A deep neural network-based method for solving obstacle problems

In this paper, we propose a method based on deep neural networks to solve obstacle problems. By introducing penalty terms, we reformulate the obstacle problem as a minimization optimization problem and utilize a deep neural network to approximate its solution. The convergence analysis is established...

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Vydáno v:	Nonlinear analysis: real world applications Ročník 72; s. 103864
Hlavní autoři:	Cheng, Xiaoliang, Shen, Xing, Wang, Xilu, Liang, Kewei
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Elsevier Ltd 01.08.2023
Témata:	Deep neural networks Error analysis Minimization optimization Obstacle problems Deep neural networks Error analysis Minimization optimization Obstacle problems
ISSN:	1468-1218, 1878-5719
On-line přístup:	Získat plný text
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Popis
Shrnutí:	In this paper, we propose a method based on deep neural networks to solve obstacle problems. By introducing penalty terms, we reformulate the obstacle problem as a minimization optimization problem and utilize a deep neural network to approximate its solution. The convergence analysis is established by decomposing the error into three parts: approximation error, statistical error and optimization error. The approximate error is bounded by the depth and width of the network, the statistical error is estimated by the number of samples, and the optimization error is reflected in the empirical loss term. Due to its unsupervised and meshless advantages, the proposed method has wide applicability. Numerical experiments illustrate the effectiveness and robustness of the proposed method and verify the theoretical proof.
ISSN:	1468-1218 1878-5719
DOI:	10.1016/j.nonrwa.2023.103864