神经微分方程的平均场和动力学描述

IF 1.7 Q2 MATHEMATICS, APPLIED Foundations of data science (Springfield, Mo.) Pub Date : 2020-01-07 DOI:10.3934/fods.2022007
M. Herty, T. Trimborn, G. Visconti
{"title":"神经微分方程的平均场和动力学描述","authors":"M. Herty, T. Trimborn, G. Visconti","doi":"10.3934/fods.2022007","DOIUrl":null,"url":null,"abstract":"Nowadays, neural networks are widely used in many applications as artificial intelligence models for learning tasks. Since typically neural networks process a very large amount of data, it is convenient to formulate them within the mean-field and kinetic theory. In this work we focus on a particular class of neural networks, i.e. the residual neural networks, assuming that each layer is characterized by the same number of neurons \\begin{document}$ N $\\end{document}, which is fixed by the dimension of the data. This assumption allows to interpret the residual neural network as a time-discretized ordinary differential equation, in analogy with neural differential equations. The mean-field description is then obtained in the limit of infinitely many input data. This leads to a Vlasov-type partial differential equation which describes the evolution of the distribution of the input data. We analyze steady states and sensitivity with respect to the parameters of the network, namely the weights and the bias. In the simple setting of a linear activation function and one-dimensional input data, the study of the moments provides insights on the choice of the parameters of the network. Furthermore, a modification of the microscopic dynamics, inspired by stochastic residual neural networks, leads to a Fokker-Planck formulation of the network, in which the concept of network training is replaced by the task of fitting distributions. The performed analysis is validated by artificial numerical simulations. In particular, results on classification and regression problems are presented.","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":" ","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2020-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Mean-field and kinetic descriptions of neural differential equations\",\"authors\":\"M. Herty, T. Trimborn, G. Visconti\",\"doi\":\"10.3934/fods.2022007\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Nowadays, neural networks are widely used in many applications as artificial intelligence models for learning tasks. Since typically neural networks process a very large amount of data, it is convenient to formulate them within the mean-field and kinetic theory. In this work we focus on a particular class of neural networks, i.e. the residual neural networks, assuming that each layer is characterized by the same number of neurons \\\\begin{document}$ N $\\\\end{document}, which is fixed by the dimension of the data. This assumption allows to interpret the residual neural network as a time-discretized ordinary differential equation, in analogy with neural differential equations. The mean-field description is then obtained in the limit of infinitely many input data. This leads to a Vlasov-type partial differential equation which describes the evolution of the distribution of the input data. We analyze steady states and sensitivity with respect to the parameters of the network, namely the weights and the bias. In the simple setting of a linear activation function and one-dimensional input data, the study of the moments provides insights on the choice of the parameters of the network. Furthermore, a modification of the microscopic dynamics, inspired by stochastic residual neural networks, leads to a Fokker-Planck formulation of the network, in which the concept of network training is replaced by the task of fitting distributions. The performed analysis is validated by artificial numerical simulations. In particular, results on classification and regression problems are presented.\",\"PeriodicalId\":73054,\"journal\":{\"name\":\"Foundations of data science (Springfield, Mo.)\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2020-01-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Foundations of data science (Springfield, Mo.)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/fods.2022007\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of data science (Springfield, Mo.)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/fods.2022007","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 4

摘要

如今,神经网络作为学习任务的人工智能模型被广泛应用于许多应用中。由于神经网络通常处理大量数据,因此在平均场和动力学理论中对其进行公式化是很方便的。在这项工作中,我们专注于一类特定的神经网络,即残差神经网络,假设每一层都由相同数量的神经元\开始{文档}$N$\结束{文档}表征,这是由数据的维度固定的。这一假设允许将残差神经网络解释为时间离散常微分方程,类似于神经微分方程。然后在无限多个输入数据的限制下获得平均场描述。这导致了描述输入数据分布演变的Vlasov型偏微分方程。我们分析了网络参数的稳态和灵敏度,即权重和偏差。在线性激活函数和一维输入数据的简单设置中,矩的研究为网络参数的选择提供了见解。此外,受随机残差神经网络的启发,对微观动力学进行了修改,得出了网络的福克-普朗克公式,其中网络训练的概念被拟合分布的任务所取代。通过人工数值模拟验证了所进行的分析。特别地,给出了关于分类和回归问题的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Mean-field and kinetic descriptions of neural differential equations
Nowadays, neural networks are widely used in many applications as artificial intelligence models for learning tasks. Since typically neural networks process a very large amount of data, it is convenient to formulate them within the mean-field and kinetic theory. In this work we focus on a particular class of neural networks, i.e. the residual neural networks, assuming that each layer is characterized by the same number of neurons \begin{document}$ N $\end{document}, which is fixed by the dimension of the data. This assumption allows to interpret the residual neural network as a time-discretized ordinary differential equation, in analogy with neural differential equations. The mean-field description is then obtained in the limit of infinitely many input data. This leads to a Vlasov-type partial differential equation which describes the evolution of the distribution of the input data. We analyze steady states and sensitivity with respect to the parameters of the network, namely the weights and the bias. In the simple setting of a linear activation function and one-dimensional input data, the study of the moments provides insights on the choice of the parameters of the network. Furthermore, a modification of the microscopic dynamics, inspired by stochastic residual neural networks, leads to a Fokker-Planck formulation of the network, in which the concept of network training is replaced by the task of fitting distributions. The performed analysis is validated by artificial numerical simulations. In particular, results on classification and regression problems are presented.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
3.30
自引率
0.00%
发文量
0
期刊最新文献
CHATGPT FOR COMPUTATIONAL TOPOLOGY. PERSISTENT PATH LAPLACIAN. Weight set decomposition for weighted rank and rating aggregation: An interpretable and visual decision support tool Hierarchical regularization networks for sparsification based learning on noisy datasets Noise calibration for SPDEs: A case study for the rotating shallow water model
×
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
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 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 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1