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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Veröffentlicht in:Nonlinear analysis: real world applications Jg. 72; S. 103864
Hauptverfasser: Cheng, Xiaoliang, Shen, Xing, Wang, Xilu, Liang, Kewei
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Ltd 01.08.2023
Schlagworte:
Deep neural networks
Error analysis
Minimization optimization
Obstacle problems
Deep neural networks
Error analysis
Minimization optimization
Obstacle problems
ISSN:1468-1218, 1878-5719
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Zusammenfassung: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