Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions

TL;DR

This paper demonstrates that deep neural networks can effectively approximate solutions to high-dimensional elliptic PDEs with boundary conditions, like the Poisson equation, without suffering from the curse of dimensionality.

Contribution

It extends neural network approximation results to PDEs on finite domains with boundary conditions, using probabilistic representations and sampling methods.

Findings

Neural networks can approximate solutions to high-dimensional elliptic PDEs with boundary conditions.

The approach avoids the curse of dimensionality in such approximations.

Probabilistic methods underpin the theoretical guarantees.

Abstract

In recent work it has been established that deep neural networks are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb{R}^d$ subject to Dirichlet boundary conditions. It is shown that deep neural networks are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.