A Natural Deep Ritz Method for Essential Boundary Value Problems

TL;DR

This paper introduces a new intrinsic method for enforcing essential boundary conditions in deep neural network solutions of PDEs, improving robustness and avoiding penalty hyper-parameter tuning.

Contribution

It proposes an intrinsic boundary condition enforcement framework for the deep Ritz method, enhancing stability and simplifying implementation compared to penalty methods.

Findings

Effective boundary condition enforcement demonstrated on Poisson problems

Method shows improved robustness and efficiency

Potential for extension to other PDEs and deep learning techniques

Abstract

Deep neural network approaches show promise in solving partial differential equations. However, unlike traditional numerical methods, they face challenges in enforcing essential boundary conditions. The widely adopted penalty-type methods, for example, offer a straightforward implementation but introduces additional complexity due to the need for hyper-parameter tuning; moreover, the use of a large penalty parameter can lead to artificial extra stiffness, complicating the optimization process. In this paper, we propose a novel, intrinsic approach to impose essential boundary conditions through a framework inspired by intrinsic structures. We demonstrate the effectiveness of this approach using the deep Ritz method applied to Poisson problems, with the potential for extension to more general equations and other deep learning techniques. Numerical results are provided to substantiate the efficiency and robustness of the proposed method.