Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions

TL;DR

This paper demonstrates that under certain conditions, shallow neural networks can efficiently approximate solutions to high-dimensional elliptic PDEs, overcoming the curse of dimensionality within Barron spaces.

Contribution

It establishes the approximation efficiency of two-layer neural networks for high-dimensional elliptic PDEs with coefficients in Barron spaces, highlighting their expressive power.

Findings

Neural networks can approximate PDE solutions efficiently in high dimensions.

Barron space assumptions enable circumventing the curse of dimensionality.

Shallow networks are sufficient for capturing complex PDE solutions.

Abstract

We study the approximation complexity of high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube, within the framework of Barron spaces. Under the assumption that the coefficients belong to suitably defined Barron spaces, we prove that the solution can be efficiently approximated by two-layer neural networks, circumventing the curse of dimensionality. Our results demonstrate the expressive power of shallow networks in capturing high-dimensional PDE solutions under appropriate structural assumptions.