Stochastic Neural Control Barrier Functions

--Control Barrier Functions (CBFs) are utilized to ensure the safety of control systems. CBFs act as safety filters in order to provide safety guarantees without compromising system performance. These safety guarantees rely on the construction of valid CBFs. Due to their complexity, CBFs can be represented by neural networks, known as neural CBFs (NCBFs). Existing works on the verification of the NCBF focus on the synthesis and verification of NCBFs in deterministic settings, leaving the stochastic NCBFs (SNCBFs) less studied. In this work, we propose a verifiably safe synthesis for SNCBFs. We consider the cases of smooth SNCBFs with twice-differentiable activation functions and SNCBFs that utilize the Rectified Linear Unit or ReLU activation function. We propose a verification-free synthesis framework for smooth SNCBFs and a verification-in-the-loop synthesis framework for both smooth and ReLU SNCBFs. Safety is one of the fundamental properties required for control systems, especially those that interact with humans and critical infrastructures. Safety violations could lead to catastrophic damage to robots, harm to humans, and economic loss [1], [2]. The safety requirements of these systems, with applications including medicine, energy, and robotics [3], have motivated recent research to design safe control policies. Safety requirements can be formulated as the positive invariance of a given safe region, meaning the system remains in the safe region for all time [4]. V arious approaches for safety-critical control have been proposed, including Hamilton-Jacobi Reachability (HJR) analysis [5]-[7] and safe reinforcement learning (RL) [8]-[10].