Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework
For researchers in dynamical systems and data-driven modeling, this provides a new theoretical framework linking wavelets to Koopman analysis, but the contribution is incremental as it builds on existing EDMD methods.
The paper introduces wavelet-based observables that are eigenfunctions of the Koopman semigroup and develops the cWDMD algorithm combining these observables with Extended Dynamic Mode Decomposition. Validation on two numerical examples shows the approach works, but no concrete performance numbers are provided.
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.