Data-driven approximation of Koopman operators and generators: Convergence rates and error bounds
University of Edinburgh（爱丁堡大学）；Lancaster University（兰卡斯特大学）；University of Salamanca（萨拉曼卡大学）；University of Manchester（曼彻斯特大学）；Heriot–Watt University（赫瑞-瓦特大学）
Liam Llamazares-Elias, Samir Llamazares-Elias, Jonas Latz, Stefan Klus
发表信息：
Journal of Computational Dynamics, 15:163-206, 2026
Global information about dynamical systems can be extracted by analysing associated infinite-dimensional transfer operators, such as Perron--Frobenius and Koopman operators as well as their infinitesimal generators. In practice, these operators typically need to be approximated from data. Popular approximation methods are extended dynamic mode decomposition (EDMD) and generator extended mode decomposition (gEDMD). We propose a unified framework that leverages Monte Carlo sampling to approximate the operator of interest on a finite-dimensional space spanned by a set of basis functions. Our framework contains EDMD and gEDMD as special cases, but can also be used to approximate more general operators. Our key contributions are proofs of the convergence of the approximating operator under relaxed conditions. We also prove that in some cases eigenpairs of the approximating operators weakly converge to eigenpairs of the exact operator, in others they do not. Moreover, we derive explicit convergence rates and account for the presence of noise in the observations. Whilst all these results are broadly applicable, they also refine previous analyses of EDMD and gEDMD. We verify the analytical results with the aid of several numerical experiments.
Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation
Department of Computing and Mathematical Sciences, Caltech（加州理工学院计算与数学科学系）；The Alan Turing Institute（艾伦·图灵研究所）；Department of Mechanical Engineering, Clemson University（克莱姆森大学机械工程系）
Numerical methods to evaluate Koopman matrix from system equations
Koopman算子有利于分析非线性和随机动力学;它是线性的但无限维的，并且它控制着可观测量的演化。扩展动态模式分解（EDMD）是Koopman算子方法中著名的方法之一。EDMD采用快照对的数据集和特定字典来评估Koopman算子的近似，即，Koopman矩阵在本研究中，我们专注于随机微分方程，并提出了一种方法来获得Koopman矩阵。该方法不需要任何数据集，它使用原始系统方程来评估Koopman矩阵的一些目标元素。所提出的方法包括组合，近似的预解，和外推。与EDMD的比较进行了嘈杂的van der Pol系统。所提出的方法产生合理的结果，即使在EDMD表现出缓慢的收敛行为的情况下。
Department of Applied Mathematics & Theoretical Physics, University of Cambridge（应用数学与理论物理系，剑桥大学）；Department of Mathematics, University of Zagreb（数学系， Zagreb大学）；Department of Mathematical Sciences, Rensselaer Polytechnic Institute（数学科学系，Rensselaer理工学院）
Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems
University of Cambridge（剑桥大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Allen Institute（艾伦研究所）；University of Washington（华盛顿大学）；Cornell University（康奈尔大学）
Department of Mathematics, Imperial College London（帝国理工学院数学系）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）
