A Rate of Convergence of Weak Adversarial Neural Networks for the Second Order Parabolic PDEs

IF 2.6 3区 物理与天体物理 Q1 PHYSICS, MATHEMATICAL Communications in Computational Physics Pub Date : 2023-06-01 DOI:10.4208/cicp.oa-2023-0063
Yuling Jiao, Jerry Zhijian Yang, Cheng Yuan null, Junyu Zhou
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Abstract

. In this paper, we give the first rigorous error estimation of the Weak Ad-versarial 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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二阶抛物型偏微分方程弱对抗神经网络的收敛速度
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来源期刊
Communications in Computational Physics
Communications in Computational Physics 物理-物理:数学物理
CiteScore
4.70
自引率
5.40%
发文量
84
审稿时长
9 months
期刊介绍: Communications in Computational Physics (CiCP) publishes original research and survey papers of high scientific value in computational modeling of physical problems. Results in multi-physics and multi-scale innovative computational methods and modeling in all physical sciences will be featured.
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