Nonlinear dynamical systems can be handily described by the associated Koopman operator, whose action evolves every observable of the system forward in time. Learning the Koopman operator and its spectral decomposition from data is enabled by a number of algorithms. In this work we present for the first time non-asymptotic learning bounds for the Koopman eigenvalues and eigenfunctions. We focus on time-reversal-invariant stochastic dynamical systems, including the important example of Langevin dynamics. We analyze two popular estimators: Extended Dynamic Mode Decomposition (EDMD) and Reduced Rank Regression (RRR). Our results critically hinge on novel {minimax} estimation bounds for the operator norm error, that may be of independent interest. Our spectral learning bounds are driven by the simultaneous control of the operator norm error and a novel metric distortion functional of the estimated eigenfunctions. The bounds indicates that both EDMD and RRR have similar variance, but EDMD suffers from a larger bias which might be detrimental to its learning rate. Our results shed new light on the emergence of spurious eigenvalues, an issue which is well known empirically. Numerical experiments illustrate the implications of the bounds in practice.
ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals
Department of Mathematical and Statistical Science, University of Alberta（阿尔伯塔大学数学与统计科学系）；International Center for Primate Brain Research, Chinese Academy Sciences（中国科学院灵长类脑研究国际中心）
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
Meta-Learning for Koopman Spectral Analysis with Short Time-series
NTT Communication Science Laboratories, NTT Corporation（NTT公司NTT通信科学实验室）；Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；Center for Advanced Intelligence Project, RIKEN（理化学研究所高级智能项目中心）
Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control
Max Planck Institute of Geoanthropology（马克斯·普朗克地理人类学研究所）；Department of Electrical Engineering and Information Technology, University of Naples（那不勒斯大学电气工程与信息学院）
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）
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）
Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systems
ElectroScience Laboratory, The Ohio State University（电科学实验室，俄亥俄州立大学）；Department of Electrical and Computer Engineering, The Ohio State University（电气与计算机工程系，俄亥俄州立大学）；SLAC National Accelerator Laboratory, Stanford University（斯坦福大学SLAC国家加速器实验室）
Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning
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）
Understanding Latent-Dimension Scaling in Dynamical-System Learning through Spectral Reliability
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Graduate School of Information Science and Technology, The University of Osaka（大阪大学信息科学与技术研究生院）
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）
DeepKoopFormer: A Koopman Enhanced Transformer Based Architecture for Time Series Forecasting
Helmholtz Center for Environmental Research-UFZ（海德堡环境研究中心-UFZ）；Delft Center for Systems and Control（代尔夫特系统与控制中心）；School of Engineering, University of Pittsburgh（匹兹堡大学工程学院）
KoopGen: Koopman Generator Networks for Representing and Predicting Dynamical Systems with Continuous Spectra
School of Mathematics and Statistics and Ministry of Education Key Lab of Intelligent Networks and Network Security（数学与统计学学院和教育部智能网络与网络安全重点实验室）；School of Mathematics（数学学院）
Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds
Centre for Mathematical Sciences, University of Cambridge, UK（剑桥大学数学科学中心，英国）；Department of Mathematics, Imperial College London, UK（伦敦帝国理工学院数学系，英国）；Department of Mechanical Engineering, University of Washington, USA（美国华盛顿大学机械工程系）
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）
SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting
Department of Electrical and Computer Engineering, McGill University, Montreal, Canada（麦吉尔大学电气与计算机工程系）；Amazon Science. This work does not relate to the author’s position at Amazon（亚马逊科学）；Mila - Quebec Artificial Intelligence Institute, Montreal, Canada（魁北克人工智能研究所）；ILLS - International Laboratory on Learning Systems, Montreal, Canada（学习系统国际实验室）
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）
Koopman early warning signals for bifurcation and rate-induced tipping
Columbia University（哥伦比亚大学）；University of Edinburgh（爱丁堡大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Pacific Northwest National Laboratory（西北太平洋国家实验室）；University of Leicester（莱斯特大学）；NASA Goddard Institute for Space Studies（美国国家航空航天局戈达德空间研究所）
