Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

TL;DR

This paper introduces a novel deep learning-based method for solving PDEs that effectively enforces essential boundary conditions using Nitsche's variational formulation, demonstrated on high-dimensional problems with proven error estimates.

Contribution

It presents a new deep Nitsche method that integrates Nitsche's formulation into deep neural network PDE solvers, enabling efficient boundary condition enforcement without extra computational costs.

Findings

Proven error estimates in the energy norm.

Effective handling of boundary conditions in high-dimensional PDEs.

Validated on problems up to 100 dimensions.

Abstract

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.