Neural Control Barrier Functions from Physics Informed Neural Networks

TL;DR

This paper tackles safety guarantees for autonomous systems by learning Neural Control Barrier Functions (NCBFs) through a PINN-based realization of Zubov's PDE. The approach produces a transformed barrier function W_N that characterizes the safe region via W_N<1 and remains well-behaved near the boundary (W_N→1), enabling stable learning and flexible level-set design. From W_N, a barrier B is obtained as B(x)=\frac{1}{2\alpha}\ln\left(\frac{1+W_N(x)}{1-W_N(x)}\right), and a QP-based controller enforces the barrier condition with minimal deviation from a nominal input, ensuring forward invariance of the safe set. The method is demonstrated on an inverted pendulum, autonomous ground navigation, and 6D aerial navigation, showing scalability to high-dimensional systems and the ability to tailor safety regions to specific requirements. Overall, the work provides a practical, scalable path to integrating physics-informed learning with formal safety guarantees in nonlinear control.

Abstract

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.

Figures (5)

• Figure 1: Contour plot of the learned barrier function $W_N(x)$ for inverted pendulum. Sub-level sets $\{ x \in \mathbb{R}^n : W_N(x) \le \gamma \}$ for increasing $\gamma$ illustrate how the safe region expands, as we converge to a sub-level set.
• Figure 2: Contour plot of $W_N(x)$ at $\psi = 0$ for the autonomous ground navigation system. Sub-level sets $\{ x \in \mathbb{R}^n : W_N(x) \le \gamma \}$ for increasing $\gamma$ illustrate how the unsafe region shrinks (leading to a larger safe area outside the contour), as we converge to a sub-level set.
• Figure 3: Simulated autonomous ground navigation robot trajectories demonstrating obstacle avoidance when a barrier-based controller is employed.
• Figure 4: Contour plot of $W_N(x)$ at $\phi = 0$ for the aerial navigation vehicle (other dimensions not shown). Sub-level sets $\{ x \in \mathbb{R}^n : W_N(x) \le \gamma \}$ for increasing $\gamma$ illustrate how the unsafe region shrinks (leading to a larger safe area outside the contour), as we converge to a sub-level set.
• Figure 5: Trajectories of the aerial navigation vehicle in the $y$-$z$ plane, using a trajectory following reference controller. Successful avoidance of the unsafe region is observed in the barrier-based controller due to safety adjustments to the control inputs.

Theorems & Definitions (5)

• Definition 1: Safe Set
• Definition 2: Reciprocal Control Barrier Function
• Definition 3: Safety Conditions for RCBF
• Theorem 1: Zubov's Characterization for Safe Regions
• Remark 1