Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions
This addresses a key limitation for applying neural networks to real-world engineering and science problems, which often involve boundary conditions, though it is incremental by extending prior work from whole domains to finite domains.
The paper tackles the problem of approximating solutions to high-dimensional elliptic PDEs with boundary conditions, specifically the Poisson equation on finite domains, and shows that deep neural networks can do so without the curse of dimensionality.
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.