Regularity-Conforming Neural Networks (ReCoNNs) for solving Partial Differential Equations

TL;DR

This paper introduces regularity-conforming neural networks (ReCoNNs) that incorporate prior regularity information of PDE solutions, improving approximation accuracy especially for solutions with discontinuities or singularities, demonstrated through 1D and 2D problems.

Contribution

The paper proposes a novel neural network architecture that explicitly encodes solution regularity, enhancing PDE approximation capabilities over classical methods.

Findings

ReCoNNs outperform classical neural networks in approximating PDE solutions with discontinuities.

Explicit regularity encoding improves the training and accuracy of neural PDE solvers.

The approach extends naturally to higher-dimensional PDE problems.

Abstract

Whilst the Universal Approximation Theorem guarantees the existence of approximations to Sobolev functions -- the natural function spaces for PDEs -- by Neural Networks (NNs) of sufficient size, low-regularity solutions may lead to poor approximations in practice. For example, classical fully-connected feed-forward NNs fail to approximate continuous functions whose gradient is discontinuous when employing strong formulations like in Physics Informed Neural Networks (PINNs). In this article, we propose the use of regularity-conforming neural networks, where a priori information on the regularity of solutions to PDEs can be employed to construct proper architectures. We illustrate the potential of such architectures via a two-dimensional (2D) transmission problem, where the solution may admit discontinuities in the gradient across interfaces, as well as power-like singularities at certain points. In particular, we formulate the weak transmission problem in a PINNs-like strong formulation with interface and continuity conditions. Such architectures are partially explainable; discontinuities are explicitly described, allowing the introduction of novel terms into the loss function. We demonstrate via several model problems in one and two dimensions the advantages of using regularity-conforming architectures in contrast to classical architectures. The ideas presented in this article easily extend to problems in higher dimensions.