Symbolic Discovery of Stochastic Differential Equations with Genetic Programming

TL;DR

This paper presents a genetic programming method for symbolic discovery of stochastic differential equations, enabling interpretable modeling of noisy dynamical systems and extending symbolic regression beyond deterministic equations.

Contribution

It introduces a novel approach for jointly discovering drift and diffusion functions in stochastic differential equations using genetic programming and maximum likelihood estimation.

Findings

Accurate recovery of governing stochastic equations

Scales efficiently to higher-dimensional systems

Robust to sparse sampling and generalizes to stochastic PDEs

Abstract

Automated scientific discovery aims to improve scientific understanding through machine learning. A central approach in this field is symbolic regression, which uses genetic programming or sparse regression to learn interpretable mathematical expressions to explain observed data. Conventionally, the focus of symbolic regression is on identifying ordinary differential equations. The general view is that noise only complicates the recovery of deterministic dynamics. However, explicitly learning a symbolic function of the noise component in stochastic differential equations enhances modelling capacity, increases knowledge gain and enables generative sampling. We introduce a method for symbolic discovery of stochastic differential equations based on genetic programming, jointly optimizing drift and diffusion functions via the maximum likelihood estimate. Our results demonstrate accurate recovery of governing equations, efficient scaling to higher-dimensional systems, robustness to sparsely sampled problems and generalization to stochastic partial differential equations. This work extends symbolic regression toward interpretable discovery of stochastic dynamical systems, contributing to the automation of science in a noisy and dynamic world.