🤖 AI Summary
The Koopman operator provides a data-driven linearization framework for nonlinear dynamical systems, but its infinite-dimensionality impedes spectral estimation convergence and undermines reliability in analyzing continuous spectra and systems lacking spectral gaps.
Method: We propose a unified residual error control framework, delivering the first elementary convergence proof for generalized Laplace analysis. We develop data-driven filtering power iteration, continuous spectrum identification, and spectral measure computation methods.
Contribution/Results: These advances significantly enhance resolution of continuous spectra and weakly decaying modes. The resulting methodology combines theoretical rigor with numerical stability, enabling verifiable long-term forecasting and spectral decomposition. We establish a structured, pedagogically accessible standard workflow for Koopman spectral analysis—applicable to both novices and experts—that advances nonlinear system modeling from empirical fitting toward interpretable, convergent quantitative analysis.
📝 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.