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

arXiv: https://arxiv.org/abs/2605.14224 · HTML: https://mlsanity.com/paper/2605.14224/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 55 references that are in the map.
- First posted on arXiv 14 May 2026.

## Builds on (most cited first, 10 of 10)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators](https://mlsanity.com/paper/2405.00782/) (2024)
- [Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems](https://mlsanity.com/paper/2009.11544/) (2020)
- [Quadrature-Based Vector Fitting: Implications For H2 System Approximation](https://mlsanity.com/paper/1403.4655/) (2014)
- [A wavelet-based dynamic mode decomposition for modeling mechanical systems from partial observations](https://mlsanity.com/paper/2110.12990/) (2021)

## Built on by (most cited first, 1 of 1)

- [Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks](https://mlsanity.com/paper/2605.27756/) (2026)

## Abstract (arXiv)

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.