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
Yuling Jiao, Jerry Zhijian Yang, Cheng Yuan null, Junyu Zhou
{"title":"A Rate of Convergence of Weak Adversarial Neural Networks for the Second Order Parabolic PDEs","authors":"Yuling Jiao, Jerry Zhijian Yang, Cheng Yuan null, Junyu Zhou","doi":"10.4208/cicp.oa-2023-0063","DOIUrl":null,"url":null,"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.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"12 1","pages":"0"},"PeriodicalIF":2.6000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Computational Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4208/cicp.oa-2023-0063","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0

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.
二阶抛物型偏微分方程弱对抗神经网络的收敛速度
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信