Computation and Formal Verification of Neural Network Contraction Metrics

IF 2.4 Q2 AUTOMATION & CONTROL SYSTEMS
Maxwell Fitzsimmons;Jun Liu
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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.
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。
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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
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