An Introductory Guide to Koopman Learning

TL;DR

The article surveys data-driven Koopman analysis as a rigorous framework for nonlinear dynamics, emphasizing convergent finite-dimensional approximations and error control via residuals. It develops a comprehensive theory around EDMD, DMD, Hankel-DMD, and ResDMD, linking finite sections to infinite-dimensional spectral properties and addressing spectral pollution, pseudospectra, and continuous spectra. A central contribution is providing an elementary convergence result for generalized Laplace analysis and outlining three robust families of spectral-measure methods (moments, eigenvalue-based, and resolvent-based) for unitary Koopman operators. The work integrates delay embeddings and Krylov subspaces to improve invariant-subspace discovery and offers practical examples in fluid dynamics and chaotic systems to illustrate reliable spectral reconstruction and mode extraction. Together, these results establish rigorous foundations and practical tools for forecasting and spectral analysis of complex, high-dimensional dynamical systems.

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.

Figures (15)

• Figure 1: Left: The singular values of $\mathbf{X}$. After truncation, $k=26$ singular values are used and $\mathbf{X} \approx U_k\Sigma_k V_k^*$. Middle: The eigenvalues computed by the DMD algorithm. Right: The DMD residuals.
• Figure 2: Left: DMD eigenvalues with corresponding residuals (see \ref{['DMD_eigs_residuals']}). Larger markers indicate higher accuracy, i.e., smaller residual (colorbar indicates $\log_{10}$ of residuals). Right: Comparison of the computed Ritz values (DMD eigenvalues) with explicitly computed eigenvalues of $e^{\Delta t\mathbf{\Omega}}$.
• Figure 3: Left: Reconstruction errors when all computed modes ($\ell=k$) are used in \ref{['eq:f_i-reconstruct-ell']}. Both methods for computing $\alpha_j$ or solving the structured least squares problem in \ref{['eq:rec-error-min']} perform well. Right: The moduli of the coefficients $\alpha_1,\ldots,\alpha_k$, computed by the two methods. The maximal relative difference between the two sets of values is $\mathcal{O}(10^{-8})$.
• Figure 4: Left: DMDSP reconstruction errors with $\ell=4$ in \ref{['eq:f_i-reconstruct-ell']}. Right: The eigenvalues $\lambda_{\varsigma_1},\ldots,\lambda_{\varsigma_{\ell}}$ selected by the sparsity-constrained optimizer in DMDSP.
• Figure 5: Left: Large $|\alpha_j|$ corresponds to small residual $r_k(j)$ (large $1/r_k(j)$). Right: The DMD eigenvalues $\lambda_{\varsigma_1}, \lambda_{\varsigma_2}, \lambda_{\varsigma_3}$ with residuals below $10^{-6}$.

Theorems & Definitions (9)

• Proposition 5.1
• proof : Proof of \ref{['new_prop']}
• Theorem 6.1
• proof : Proof of \ref{['thm:weak_conv_gq']}
• Theorem 6.2
• proof : Proof of \ref{['thm:weak_conv_fs']}
• Theorem 6.3: Convergence of filtered Fourier series
• Theorem 6.4: Convergence properties of mpEDMD
• Theorem 6.5: Convergence of smoothed measures