{"title":"Control of neural transport for normalising flows","authors":"Domènec Ruiz-Balet , Enrique Zuazua","doi":"10.1016/j.matpur.2023.10.005","DOIUrl":null,"url":null,"abstract":"<div><p><span>Inspired by normalising flows, we analyse the bilinear control of neural transport equations by means of time-dependent velocity fields<span> restricted to fulfil, at any time instance, a simple neural network ansatz. The </span></span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span><span> approximate controllability property is proved, showing that any probability density can be driven arbitrarily close to any other one in any time horizon. The control vector fields are built explicitly and inductively and this provides quantitative estimates on their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.</span></p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"181 ","pages":"Pages 58-90"},"PeriodicalIF":2.3000,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782423001460","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
Inspired by normalising flows, we analyse the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfil, at any time instance, a simple neural network ansatz. The approximate controllability property is proved, showing that any probability density can be driven arbitrarily close to any other one in any time horizon. The control vector fields are built explicitly and inductively and this provides quantitative estimates on their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.
受归一化流的启发，我们利用时间相关的速度场来分析神经传递方程的双线性控制，这些速度场被限制在任何时间实例上，以满足一个简单的神经网络分析。证明了L1近似可控性，表明任意概率密度都可以在任意时间范围内被驱动到任意另一个概率密度附近。控制向量场是显式地和归纳地建立的，这提供了对它们的复杂性和幅度的定量估计。当只有目标概率密度的随机样本可用时，这也会导致统计误差界限。inspirs samsams par flux normalisaturs, nous分析为contrôle bilinsamaire des samsamas de transport neuron或moyen de champs de vitesse dsampendant du temps and limites samsamas vsamrifier, chaque instance temporelle，简单分析为samsamseau neuron。仲裁解决办法:固有的薪金薪金，固有的薪金薪金，固有的薪金薪金，的薪金薪金，或然的薪金，的薪金，的薪金，的薪金，的薪金，的薪金，的薪金，的薪金。Les champs de vecteurs de contrôle sont construcits de maniires explicit et归纳，ce qui permet d'obtenir des estimations de leur complex itures de leur amplitude。从统计数据上看，所有的数据都有可能是不可靠的，因为所有的数据都可能是不可靠的。
期刊介绍：
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.
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