@Article{CiCP-34-3,
author = {Yuling, Jiao and Zhijian, Yang, Jerry and Yuan, Cheng and Zhou, Junyu},
title = {A Rate of Convergence of Weak Adversarial Neural Networks for the Second Order Parabolic PDEs},
journal = {Communications in Computational Physics},
year = {2023},
volume = {34},
number = {3},
pages = {813--836},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2023-0063},
url = {https://global-sci.com/article/79382/a-rate-of-convergence-of-weak-adversarial-neural-networks-for-the-second-order-parabolic-pdes}
}