As autonomous systems become increasingly prevalent in daily life, ensuring their safety is paramount. Control Barrier Functions (CBFs) have emerged as an effective tool for guaranteeing safety; however, manually designing them for specific applications remains a significant challenge. With the advent of deep learning techniques, recent research has explored synthesizing CBFs using neural networks-commonly referred to as neural CBFs. This paper introduces a novel class of neural CBFs that leverages a physics-inspired neural network framework by incorporating Zubov's Partial Differential Equation (PDE) within the context of safety. This approach provides a scalable methodology for synthesizing neural CBFs applicable to high-dimensional systems. Furthermore, by utilizing reciprocal CBFs instead of zeroing CBFs, the proposed framework allows for the specification of flexible, user-defined safe regions. To validate the effectiveness of the approach, we present case studies on three different systems: an inverted pendulum, autonomous ground navigation, and aerial navigation in obstacle-laden environments.

本研究解决了在特定应用中手动设计控制安全函数（CBF）的挑战，通过引入一种基于物理启发的神经网络框架，提出了一类新的神经CBF。这种方法能够合成适用于高维系统的神经CBF，并通过允许用户定义的安全区域来增强灵活性，有效性通过对倒立摆、自治地面导航和障碍环境中的空中导航的案例研究得到了验证。

基于物理信息神经网络的神经控制安全函数