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Learning Approximate Symbolic Solutions to Burgers' Equation using Symbolic Regression
Benjamin G Cohen1, Burcu Beykal1, George M Bollas1
1Department of Chemical and Biomolecular Engineering, University of Connecticut, Storrs, CT 60629 USA.
Ifac-Papersonline
|September 15, 2025
Summary
This study uses symbolic regression to discover physics equations without data. A stepwise approach, incorporating domain knowledge, successfully modeled the diffusion and Burgers
Area of Science:
- * Computational Physics
- * Applied Mathematics
- * Machine Learning
Background:
- * Traditional methods for solving partial differential equations (PDEs) often require extensive data or complex numerical simulations.
- * Symbolic regression offers a data-driven approach to discover governing equations, but can struggle with complex systems.
- * Integrating domain knowledge can guide symbolic regression towards more efficient and interpretable solutions.
Purpose of the Study:
- * To develop and demonstrate a stepwise symbolic regression strategy for learning solutions to PDEs without requiring prior data.
- * To investigate the efficacy of incorporating domain knowledge to simplify the search space and improve model discovery.
- * To generate interpretable symbolic solutions for the diffusion equation and Burgers' equation.
Main Methods:
- * A stepwise symbolic regression approach was employed, starting with learning a partial model of the system's physics.
- * Domain knowledge was leveraged to define initial primitives and guide the discovery of a complete physical model.
- * The method was applied to the diffusion equation and Burgers' equation with varying convection coefficients.
Main Results:
- * The method achieved an R-squared value of 0.99 for the diffusion equation model.
- * Symbolic models for Burgers' equation were generated with R-squared values exceeding 0.98 across different convection coefficients.
- * Discovered solutions for Burgers' equation were represented as transformations of the diffusion equation solution.
Conclusions:
- * Stepwise symbolic regression, augmented with domain knowledge, can effectively learn symbolic solutions to dynamical systems without data.
- * This approach enhances interpretability and demonstrates the synergy between expert intuition and automated discovery.
- * The findings highlight a promising direction for developing more efficient and understandable physics-informed machine learning models.
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