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Volume 31, Issue 3
An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

Jianguo Huang, Haoqin Wang & Tao Zhou

Commun. Comput. Phys., 31 (2022), pp. 966-986.

Published online: 2022-03

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  • Abstract

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method.

  • AMS Subject Headings

65N25, 65N30, 68U99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-966, author = {Huang , JianguoWang , Haoqin and Zhou , Tao}, title = {An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {3}, pages = {966--986}, abstract = {

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0176}, url = {http://global-sci.org/intro/article_detail/cicp/20305.html} }
TY - JOUR T1 - An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions AU - Huang , Jianguo AU - Wang , Haoqin AU - Zhou , Tao JO - Communications in Computational Physics VL - 3 SP - 966 EP - 986 PY - 2022 DA - 2022/03 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0176 UR - https://global-sci.org/intro/article_detail/cicp/20305.html KW - The augmented Lagrangian method, deep learning, variational problems, saddle point problems, essential boundary conditions. AB -

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method.

Jianguo Huang, Haoqin Wang & Tao Zhou. (2022). An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions. Communications in Computational Physics. 31 (3). 966-986. doi:10.4208/cicp.OA-2021-0176
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