Sparse learning of stochastic dynamic equations

TL;DR

This paper extends the SINDy framework to stochastic dynamical systems, providing theoretical guarantees, practical algorithms, and demonstrating effectiveness on diffusion models.

Contribution

It introduces a stochastic version of SINDy, proves its asymptotic correctness, and discusses algorithms with cross validation for sparse regression.

Findings

Proved asymptotic correctness of stochastic SINDy.

Developed algorithms incorporating cross validation.

Validated on diffusion system examples.

Abstract

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.