Neural Network Methods for Boundary Value Problems Defined in Arbitrarily Shaped Domains

TL;DR

This paper introduces neural network approaches, including multilayer perceptrons and radial basis function networks, to solve boundary value problems with complex, arbitrarily shaped domains, demonstrating accurate results in 2D and 3D cases.

Contribution

It extends neural network methods to handle complex boundary geometries by using point-based boundary representations and combines different network types for improved boundary condition satisfaction.

Findings

Successful application to 2D PDEs with complex boundaries

Effective use of RBF networks for boundary conditions

Achieved accurate solutions in 3D PDEs

Abstract

Partial differential equations (PDEs) with Dirichlet boundary conditions defined on boundaries with simple geometry have been succesfuly treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of complex boundary geometry, where the boundary is determined by a number of points that belong to it and are closely located, so as to offer a reasonable representation. Two networks are employed: a multilayer perceptron and a radial basis function network. The later is used to account for the satisfaction of the boundary conditions. The method has been successfuly tested on two-dimensional and three-dimensional PDEs and has yielded accurate solutions.