Sparse inference and active learning of stochastic differential equations from data

TL;DR

This paper introduces a Bayesian sparse inference method for directly identifying differential equations from data, including an active learning approach that enhances the discovery of stochastic dynamics.

Contribution

It presents a novel Bayesian framework with a Laplacian prior for sparse, accurate inference of differential equations and develops an active learning loop for stochastic systems.

Findings

Method achieves high accuracy for ordinary, partial, and stochastic differential equations.

Active learning improves the inference of Langevin-type stochastic processes.

Demonstrated robustness and efficiency across various simulated cases.

Abstract

Automatic machine learning of empirical models from experimental data has recently become possible as a result of increased availability of computational power and dedicated algorithms. Despite the successes of non-parametric inference and neural-network-based inference for empirical modelling, a physical interpretation of the results often remains challenging. Here, we focus on direct inference of governing differential equations from data, which can be formulated as a linear inverse problem. A Bayesian framework with a Laplacian prior distribution is employed for finding sparse solutions efficiently. The superior accuracy and robustness of the method is demonstrated for various cases, including ordinary, partial, and stochastic differential equations. Furthermore, we develop an active learning procedure for the automated discovery of stochastic differential equations. In this procedure, learning of the unknown dynamical equations is coupled to the application of perturbations to the measured system in a feedback loop. We demonstrate with simulations that the active learning procedure improves the inference of empirical, Langevin-type descriptions of stochastic processes.