Shallow Neural Networks and Laplace's Equation on the Half-Space with Dirichlet Boundary Data

In this paper we investigate the ability of Shallow Neural Networks i.e. neural networks with one hidden layer, to solve Laplace’s equation on the half space. We are interested in answering the question if it is possible to fit the boundary value using a neural network then is it possible to learn the solution to the PDE in the entire region using the same network? Our analysis is done primarily in Barron Spaces, which are function spaces designed to include neural networks with a single hidden layer and infinite width. Our results indicate in general the solution is not in the Barron space even if the boundary values are. However, the solution can be approximated to \(\sim \varepsilon^2\) accuracy with functions of a low Barron norm. We implement a Physics Informed Neural Network with a custom loss function to demonstrate some of the theoretical results shown before.