Data-Driven Modeling of Nonlinear Dynamical Systems with Machine Learning

Abstract

Data-driven modelling of nonlinear dynamical systems based on machine learning is reviewed in this paper. Nonlinear dynamics refer to all sorts of complicated and irregular phenomena in mathematics, physics, engineering and life that are not well explained by linear models. Based on the accumulation of observation data from sensors, experiments and high-fidelity simulations, machine learning has recently begun to be applied in system identification, state estimation and the discovery of governing equations. This paper systematically presents the basic ideas of dynamical systems and then introduces several data-driven methods, such as Sparse Identification of Nonlinear Dynamics (SINDy), Neural Ordinary Differential Equations (Neural ODEs) and Koopman operator theory. Based on the above research, a number of methods have been proposed to solve the problems of noise, irregular sampling, etc., in high-dimensional chaos, and their respective advantages are introduced below. At present, the problems of poor data quality, partial observability and lack of interpretability in deep learning models are well-known. In the future, Physics-Informed Machine Learning, uncertainty quantification and robust model methods will be applied in engineering. Data science and applied mathematics will be combined in this paper to offer a comprehensive introduction to the problems and models of complex systems.