We introduce a novel neural network architecture, termed Taylor-neural Lyapunov functions, designed to approximate Lyapunov functions with formal certification. This architecture innovatively encodes local approximations and extends them globally by leveraging neural networks to approximate the residuals. Our method recasts the problem of estimating the largest region of attraction - specifically for maximal Lyapunov functions - into a learning problem, ensuring convergence around the origin through robust control theory. Physics-informed machine learning techniques further refine the estimation of the largest region of attraction. Remarkably, this method is versatile, operating effectively even without simulated data points. We validate the efficacy of our approach by providing numerical certificates of convergence across multiple examples. Our proposed methodology not only competes closely with state-of-the-art approaches, such as sum-of-squares and LyZNet, but also achieves comparable results even in the absence of simulated data. This work represents a significant advancement in control theory, with broad potential applications in the design of stable control systems and beyond.

本研究解决了李雅普诺夫函数的近似与形式认证之间的空白，提出了一种新颖的泰勒-神经李雅普诺夫函数架构，通过神经网络全局编码局部近似。研究表明，该方法在缺乏模拟数据的情况下依然有效，提供了多个示例的收敛数值认证，并在控制理论中取得了显著进展，具有广泛的应用潜力。

学习与验证最大泰勒-神经李雅普诺夫函数