Volume 18, Issue 4
A Deep Learning Galerkin Method for the Second-Order Linear Elliptic Equations

Int. J. Numer. Anal. Mod., 18 (2021), pp. 427-441.

Published online: 2021-05

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• Abstract

In this paper we propose a Deep Learning Galerkin Method (DGM) based on the deep neural network learning algorithm to approximate the general second-order linear elliptic problem. This method is a combination of Galerkin Method and machine learning. The DGM uses the deep neural network instead of the linear combination of basis functions. Our algorithm is meshfree and we train the neural network by randomly sampling the space points and using the gradient descent algorithm to satisfy the differential operators and boundary conditions. Moreover, the approximate ability of a neural networks' solution to the exact solution is proved by the convergence of the loss function and the convergence of the neural network to the exact solution in $L^2$ norm under certain conditions. Finally, some numerical experiments reflect the approximation ability of the neural networks intuitively.

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@Article{IJNAM-18-427, author = {Li , ​JianZhang , Wen and Yue , Jing}, title = {A Deep Learning Galerkin Method for the Second-Order Linear Elliptic Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2021}, volume = {18}, number = {4}, pages = {427--441}, abstract = {

In this paper we propose a Deep Learning Galerkin Method (DGM) based on the deep neural network learning algorithm to approximate the general second-order linear elliptic problem. This method is a combination of Galerkin Method and machine learning. The DGM uses the deep neural network instead of the linear combination of basis functions. Our algorithm is meshfree and we train the neural network by randomly sampling the space points and using the gradient descent algorithm to satisfy the differential operators and boundary conditions. Moreover, the approximate ability of a neural networks' solution to the exact solution is proved by the convergence of the loss function and the convergence of the neural network to the exact solution in $L^2$ norm under certain conditions. Finally, some numerical experiments reflect the approximation ability of the neural networks intuitively.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/19114.html} }
TY - JOUR T1 - A Deep Learning Galerkin Method for the Second-Order Linear Elliptic Equations AU - Li , ​Jian AU - Zhang , Wen AU - Yue , Jing JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 427 EP - 441 PY - 2021 DA - 2021/05 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/19114.html KW - Deep learning Galerkin method, deep neural network, second-order linear elliptic equations, convergence, numerical experiments. AB -

In this paper we propose a Deep Learning Galerkin Method (DGM) based on the deep neural network learning algorithm to approximate the general second-order linear elliptic problem. This method is a combination of Galerkin Method and machine learning. The DGM uses the deep neural network instead of the linear combination of basis functions. Our algorithm is meshfree and we train the neural network by randomly sampling the space points and using the gradient descent algorithm to satisfy the differential operators and boundary conditions. Moreover, the approximate ability of a neural networks' solution to the exact solution is proved by the convergence of the loss function and the convergence of the neural network to the exact solution in $L^2$ norm under certain conditions. Finally, some numerical experiments reflect the approximation ability of the neural networks intuitively.

​Jian Li, WenZhang & JingYue. (2021). A Deep Learning Galerkin Method for the Second-Order Linear Elliptic Equations. International Journal of Numerical Analysis and Modeling. 18 (4). 427-441. doi:
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