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

• Published in: IMA journal of numerical analysis Vol. 42; no. 3; pp. 2055 - 2082
• Main Authors: Grohs, Philipp, Herrmann, Lukas
• Format: Journal Article
• Language: English
• Published: Oxford University Press 22.07.2022
• Subjects: neural network approximation; Monte Carlo methods; high-dimensional approximation
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/drab031

Abstract In recent work it has been established that deep neural networks (DNNs) 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 in 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 DNNs 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.