An Introductory Guide to Koopman Learning

TL;DR

This paper provides a comprehensive introduction to Koopman learning, focusing on rigorous, convergent data-driven methods for analyzing nonlinear dynamical systems through spectral analysis and forecasting.

Contribution

It offers a unified, rigorous framework for error control, convergence proofs, and reviews advanced techniques for spectral computation in Koopman analysis.

Findings

Provides an elementary proof of convergence for generalized Laplace analysis.

Reviews state-of-the-art methods for computing continuous spectra.

Offers a structured overview suitable for both newcomers and experts.

Abstract

Koopman operators provide a linear framework for data-driven analyses of nonlinear dynamical systems, but their infinite-dimensional nature presents major computational challenges. In this article, we offer an introductory guide to Koopman learning, emphasizing rigorously convergent data-driven methods for forecasting and spectral analysis. We provide a unified account of error control via residuals in both finite- and infinite-dimensional settings, an elementary proof of convergence for generalized Laplace analysis -- a variant of filtered power iteration that works for operators with continuous spectra and no spectral gaps -- and review state-of-the-art approaches for computing continuous spectra and spectral measures. The goal is to provide both newcomers and experts with a clear, structured overview of reliable data-driven techniques for Koopman spectral analysis.