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

2026-05-13

AI Summary

This work addresses the efficient approximation of the Koopman operator for nonlinear dynamical systems by proposing a novel construction of observables based on the continuous wavelet transform. It establishes, for the first time, a rigorous proof that these wavelet-based observables serve as eigenfunctions of the Koopman semigroup in a specific Banach space, and derives closed-form expressions for both the operator’s action and its resolvent. Building upon this theoretical foundation, the authors integrate the approach with Extended Dynamic Mode Decomposition (EDMD) to formulate a new algorithmic framework, termed cWDMD. Numerical experiments demonstrate that the proposed method achieves high-accuracy approximations of the Koopman operator, significantly enhancing both the precision and computational efficiency of spectral analysis for nonlinear systems.

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.

Problem

• Koopman analysis
• wavelet-based observables
• Extended Dynamic Mode Decomposition
• Koopman semigroup
• dynamic systems

Innovation

• wavelet-based observables
• Koopman semigroup
• Extended Dynamic Mode Decomposition
• continuous wavelet transform
• cWDMD