A Neural Network Based Method to Solve Boundary Value Problems

TL;DR

This paper introduces a neural network-based numerical approach for solving boundary value problems, offering advantages over traditional methods by handling unstructured data points without meshing issues.

Contribution

It formulates and validates a neural network method for BVPs, demonstrating its effectiveness and limitations through numerical experiments on Laplace and Poisson equations.

Findings

Neural network method successfully solves BVPs with Dirichlet and mixed boundary conditions.

The method handles unstructured data points, avoiding meshing issues.

Numerical results validate the approach's efficacy and reveal its limitations.

Abstract

A Neural Network (NN) based numerical method is formulated and implemented for solving Boundary Value Problems (BVPs) and numerical results are presented to validate this method by solving Laplace equation with Dirichlet boundary condition and Poisson's equation with mixed boundary conditions. The principal advantage of NN based numerical method is the discrete data points where the field is computed, can be unstructured and do not suffer from issues of meshing like traditional numerical methods such as Finite Difference Time Domain or Finite Element Method. Numerical investigations are carried out for both uniform and non-uniform training grid distributions to understand the efficacy and limitations of this method and to provide qualitative understanding of various parameters involved.