Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions (Communications in Computational Physics, 2021)
We propose a new method to deal with the essential boundary conditionsencountered 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 severalrepresentative problems posed in at most 100 dimension.