Automated differential equation solver based on the parametric approximation optimization

TL;DR

This paper introduces an automated differential equation solver that employs parametric approximation optimization, enabling broad applicability without manual parameter tuning, though with potentially less precision.

Contribution

The paper presents a novel optimization-based method for solving differential equations automatically, reducing the need for manual parameter adjustments across diverse equation types.

Findings

Enables automated solutions for a wide class of differential equations.

Reduces the need for manual parameter tuning in numerical methods.

Provides solutions with acceptable accuracy despite less precision.

Abstract

The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of the equations, on which the convergence with a given parameter set or range is proved. Only a few "cheap and dirty" numerical methods converge on a wide class of equations without parameter tuning with the lower approximation order price. The article presents a method that uses an optimization algorithm to obtain a solution using the parameterized approximation. The result may not be as precise as an expert one. However, it allows solving the wide class of equations in an automated manner without the algorithm's parameters change.