Year: 2021
Author: Jian Li, Wen Zhang, Jing Yue
International Journal of Numerical Analysis and Modeling, Vol. 18 (2021), Iss. 4 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2021-IJNAM-19114
International Journal of Numerical Analysis and Modeling, Vol. 18 (2021), Iss. 4 : pp. 427–441
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
Keywords: Deep learning Galerkin method deep neural network second-order linear elliptic equations convergence numerical experiments.