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An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions
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Year:    2022

Author:    Jianguo Huang, Haoqin Wang, Tao Zhou

Communications in Computational Physics, Vol. 31 (2022), Iss. 3 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2021-0176

Communications in Computational Physics, Vol. 31 (2022), Iss. 3 : pp. 966–986

Published online:    2022-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    21

Keywords:    The augmented Lagrangian method deep learning variational problems saddle point problems essential boundary conditions.

Author Details

Jianguo Huang

Haoqin Wang

Tao Zhou

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