# Reachability Barrier Networks: Learning Hamilton-Jacobi Solutions for Smooth and Flexible Control Barrier Functions

arXiv: https://arxiv.org/abs/2505.11755 · HTML: https://mlsanity.com/paper/2505.11755/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 36 references that are in the map.
- First posted on arXiv 16 May 2025.

## Builds on (most cited first, 27 of 27)

- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [Learning Control Barrier Functions from Expert Demonstrations](https://mlsanity.com/paper/2004.03315/) (2020)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systems](https://mlsanity.com/paper/2309.05837/) (2023)
- [Learning for Safety-Critical Control with Control Barrier Functions](https://mlsanity.com/paper/1912.10099/) (2019)
- [How Should a Robot Assess Risk? Towards an Axiomatic Theory of Risk in Robotics](https://mlsanity.com/paper/1710.11040/) (2017)
- [Robust Control Barrier-Value Functions for Safety-Critical Control](https://mlsanity.com/paper/2104.02808/) (2021)
- [Synthesis of Control Barrier Functions Using a Supervised Machine Learning Approach](https://mlsanity.com/paper/2003.04950/) (2020)
- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction](https://mlsanity.com/paper/2312.08604/) (2023)
- [Refining Control Barrier Functions through Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2204.12507/) (2022)
- [Online Update of Safety Assurances Using Confidence-Based Predictions](https://mlsanity.com/paper/2210.01199/) (2022)
- [BarrierNet: A Safety-Guaranteed Layer for Neural Networks](https://mlsanity.com/paper/2111.11277/) (2021)
- [Parameter-Conditioned Reachable Sets for Updating Safety Assurances Online](https://mlsanity.com/paper/2209.14976/) (2022)
- [Safety Filters for Black-Box Dynamical Systems by Learning Discriminating Hyperplanes](https://mlsanity.com/paper/2402.05279/) (2024)
- [Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation](https://mlsanity.com/paper/2410.16281/) (2024)
- [Generating Formal Safety Assurances for High-Dimensional Reachability](https://mlsanity.com/paper/2209.12336/) (2022)
- [On Safety and Liveness Filtering Using Hamilton-Jacobi Reachability Analysis](https://mlsanity.com/paper/2312.15347/) (2023)
- [Quantifying Safety of Learning-based Self-Driving Control Using Almost-Barrier Functions](https://mlsanity.com/paper/2207.13891/) (2022)
- [One Filter to Deploy Them All: Robust Safety for Quadrupedal Navigation in Unknown Environments](https://mlsanity.com/paper/2412.09989/) (2024)
- [Learning Control Barrier Functions with High Relative Degree for Safety-Critical Control](https://mlsanity.com/paper/2011.10721/) (2020)
- [Learning to Refine Input Constrained Control Barrier Functions via Uncertainty-Aware Online Parameter Adaptation](https://mlsanity.com/paper/2409.14616/) (2024)
- [Safety-Critical Control with Bounded Inputs via Reduced Order Models](https://mlsanity.com/paper/2303.03247/) (2023)
- [Patching Approximately Safe Value Functions Leveraging Local Hamilton-Jacobi Reachability Analysis](https://mlsanity.com/paper/2304.09850/) (2023)
- [Linear Supervision for Nonlinear, High-Dimensional Neural Control and Differential Games](https://mlsanity.com/paper/2412.02033/) (2024)

## Built on by (most cited first, 3 of 3)

- [How to Train Your Latent Control Barrier Function: Smooth Safety Filtering Under Hard-to-Model Constraints](https://mlsanity.com/paper/2511.18606/) (2025)
- [Statistically Assuring Safety of Control Systems using Ensembles of Safety Filters and Conformal Prediction](https://mlsanity.com/paper/2511.07899/) (2025)
- [Safe Control using Learned Safety Filters and Adaptive Conformal Inference](https://mlsanity.com/paper/2604.18482/) (2026)

## Abstract (arXiv)

Recent developments in autonomous driving and robotics underscore the necessity of safety-critical controllers. Control barrier functions (CBFs) are a popular method for appending safety guarantees to a general control framework, but they are notoriously difficult to generate beyond low dimensions. Existing methods often yield non-differentiable or inaccurate approximations that lack integrity, and thus fail to ensure safety. In this work, we use physics-informed neural networks (PINNs) to generate smooth approximations of CBFs by computing Hamilton-Jacobi (HJ) optimal control solutions. These reachability barrier networks (RBNs) avoid traditional dimensionality constraints and support the tuning of their conservativeness post-training through a parameterized discount term. To ensure robustness of the discounted solutions, we leverage conformal prediction methods to derive probabilistic safety guarantees for RBNs. We demonstrate that RBNs are highly accurate in low dimensions, and safer than the standard neural CBF approach in high dimensions. Namely, we showcase the RBNs in a 9D multi-vehicle collision avoidance problem where it empirically proves to be 5.5x safer and 1.9x less conservative than the neural CBFs, offering a promising method to synthesize CBFs for general nonlinear autonomous systems.