Volume 29, Issue 5
Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

Yulei Liao & Pingbing Ming

Commun. Comput. Phys., 29 (2021), pp. 1365-1384.

Published online: 2021-03

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  • 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.

  • Keywords

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

  • AMS Subject Headings

65N30, 65M12, 41A46, 35J25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-1365, author = {Liao , Yulei and Ming , Pingbing}, title = {Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {5}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0219}, url = {http://global-sci.org/intro/article_detail/cicp/18717.html} }
TY - JOUR T1 - Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions AU - Liao , Yulei AU - Ming , Pingbing JO - Communications in Computational Physics VL - 5 SP - 1365 EP - 1384 PY - 2021 DA - 2021/03 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0219 UR - https://global-sci.org/intro/article_detail/cicp/18717.html KW - Deep Nitsche Method, Deep Ritz Method, neural network approximation, mixed boundary conditions, curse of dimensionality. AB -

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.

Yulei Liao & Pingbing Ming. (2021). Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions. Communications in Computational Physics. 29 (5). 1365-1384. doi:10.4208/cicp.OA-2020-0219
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