Year: 2023
Author: Yuling Jiao, Jerry Zhijian Yang, Cheng Yuan, Junyu Zhou
Communications in Computational Physics, Vol. 34 (2023), Iss. 3 : pp. 813–836
Abstract
In this paper, we give the first rigorous error estimation of the Weak Adversarial Neural Networks (WAN) in solving the second order parabolic PDEs. By decomposing the error into approximation error and statistical error, we first show the weak solution can be approximated by the $ReLU^2$ with arbitrary accuracy, then prove that the statistical error can also be efficiently bounded by the Rademacher complexity of the network functions, which can be further bounded by some integral related with the covering numbers and pseudo-dimension of $ReLU^2$ space. Finally, by combining the two bounds, we prove that the error of the WAN method can be well controlled if the depth and width of the neural network as well as the sample numbers have been properly selected. Our result also reveals some kind of freedom in choosing sample numbers on $∂Ω$ and in the time axis.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2023-0063
Communications in Computational Physics, Vol. 34 (2023), Iss. 3 : pp. 813–836
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Weak Adversarial Networks second order parabolic PDEs error analysis.