  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.
