--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].

Safety is a fundamental requirement of control systems. Control Barrier Functions (CBFs) are proposed to ensure the safety of the control system by constructing safety filters or synthesizing control inputs. However, the safety guarantee and performance of safe controllers rely on the construction of valid CBFs. Inspired by universal approximatability, CBFs are represented by neural networks, known as neural CBFs (NCBFs). This paper presents an algorithm for synthesizing formally verified continuous-time neural Control Barrier Functions in stochastic environments in a single step. The proposed training process ensures efficacy across the entire state space with only a finite number of data points by constructing a sample-based learning framework for Stochastic Neural CBFs (SNCBFs). Our methodology eliminates the need for post hoc verification by enforcing Lipschitz bounds on the neural network, its Jacobian, and Hessian terms. We demonstrate the effectiveness of our approach through case studies on the inverted pendulum system and obstacle avoidance in autonomous driving, showcasing larger safe regions compared to baseline methods.