Learning Approximate Symbolic Solutions to Burgers’ Equation using Symbolic Regression

This work explores the application of symbolic regression to learn symbolic solutions to Burgers' equation without data. We demonstrate a stepwise symbolic regression strategy that explores models that provide tractable logic from coordinates to state estimates. The first step is to learn a model representing part of the system's physics. This partial model is then used to help discover a model capturing the entire physics of the system. The method was able to learn a model of the solution of the diffusion equation with an R-squared value of 0.99 and produced models for Burgers' equation with different values of the convection coefficient, all with R-squared values greater than 0.98. These solutions to Burgers' equation, represented as transformations of the solution to the diffusion equation, demonstrate the potential of leveraging domain knowledge to simplify the symbol space and build useful primitives for symbolic regression. This work highlights how domain knowledge, expert intuition, and symbolic regression can complement each other to create more interpretable solutions to dynamical system models.