# Learning a Formally Verified Control Barrier Function in Stochastic Environment

arXiv: https://arxiv.org/abs/2403.19332 · HTML: https://mlsanity.com/paper/2403.19332/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Mar 2024.

## Built on by (most cited first, 11 of 11)

- [Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation](https://mlsanity.com/paper/2410.16281/) (2024)
- [A Physics-Informed Machine Learning Framework for Safe and Optimal Control of Autonomous Systems](https://mlsanity.com/paper/2502.11057/) (2025)
- [V-OCBF: Learning Safety Filters from Offline Data via Value-Guided Offline Control Barrier Functions](https://mlsanity.com/paper/2512.10822/) (2025)
- [CP-NCBF: A Conformal Prediction-based Approach to Synthesize Verified Neural Control Barrier Functions](https://mlsanity.com/paper/2503.17395/) (2025)
- [Learning Ensembles of Vision-based Safety Control Filters](https://mlsanity.com/paper/2412.02029/) (2024)
- [Safety Certification in the Latent space using Control Barrier Functions and World Models](https://mlsanity.com/paper/2507.13871/) (2025)
- [Epigraph-Guided Flow Matching for Safe and Performant Offline Reinforcement Learning](https://mlsanity.com/paper/2602.08054/) (2026)
- [Neural Control Barrier Functions from Physics Informed Neural Networks](https://mlsanity.com/paper/2504.11045/) (2025)
- [Formal Verification of Neural Certificates Done Dynamically](https://mlsanity.com/paper/2507.11987/) (2025)
- [MAD-PINN: A Decentralized Physics-Informed Machine Learning Framework for Safe and Optimal Multi-Agent Control](https://mlsanity.com/paper/2509.23960/) (2025)
- [Efficient Verification of Neural Control Barrier Functions with Smooth Nonlinear Activations](https://mlsanity.com/paper/2605.07757/) (2026)

## Abstract (arXiv)

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.