Computation and Formal Verification of Neural Network Contraction Metrics

IF 2.4 Q2 AUTOMATION & CONTROL SYSTEMS IEEE Control Systems Letters Pub Date : 2024-10-11 DOI:10.1109/LCSYS.2024.3478272
Maxwell Fitzsimmons;Jun Liu
{"title":"Computation and Formal Verification of Neural Network Contraction Metrics","authors":"Maxwell Fitzsimmons;Jun Liu","doi":"10.1109/LCSYS.2024.3478272","DOIUrl":null,"url":null,"abstract":"A contraction metric defines a differential Lyapunov-like function that robustly captures the convergence between trajectories. In this letter, we investigate the use of neural networks for computing verifiable contraction metrics. We first prove the existence of a smooth neural network contraction metric within the domain of attraction of an exponentially stable equilibrium point. We then focus on the computation of a neural network contraction metric over a compact invariant set within the domain of attraction certified by a physics-informed neural network Lyapunov function. We consider both partial differential inequality (PDI) and equation (PDE) losses for computation. We show that sufficiently accurate neural approximate solutions to the PDI and PDE are guaranteed to be a contraction metric under mild technical assumptions. We rigorously verify the computed neural network contraction metric using a satisfiability modulo theories solver. Through numerical examples, we demonstrate that the proposed approach outperforms traditional semidefinite programming methods for finding sum-of-squares polynomial contraction metrics.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"IEEE Control Systems Letters","FirstCategoryId":"1085","ListUrlMain":"https://ieeexplore.ieee.org/document/10714396/","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0

Abstract

A contraction metric defines a differential Lyapunov-like function that robustly captures the convergence between trajectories. In this letter, we investigate the use of neural networks for computing verifiable contraction metrics. We first prove the existence of a smooth neural network contraction metric within the domain of attraction of an exponentially stable equilibrium point. We then focus on the computation of a neural network contraction metric over a compact invariant set within the domain of attraction certified by a physics-informed neural network Lyapunov function. We consider both partial differential inequality (PDI) and equation (PDE) losses for computation. We show that sufficiently accurate neural approximate solutions to the PDI and PDE are guaranteed to be a contraction metric under mild technical assumptions. We rigorously verify the computed neural network contraction metric using a satisfiability modulo theories solver. Through numerical examples, we demonstrate that the proposed approach outperforms traditional semidefinite programming methods for finding sum-of-squares polynomial contraction metrics.
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
神经网络收缩指标的计算与形式验证
收缩度量定义了一个类似于李雅普诺夫的微分函数,它能稳健地捕捉轨迹之间的收敛性。在这封信中,我们研究了如何利用神经网络计算可验证的收缩度量。我们首先证明了在指数稳定平衡点的吸引域内存在平滑的神经网络收缩度量。然后,我们将重点放在计算由物理信息神经网络 Lyapunov 函数证明的吸引力域内紧凑不变集上的神经网络收缩度量。我们考虑了计算中的偏微分不等式(PDI)和方程(PDE)损失。我们证明,在温和的技术假设条件下,足够精确的偏微分不等式和偏微分方程的神经近似解保证是一个收缩度量。我们使用满足性模态理论求解器严格验证了计算出的神经网络收缩度量。通过数值示例,我们证明了在寻找平方和多项式收缩指标方面,所提出的方法优于传统的半有限编程方法。
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 去求助
来源期刊
IEEE Control Systems Letters
IEEE Control Systems Letters Mathematics-Control and Optimization
CiteScore
4.40
自引率
13.30%
发文量
471
期刊最新文献
Rationality of Learning Algorithms in Repeated Normal-Form Games Impact of Opinion on Disease Transmission With Waterborne Pathogen and Stubborn Community Numerical and Lyapunov-Based Investigation of the Effect of Stenosis on Blood Transport Stability Using a Control-Theoretic PDE Model of Cardiovascular Flow Almost Sure Convergence and Non-Asymptotic Concentration Bounds for Stochastic Mirror Descent Algorithm Opinion Dynamics With Set-Based Confidence: Convergence Criteria and Periodic Solutions
×
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