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Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions
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Year:    2021

Author:    Yulei Liao, Pingbing Ming

Communications in Computational Physics, Vol. 29 (2021), Iss. 5 : pp. 1365–1384

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2020-0219

Communications in Computational Physics, Vol. 29 (2021), Iss. 5 : pp. 1365–1384

Published online:    2021-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    20

Keywords:    Deep Nitsche Method Deep Ritz Method neural network approximation mixed boundary conditions curse of dimensionality.

Author Details

Yulei Liao

Pingbing Ming

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