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

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