FluxDisco: Symbolic Regression for Stoichiometric Dynamical Systems via Monte Carlo Graph Search

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.

Symbolic regression is a powerful tool for knowledge discovery, enabling the extraction of interpretable mathematical expressions directly from data. However, conventional symbolic discovery typically follows an end-to-end, "one-step" process, which often generates lengthy and physically meaningless expressions when dealing with real physical systems, leading to poor model generalization. This limitation fundamentally stems from its deviation from the basic path of scientific discovery: physical laws do not exist in a single form but follow a hierarchical and progressive pattern from simplicity to complexity. Motivated by this principle, we propose Chain of Symbolic Regression (CoSR), a novel framework that models the discovery of physical laws as a chain of symbolic knowledge. This knowledge chain is formed by progressively combining multiple knowledge units with clear physical meanings along a specific logic, ultimately enabling the precise discovery of the underlying physical laws from data. CoSR fully recapitulates the progressive discovery path from Kepler's third law to the law of universal gravitation in classical mechanics, and is applied to three types of problems: turbulent Rayleigh-Benard convection, viscous flows in a circular pipe, and laser-metal interaction, demonstrating its ability to improve classical scaling theories. Finally, CoSR showcases its capability to discover new knowledge in the complex engineering problem of aerodynamic coefficients scaling for different aircraft.

Discovering the governing equations of dynamical systems is a central problem across many scientific disciplines. As experimental data become increasingly available, automated equation discovery methods offer a promising data-driven approach to accelerate scientific discovery. Among these methods, genetic programming (GP) has been widely adopted due to its flexibility and interpretability. However, GP-based approaches often suffer from inefficient exploration of the symbolic search space, leading to slow convergence and suboptimal solutions. To address these limitations, we propose LLM-ODE, a large language model-aided model discovery framework that guides symbolic evolution using patterns extracted from elite candidate equations. By leveraging the generative prior of large language models, LLM-ODE produces more informed search trajectories while preserving the exploratory strengths of evolutionary algorithms. Empirical results on 91 dynamical systems show that LLM-ODE variants consistently outperform classical GP methods in terms of search efficiency and Pareto-front quality. Overall, our results demonstrate that LLM-ODE improves both efficiency and accuracy over traditional GP-based discovery and offers greater scalability to higher-dimensional systems compared to linear and Transformer-only model discovery methods.