Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

作者：Yulei Liao, Pingbing Ming · 发表于：Communications in Computational Physics · 年份：2021 · DOI：10.4208/cicp.oa-2020-0219 · 被引用次数：137 · 研究领域：Model Reduction and Neural Networks、Advanced Numerical Methods in Computational Mathematics、Numerical methods in engineering

We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are non-interpolatory, which makes the enforcement of the essential boundary conditions a nontrivial matter. Our method resorts to Nitsche's variational formulation to deal with this difficulty, which is consistent, and does not require significant extra computational costs. We prove the error estimate in the energy norm and illustrate the method on several representative problems posed in at most 100 dimension.