Partial differential equation solver based on optimization methods

TL;DR

This paper introduces an optimization-based approach to solving partial differential equations, aiming to provide a versatile initial guess method that broadens applicability beyond traditional numerical techniques.

Contribution

It proposes a novel optimization method for PDE solutions that can serve as a starting point for a wide range of equations, overcoming limitations of existing methods.

Findings

The method offers a convergent initial guess for various PDEs.

It reduces the restrictions on the class of equations solvable by numerical methods.

The approach is computationally efficient compared to traditional techniques.

Abstract

The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods usually have the restricted class of the equations, on which the convergence is proved. Only a small amount of "cheap and dirty" numerical methods converge on a wide class of equations with the lower approximation order price. In the article, we present a method that uses an optimization algorithm to obtain a solution that could be used as the initial guess for the wide class of equations.