Sparse learning of stochastic dynamic equations

With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics (SINDy) has been introduced to identify the governing equations of dynamical systems from simulation data. In this study, we extend SINDy to stochastic dynamical systems, which are frequently used to model biophysical processes. We prove the asymptotic correctness of stochastics SINDy in the infinite data limit, both in the original and projected variables. We discuss algorithms to solve the sparse regression problem arising from the practical implementation of SINDy, and show that cross validation is an essential tool to determine the right level of sparsity. We demonstrate the proposed methodology on two test systems, namely, the diffusion in a one-dimensional potential, and the projected dynamics of a two-dimensional diffusion process. 1 I. INTRODUCTION The last decade has seen a dramatic increase in our ability to collect or produce large amounts of high resolution and high dimensional data associated with complex physical and chemical systems, both by means of experimental measurements or computer simulations. In many different scientific fields, ranging from high energy physics to neuroscience, the "bigdata" problem has spurred interest in data analysis methods that can condense massive datasets into a minimal amount of essential information and/or can detect relevant patterns and anomalies in the distribution of the data.