Automated Differential Equation Solver Based on the Parametric Approximation Optimization

Abstract

The classical numerical methods for differential equations are a well-studied field. Nevertheless, these numerical methods are limited in their scope to certain classes of equations. Modern machine learning applications, such as equation discovery, may benefit from having the solution to the discovered equations. The solution to an arbitrary equation typically requires either an expert system that chooses the proper method for a given equation, or a method with a wide range of equation types. Machine learning methods may provide the needed versatility. This article presents a method that uses an optimization algorithm for a parameterized approximation to find a solution to a given problem. We take an agnostic approach without dividing equations by their type or boundary conditions, which allows for fewer restrictions on the algorithm. The results may not be as precise as those of an expert; however, our method enables automated solutions for a wide range of equations without the algorithm’s parameters changing. In this paper, we provide examples of the Legendre equation, Painlevé transcendents, wave equation, heat equation, and Korteweg–de Vries equation, which are solved in a unified manner without significant changes to the algorithm’s parameters.