We propose a physics-constrained machine learning method-based on reservoir computing- to time-accurately predict extreme events and long-term velocity statistics in a model of turbulent shear flow. The method leverages the strengths of two different approaches: empirical modelling based on reservoir computing, which it learns the chaotic dynamics from data only, and physical modelling based on conservation laws, which extrapolates the dynamics when training data becomes unavailable. We show that the combination of the two approaches is able to accurately reproduce the velocity statistics and to predict the occurrence and amplitude of extreme events in a model of self-sustaining process in turbulence. In this flow, the extreme events are abrupt transitions from turbulent to quasi-laminar states, which are deterministic phenomena that cannot be traditionally predicted because of chaos. Furthermore, the physics-constrained machine learning method is shown to be robust with respect to noise. This work opens up new possibilities for synergistically enhancing data-driven methods with physical knowledge for the time-accurate prediction of chaotic flows.
Minimal Deterministic Echo State Networks Outperform Random Reservoirs in Learning Chaotic Dynamics
Center for Scalable Data Analytics；Artificial Intelligence (ScaDS.AI), Leipzig University, Leipzig, Germany；Institute for Earth System Science \& Remote Sensing, Leipzig University, Leipzig, Germany
Prediction of Nonlinear Oscillations in a Jumping Quarter-Car Model Using Reservoir Computing
Tokyo University of Agriculture and Technology（东京农工大学）；Amity University Haryana - Gurugram（Amity大学哈里亚纳邦-古鲁格拉姆校区）；Nagoya Institute of Technology（名古屋工业大学）；Central University of Rajasthan（中央拉贾斯坦大学）
Designing Chaotic Attractors: A Semi-supervised Approach
Graduate School of Information Science and Technology, University of Tokyo（东京大学情报理工学系研究科）；International Research Center for Neurointelligence, University of Tokyo Institutes for Advanced Study, University of Tokyo（东京大学高等研究院国际神经智能研究中心）
Inferring bifurcation diagrams of two distinct chaotic systems by a single machine
School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China（陕西师范大学物理与信息技术学院，西安710062，中国）；College of Physics and Electronic Information Engineering, Qinghai Normal University, Xining 810008, China（青海师范大学物理与电子信息技术学院，西宁810008，中国）；School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China（华东师范大学物理与电子科学学院，上海200062，中国）
Model-free inference of unseen attractors: Reconstructing phase space features from a single noisy trajectory using reservoir computing
Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB)（跨学科物理与复杂系统研究所 (IFISC (CSIC-UIB))）；The University of Tokyo（东京大学）；The Ohio State University（俄亥俄州立大学）；ResCon Technologies, LLC（ResCon Technologies公司）
Combining Ensemble Kalman Filter and Reservoir Computing to predict spatio-temporal chaotic systems from imperfect observations and models
School of Engineering, the University of Tokyo（东京大学工程学院）；Meteorological Research Institute, Japan Meteorological Agency（日本气象厅气象研究所）；RIKEN Center for Computational Science（理化学研究所计算科学中心）
Oscillations enhance time-series prediction in reservoir computing with feedback
Symbiotic Intelligent Systems Research Center, Institute for Open and Transdisciplinary Research Initiatives, Osaka University（共生智能系统研究中心，开放与跨学科研究倡议所，大阪大学）；Academy of Emerging Sciences, Chubu University（新兴科学学院，名古屋大学）；Center for Information and Neural Networks, National Institute of Information and Communications Technology（信息与神经网络中心，信息与通信技术国家研究所）；International Professional University of Technology in Osaka（大阪国际技术专业大学）
Versatile Reservoir Computing for Heterogeneous Complex Networks
School of Physics and Information Technology, Shaanxi Normal University（物理与信息科技学院，陕西师范大学）；School of Science, Xi’an University of Posts and Telecommunications（科学学院，西安邮电大学）
Hybridizing Traditional and Next-Generation Reservoir Computing to Accurately and Efficiently Forecast Dynamical Systems
University of Maryland（马里兰大学）；University of Chicago（芝加哥大学）；Institute for Research in Electronics and Applied Physics (IREAP)（电子与应用物理研究所）；Institute for Physical Science and Technology (IPST)（物理科学与技术研究所）；Santa Fe Institute（圣达菲研究所）
Reservoir Computing meets Recurrent Kernels and Structured Transforms
Ecole Normale Supérieure（巴黎高等师范学院）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université de Paris（巴黎大学）；Collège de France（法兰西公学院）；LightOn（LightOn公司）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Prediction of Unobserved Bifurcation by Unsupervised Extraction of Slowly Time-Varying System Parameter Dynamics from Time Series Using Reservoir Computing
Faculty of Health Data science, Juntendo University（顺天堂大学健康数据科学学部）；The School of Systems Information Science, Future University Hakodate（函馆未来大学系统信息科学学院）
Wyant College of Optical Sciences, University of Arizona（亚利桑那大学怀恩特光学科学学院）；School of Electrical Engineering and Telecommunications, University of New South Wales（新南威尔士大学电气工程与电信学院）
