@Article{IJNAM-18-4,
author = {​Jian, Li and Zhang, 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 = {<p style="text-align: justify;">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&#39; 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.</p>},
issn = {2617-8710},
doi = {https://doi.org/2021-IJNAM-19114},
url = {https://global-sci.com/article/82989/a-deep-learning-galerkin-method-for-the-second-order-linear-elliptic-equations}
}