Symbolic Discovery of Stochastic Differential Equations with Genetic Programming

2026-03-10

AI Summary

Traditional symbolic regression struggles to model stochastic dynamical systems corrupted by noise. This work pioneers the extension of symbolic regression to stochastic differential equations (SDEs) by introducing a genetic programming–based approach that jointly optimizes both the drift and diffusion terms of an SDE. The method employs maximum likelihood estimation to learn interpretable stochastic dynamical models directly from data. It accurately recovers the underlying governing equations, exhibits robustness to sparse sampling, and scales efficiently to high-dimensional systems. Furthermore, the framework successfully generalizes to stochastic partial differential equations, substantially enhancing the expressiveness, interpretability, and generative capacity of the learned models.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• symbolic regression
• stochastic differential equations
• genetic programming
• noise modeling
• interpretable discovery

Innovation

• symbolic regression
• stochastic differential equations
• genetic programming
• drift and diffusion
• interpretable discovery