A Data-Free Symbolic Regression Approach for Solving Equations

2026-06-05

AI Summary

Many equations in science lack closed-form solutions, and existing symbolic regression methods rely on input-output data, making it difficult to solve equations using only their mathematical form. This work proposes a Symbolic Equation Solver (SES), which, for the first time, frames equation solving as a data-free optimization problem over a differentiable symbolic model. SES constructs an objective function solely from the equation’s structure and its initial or boundary conditions, without requiring any observed data. By leveraging symbolic expression generation techniques, the method directly recovers compact and accurate analytical solutions from the equation itself. It demonstrates success across diverse equation types—including algebraic equations, those involving transcendental terms, ordinary differential equations, and partial differential equations—thereby overcoming the traditional dependence of symbolic regression on training data.

Abstract

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.

Problem

• symbolic regression
• equation solving
• data-free
• symbolic expressions
• analytical solutions

Innovation

• Data-Free
• Symbolic Regression
• Equation Solving
• Differentiable Symbolic Models
• Analytical Solutions