The process of transforming observed data into predictive mathematical models of the physical world has always been paramount in science and engineering. Although data is currently being collected at an ever-increasing pace, devising meaningful models out of such observations in an automated fashion still remains an open problem. In this work, we put forth a machine learning approach for identifying nonlinear dynamical systems from data. Specifically, we blend classical tools from numerical analysis, namely the multi-step time-stepping schemes, with powerful nonlinear function approximators, namely deep neural networks, to distill the mechanisms that govern the evolution of a given data-set. We test the effectiveness of our approach for several benchmark problems involving the identification of complex, nonlinear and chaotic dynamics, and we demonstrate how this allows us to accurately learn the dynamics, forecast future states, and identify basins of attraction. In particular, we study the Lorenz system, the fluid flow behind a cylinder, the Hopf bifurcation, and the Glycoltic oscillator model as an example of complicated nonlinear dynamics typical of biological systems.
Improved deep learning of chaotic dynamical systems with multistep penalty losses
School of Information Sciences and Technology（信息科学与技术学院）；Pennsylvania State University（宾夕法尼亚州立大学）；Center for Applied Scientific Computing（应用科学计算中心）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；University of California Santa Cruz（加州大学圣克鲁兹分校）
Spatiotemporal information conversion machine for time-series prediction
South China University of Technology（华南理工大学）；Pazhou Lab（琶洲实验室）；Center for Excellence in Molecular Cell Science, Chinese Academy of Sciences（中国科学院分子细胞科学卓越创新中心）；Center for Excellence in Animal Evolution and Genetics, Chinese Academy of Sciences（中国科学院动物进化与遗传学卓越创新中心）；Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Chinese Academy of Sciences（中国科学院大学杭州高等研究院）；ShanghaiTech University（上海科技大学）
Nonlocal Kramers-Moyal formulas and data-driven discovery of stochastic dynamical systems with multiplicative Lévy noise
School of Automation, Nanjing University of Science and Technology（南京理工大学自动化学院）；Department of Mathematics and Department of Physics, Great Bay University（大湾区大学数学系和物理系）
Data-driven model discovery with Kolmogorov-Arnold networks
School of Electrical, Computer, and Energy Engineering, Arizona State University（亚利桑那州立大学电气、计算机与能源工程学院）；Clarkson University（克拉克森大学）；Department of Physics, Arizona State University（亚利桑那州立大学物理系）
SODAs: Sparse Optimization for the Discovery of Differential and Algebraic Equations
Department of Engineering Sciences and Applied Mathematics, Northwestern University（工程科学与应用数学系，西北大学）；Department of Physics and Astronomy, Northwestern University（物理与天文学系，西北大学）
How Low Can You Go? Active Learning for Sparse Model Discovery in the Ultra-Low-Data Limit
Department of Mechanical Engineering, University of Washington（华盛顿大学机械工程系）；NSF AI Institute in Dynamic Systems, University of Washington（华盛顿大学NSF动态系统人工智能研究所）；Department of Aeronautics, Imperial College London（伦敦帝国理工学院航空系）
Machine Learning of Nonlinear Waves: Data-Driven Methods for Computer-Assisted Discovery of Equations, Symmetries, Conservation Laws, and Integrability
School of Mathematics, Georgia Institute of Technology（佐治亚理工学院数学学院）
