A Neuro-Symbolic Method for Solving Differential and Functional Equations

TL;DR

This paper presents a scalable neuro-symbolic approach that generates interpretable symbolic solutions to differential and functional equations, combining deep learning with symbolic mathematics without requiring language models.

Contribution

The authors introduce a novel neural architecture that learns to produce valid symbolic expressions for differential equations, enabling scalable and adaptable solutions without language model training.

Findings

Successfully generates symbolic solutions for various differential equations.

Produces useful symbolic approximations when exact solutions are unavailable.

Generalizes to other mathematical tasks like integration and functional equations.

Abstract

When neural networks are used to solve differential equations, they usually produce solutions in the form of black-box functions that are not directly mathematically interpretable. We introduce a method for generating symbolic expressions to solve differential equations while leveraging deep learning training methods. Unlike existing methods, our system does not require learning a language model over symbolic mathematics, making it scalable, compact, and easily adaptable for a variety of tasks and configurations. As part of the method, we propose a novel neural architecture for learning mathematical expressions to optimize a customizable objective. The system is designed to always return a valid symbolic formula, generating a useful approximation when an exact analytic solution to a differential equation is not or cannot be found. We demonstrate through examples how our method can be applied on a number of differential equations, often obtaining symbolic approximations that are useful or insightful. Furthermore, we show how the system can be effortlessly generalized to find symbolic solutions to other mathematical tasks, including integration and functional equations.