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

07/10/2020

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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⊂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.

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

Numerical solution of Poisson partial differential equations in high dimension using deep neural networks

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

Deep nurbs – admissible neural networks

Poisson CNN: Convolutional Neural Networks for the Solution of the Poisson Equation with Varying Meshes and Dirichlet Boundary Conditions

DS-GPS : A Deep Statistical Graph Poisson Solver (for faster CFD simulations)

Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks

Handling Neumann and Robin boundary conditions in a fictitious domain volume penalization framework

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

Related Research

Numerical solution of Poisson partial differential equations in high dimension using deep neural networks

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

Deep nurbs – admissible neural networks

Poisson CNN: Convolutional Neural Networks for the Solution of the Poisson Equation with Varying Meshes and Dirichlet Boundary Conditions

DS-GPS : A Deep Statistical Graph Poisson Solver (for faster CFD simulations)

Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks

Handling Neumann and Robin boundary conditions in a fictitious domain volume penalization framework