Solving the Poisson Equation with Dirichlet data by shallow ReLU$^\alpha$-networks: A regularity and approximation perspective

TL;DR

This paper investigates how shallow ReLU$^eta$ neural networks can effectively approximate solutions to the Poisson equation with Dirichlet boundary conditions, emphasizing regularity and approximation capabilities.

Contribution

It provides a theoretical analysis of the approximation power of shallow ReLU$^eta$ networks for elliptic PDE solutions, focusing on boundary value problems.

Findings

Neural networks can approximate solutions with certain regularity assumptions.

The approximation capacity depends on the regularity of the solution and the network architecture.

The study offers insights into the regularity requirements for effective neural PDE solvers.

Abstract

For several classes of neural PDE solvers (Deep Ritz, PINNs, DeepONets), the ability to approximate the solution or solution operator to a partial differential equation (PDE) hinges on the abilitiy of a neural network to approximate the solution in the spatial variables. We analyze the capacity of neural networks to approximate solutions to an elliptic PDE assuming that the boundary condition can be approximated efficiently. Our focus is on the Laplace operator with Dirichlet boundary condition on a half space and on neural networks with a single hidden layer and an activation function that is a power of the popular ReLU activation function.