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In applications of nonlinear and complex dynamical systems, a commonsituation is that the system can be measured but its structure and the detailedrules of dynamical evolution are unknown. The inverse problem is to determinethe system equations and structure based solely on measured time series.Recently, methods based on sparse optimization have been developed. Forexample, the principle of exploiting sparse optimization such as compressivesensing to find the equations of nonlinear dynamical systems from data wasarticulated in 2011 by the Nonlinear Dynamics Group at Arizona StateUniversity. This article presents a brief review of the recent progress in thisarea. The basic idea is to expand the equations governing the dynamicalevolution of the system into a power series or a Fourier series of a finitenumber of terms and then to determine the vector of the expansion coefficientsbased solely on data through sparse optimization. Examples discussed hereinclude discovering the equations of stationary or nonstationary chaoticsystems to enable prediction of dynamical events such as critical transitionand system collapse, inferring the full topology of complex networks ofdynamical oscillators and social networks hosting evolutionary game dynamics,and identifying partial differential equations for spatiotemporal dynamicalsystems. Situations where sparse optimization is effective and those in whichthe method fails are discussed. Comparisons with the traditional method ofdelay coordinate embedding in nonlinear time series analysis are given and therecent development of model-free, data driven prediction framework based onmachine learning is briefly introduced.

Finding nonlinear system equations and complex network structures from data: a sparse optimization approach