教人用线性方法分析非线性系统，解决计算难题。
An Introductory Guide to Koopman Learning
- 基于残差控制误差，统一处理有限与无限维情况
- 证明了广义拉普拉斯分析的收敛性，适用于连续谱无间隙算子
- 提供最新谱分析与谱测度计算方法，适合动力系统研究者
Koopman 算子为数据驱动分析非线性动力系统提供了线性框架，但其无穷维特性带来重大计算挑战。本文提供一份关于 Koopman 学习的入门指南，重点强调严格收敛的数据驱动方法，用于预测与谱分析。文章统一阐述了在有限与无限维情形下通过残差实现误差控制的方法，给出广义拉普拉斯分析（一种适用于具有连续谱且无谱间隙算子的滤波幂迭代变体）收敛性的初等证明，并综述了当前最先进的连续谱与谱测度计算方法。目标是为新手和专家提供可靠数据驱动 Koopman 谱分析技术的清晰、结构化概述。
原文摘要 · Abstract (English)
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.
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