A Data-Free Symbolic Regression Approach for Solving Equations
For scientists and engineers who need explicit symbolic solutions to equations that cannot be solved analytically, SES provides a novel method to obtain such expressions without requiring numerical data.
The authors propose a data-free symbolic regression framework (SES) that solves equations by optimizing differentiable symbolic models directly from the equation and boundary conditions, eliminating the need for paired training data. SES recovers compact symbolic expressions matching analytical solutions across algebraic and differential equations.
Many equations arising in science currently cannot be solved by available analytical techniques and are therefore solved numerically, without yielding explicit symbolic expressions. Existing symbolic regression approaches can recover symbolic expressions, but require training data obtained from the underlying process, rather than the governing equation alone. We propose the Symbolic Equation Solver (SES), a framework that formulates equation solving as an optimization problem over differentiable symbolic models. SES constructs its objective from the equation together with initial or boundary conditions, eliminating the need for paired input-output data. The learned model is expressed in explicit symbolic form, enabling further analysis. We evaluate SES on representative algebraic and differential equations, including a system of algebraic equations, an equation with transcendental terms, an ordinary differential equation, and partial differential equations with different initial or boundary conditions. Across these settings, SES recovers compact symbolic expressions that match the corresponding analytical solutions.