The study of mathematical connections between operator-theoretic formulations of classical dynamics and quantum mechanics began at least as early as the 1930s in work of Koopman and von Neumann and was developed in later decades by many authors, often independently, into a framework now broadly known as Koopman-von Neumann representation of classical dynamics. This article surveys aspects of this framework for measure-preserving ergodic dynamical systems and connects it with recent approximation techniques for Koopman and transfer operators that are amenable to data-driven numerical implementation. In broad terms, these methods are based on representations of (i) classical observables as elements of an algebra of operators acting on a Hilbert space; and (ii) classical probability measures as elements of the state space of that algebra, with lifted versions of the Koopman and transfer operators inducing dynamical evolution of observables and states, respectively. A common theme underlying the techniques surveyed here is the use of reproducing kernel Hilbert spaces with coalgebra structure (so-called "reproducing kernel Hilbert algebras'') that aids the quantum representation of classical objects, as well as the use of Fock spaces to build approximation schemes with high expressivity and structure preservation properties (notably, preservation of positivity and multiplicativity of composition operators). Applications to quantum algorithms for approximating the Koopman evolution of observables in systems with pure point spectra are also discussed.
Generative Modeling through Koopman Spectral Analysis: An Operator-Theoretic Perspective
Center for Science Adventure；Collaborative Research Advancement, Graduate School of Science, Kyoto University, Kyoto, Japan；Massachusetts Institute of Technology, Cambridge, MA；LinkedIn Corporation, Sunnyvale, CA；Dynamic Systems Lab, University College London, London, UK；Graduate School of Information Science；Technology, The University of Osaka, Osaka, Japan；RIKEN Center for Advanced Intelligence Project, Tokyo, Japan
Numerical approximation of the Koopman-von Neumann equation: Operator learning and quantum computing
School of Mathematical & Computer Sciences, Heriot–Watt University（爱丁堡赫瑞-瓦特大学数学与计算机科学学院）；Max-Planck-Institute for Dynamics of Complex Technical Systems（马普复杂技术系统动力学研究所）；Zuse Institute Berlin（柏林祖斯研究所）
Data-driven Stabilization of Discrete-time Control-affine Nonlinear Systems: A Koopman Operator Approach
近年来，数据驱动的动力系统分析引起了人们的广泛关注，Perron-Frobenius算子和Koopman算子等传递算子技术也得到了广泛的应用。由于数据总是在离散时间获得的，在本文中，我们提出了一个纯粹的数据驱动的方法设计的稳定反馈控制律的一般类的离散时间控制仿射非线性系统。特别是，我们使用Koopman算子提升一个控制仿射系统到一个高维空间，其中控制系统的演化是双线性的。我们分析了提升的双线性系统的可控性，并将其与潜在的非线性系统的可控性联系起来。然后，我们利用控制李雅普诺夫函数（CLF）的概念，设计一个状态反馈法，稳定的原点。此外，我们证明了该方法的有效性，以稳定的起源的Van der Pol振荡器和混沌Henon映射的时间序列数据。
Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces
Istituto Italiano di Tecnologia（意大利理工学院）；University of Novi Sad（诺维萨德大学）；University College London（伦敦大学学院）；University of Genova（热那亚大学）；Massachusetts Institute of Technology（麻省理工学院）
Transfer operators on graphs: Spectral clustering and beyond
School of Mathematical & Computer Sciences, Heriot–Watt University（赫瑞瓦特大学数学与计算机科学学院）；Maxwell Institute for Mathematical Sciences, University of Edinburgh and Heriot–Watt University（爱丁堡大学与赫瑞瓦特大学麦克斯韦数学科学学院）
Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators
Technische Universität Ilmenau（伊尔梅瑙工业大学）；Freie Universität Berlin（柏林自由大学）；Chemnitz University of Technology（开姆尼茨工业大学）
Dynamic robotic cloth folding with efficient Koopman operator-based model predictive control
Istituto Italiano di Tecnologia（意大利技术研究院）；Institut de Robòtica i Informàtica Industrial（机器人与信息技术研究所）；MaLGa Center, DIBRIS, Università degli Studi di Genova（MaLGa中心，DIBRIS，热那亚大学）
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）
How deep is your network? Deep vs. shallow learning of transfer operators
Maxwell Institute for Mathematical Sciences, University of Edinburgh and Heriot–Watt University（数学科学研究所、爱丁堡大学和赫里奥特-沃斯大学）；School of Mathematics, University of Edinburgh（数学学院、爱丁堡大学）；School of Mathematical and Computer Sciences, Heriot–Watt University（数学与计算机科学学院、赫里奥特-沃斯大学）
