An Introductory Guide to Koopman Learning

• URL: http://arxiv.org/abs/2510.22002v1
• Date: Fri, 24 Oct 2025 20:09:22 GMT
• Title: An Introductory Guide to Koopman Learning
• Authors: Matthew J. Colbrook, Zlatko Drmač, Andrew Horning,
• Abstract summary: Koopman operators provide a linear framework for data-driven analyses of nonlinear dynamical systems.<n>This article emphasizes rigorously convergent data-driven methods for forecasting and spectral analysis.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• KoopGen: Koopman Generator Networks for Representing and Predicting Dynamical Systems with Continuous Spectra [65.11254608352982]
  We introduce a generator-based neural Koopman framework that models dynamics through a structured, state-dependent representation of Koopman generators.<n>By exploiting the intrinsic Cartesian decomposition into skew-adjoint and self-adjoint components, KoopGen separates conservative transport from irreversible dissipation.
  arXiv  Detail & Related papers  (2026-02-15T06:32:23Z)
• Toeplitz Based Spectral Methods for Data-driven Dynamical Systems [27.882467972636316]
  We introduce a Toeplitz-based framework for data-driven spectral estimation of linear evolution operators in dynamical systems.<n>Our method applies Toeplitz filters to the infinitesimal generator to extract eigenvalues, eigenfunctions, and spectral measures.
  arXiv  Detail & Related papers  (2026-02-10T13:53:21Z)
• Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control [1.2604738912025477]
  We introduce four learnable Koopman variants-scalar-gated, per-mode, gated, linear-shaped spectral mapping, and low-rank Koopman operators.<n>Our formulation enables explicit control over the spectrum, stability, and rank of the linear transition operator.<n>Our results demonstrate that learnable Koopman operators are effective, stable, and theoretically principled components for deep forecasting.
  arXiv  Detail & Related papers  (2026-02-01T09:13:49Z)
• Avoiding spectral pollution for transfer operators using residuals [0.6116681488656472]
  We present algorithms for computing spectral properties of transfer operators without spectral pollution.<n>Case studies range from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding.<n>Our methods offer robust tools for spectral estimation across a broad range of applications.
  arXiv  Detail & Related papers  (2025-07-22T18:01:05Z)
• Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems [51.54065545849027]
  The Koopman operator provides a principled framework for analyzing nonlinear dynamical systems.<n>VAMPnet and DPNet have been proposed to learn the leading singular subspaces of the Koopman operator.<n>We propose a scalable and conceptually simple method for learning the top-$k$ singular functions of the Koopman operator.
  arXiv  Detail & Related papers  (2025-07-09T18:55:48Z)
• ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals [1.8570740863168362]
  Methods for approximating spectral components of high-dimensional dynamical systems often face theoretical limitations.<n>We introduce ResKoopNet, which explicitly minimizes the emphspectral residual to compute Koopman eigenpairs.<n>Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods.
  arXiv  Detail & Related papers  (2025-01-01T02:19:42Z)
• Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space [54.13671100638092]
  Holistic Physics Mixer (HPM) is a framework for integrating spectral and physical information in a unified space.<n>We show that HPM consistently outperforms state-of-the-art methods in both accuracy and computational efficiency.
  arXiv  Detail & Related papers  (2024-10-15T08:19:39Z)
• Spectral Decomposition Representation for Reinforcement Learning [100.0424588013549]
  We propose an alternative spectral method, Spectral Decomposition Representation (SPEDER), that extracts a state-action abstraction from the dynamics without inducing spurious dependence on the data collection policy. A theoretical analysis establishes the sample efficiency of the proposed algorithm in both the online and offline settings. An experimental investigation demonstrates superior performance over current state-of-the-art algorithms across several benchmarks.
  arXiv  Detail & Related papers  (2022-08-19T19:01:30Z)
• Reinforcement Learning from Partial Observation: Linear Function Approximation with Provable Sample Efficiency [111.83670279016599]
  We study reinforcement learning for partially observed decision processes (POMDPs) with infinite observation and state spaces. We make the first attempt at partial observability and function approximation for a class of POMDPs with a linear structure.
  arXiv  Detail & Related papers  (2022-04-20T21:15:38Z)
• Unsupervised Machine Learning for Exploratory Data Analysis of Exoplanet Transmission Spectra [68.8204255655161]
  We focus on unsupervised techniques for analyzing spectral data from transiting exoplanets. We show that there is a high degree of correlation in the spectral data, which calls for appropriate low-dimensional representations. We uncover interesting structures in the principal component basis, namely, well-defined branches corresponding to different chemical regimes.
  arXiv  Detail & Related papers  (2022-01-07T22:26:33Z)
• Neural Dynamic Mode Decomposition for End-to-End Modeling of Nonlinear Dynamics [49.41640137945938]
  We propose a neural dynamic mode decomposition for estimating a lift function based on neural networks. With our proposed method, the forecast error is backpropagated through the neural networks and the spectral decomposition. Our experiments demonstrate the effectiveness of our proposed method in terms of eigenvalue estimation and forecast performance.
  arXiv  Detail & Related papers  (2020-12-11T08:34:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.