Extracting Structured Dynamical Systems Using Sparse Optimization With Very Few Samples

Hayden Schaeffer, Giang Tran, Rachel Ward, Linan Zhang

Abstract: Learning governing equations allows for deeper understanding of the structure and dynamics of data. We present a random sampling method for learning structured dynamical systems from undersampled and possibly noisy state-space measurements. The learning problem takes the form of a sparse least-squares fitting over a large set of candidate functions. Based on a Bernstein-like inequality for partly dependent random variables, we provide theoretical guarantees on the recovery rate of the sparse coefficients and the identification of the candidate functions for the corresponding problem. Computational results are demonstrated on datasets generated by the Lorenz 96 equation, the viscous Burgers' equation, and the two-component reaction-diffusion equations. Our formulation includes theoretical guarantees of success and is shown to be efficient with respect to the ambient dimension and the number of candidate functions.