation.

Contribution

It develops a novel wavelet-based observable framework for Koopman analysis and combines it with EDMD to create the cWDMD algorithm.

Findings

Wavelet observables are eigenfunctions of the Koopman semigroup.

Closed-form expressions for Koopman operator actions are derived.

The cWDMD algorithm effectively approximates Koopman semigroup dynamics.

Abstract

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.