In applications of nonlinear and complex dynamical systems, a common situation is that the system can be measured but its structure and the detailed rules of dynamical evolution are unknown. The inverse problem is to determine the system equations and structure based solely on measured time series. Recently, methods based on sparse optimization have been developed. For example, the principle of exploiting sparse optimization such as compressive sensing to find the equations of nonlinear dynamical systems from data was articulated in 2011 by the Nonlinear Dynamics Group at Arizona State University. This article presents a brief review of the recent progress in this area. The basic idea is to expand the equations governing the dynamical evolution of the system into a power series or a Fourier series of a finite number of terms and then to determine the vector of the expansion coefficients based solely on data through sparse optimization. Examples discussed here include discovering the equations of stationary or nonstationary chaotic systems to enable prediction of dynamical events such as critical transition and system collapse, inferring the full topology of complex networks of dynamical oscillators and social networks hosting evolutionary game dynamics, and identifying partial differential equations for spatiotemporal dynamical systems. Situations where sparse optimization is effective and those in which the method fails are discussed. Comparisons with the traditional method of delay coordinate embedding in nonlinear time series analysis are given and the recent development of model-free, data driven prediction framework based on machine learning is briefly introduced.
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