# BarrierNet: A Safety-Guaranteed Layer for Neural Networks

arXiv: https://arxiv.org/abs/2111.11277 · HTML: https://mlsanity.com/paper/2111.11277/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 46 references that are in the map.
- First posted on arXiv 22 Nov 2021.

## Builds on (most cited first, 30 of 32)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Liquid Time-constant Networks](https://mlsanity.com/paper/2006.04439/) (2020)
- [Learning Control Barrier Functions from Expert Demonstrations](https://mlsanity.com/paper/2004.03315/) (2020)
- [Learning Long-Term Dependencies in Irregularly-Sampled Time Series](https://mlsanity.com/paper/2006.04418/) (2020)
- [Learning for Safety-Critical Control with Control Barrier Functions](https://mlsanity.com/paper/1912.10099/) (2019)
- [Reinforcement Learning for Safety-Critical Control under Model Uncertainty, using Control Lyapunov Functions and Control Barrier Functions](https://mlsanity.com/paper/2004.07584/) (2020)
- [Safe Learning of Quadrotor Dynamics Using Barrier Certificates](https://mlsanity.com/paper/1710.05472/) (2017)
- [Synthesis of Control Barrier Functions Using a Supervised Machine Learning Approach](https://mlsanity.com/paper/2003.04950/) (2020)
- [Neural Certificates for Safe Control Policies](https://mlsanity.com/paper/2006.08465/) (2020)
- [Robustness of Control Barrier Functions for Safety Critical Control](https://mlsanity.com/paper/1612.01554/) (2016)
- [Adaptive Safety with Control Barrier Functions](https://mlsanity.com/paper/1910.00555/) (2019)
- [On The Verification of Neural ODEs with Stochastic Guarantees](https://mlsanity.com/paper/2012.08863/) (2020)
- [Causal Navigation by Continuous-time Neural Networks](https://mlsanity.com/paper/2106.08314/) (2021)
- [Safe Optimal Control Using Stochastic Barrier Functions and Deep Forward-Backward SDEs](https://mlsanity.com/paper/2009.01196/) (2020)
- [Adversarial Training is Not Ready for Robot Learning](https://mlsanity.com/paper/2103.08187/) (2021)
- [Learning Safe Neural Network Controllers with Barrier Certificates](https://mlsanity.com/paper/2009.09826/) (2020)
- [ShieldNN: A Provably Safe NN Filter for Unsafe NN Controllers](https://mlsanity.com/paper/2006.09564/) (2020)
- [GoTube: Scalable Stochastic Verification of Continuous-Depth Models](https://mlsanity.com/paper/2107.08467/) (2021)
- [Lagrangian Reachtubes: The Next Generation](https://mlsanity.com/paper/2012.07458/) (2020)
- [Training Neural Network Controllers Using Control Barrier Functions in the Presence of Disturbances](https://mlsanity.com/paper/2001.08088/) (2020)
- [A Control Barrier Perspective on Episodic Learning via Projection-to-State Safety](https://mlsanity.com/paper/2003.08028/) (2020)
- [Episodic Learning for Safe Bipedal Locomotion with Control Barrier Functions and Projection-to-State Safety](https://mlsanity.com/paper/2105.01697/) (2021)
- [Bridging the Gap between Optimal Trajectory Planning and Safety-Critical Control with Applications to Autonomous Vehicles](https://mlsanity.com/paper/2008.07632/) (2020)
- [Comparison between safety methods control barrier function vs. reachability analysis](https://mlsanity.com/paper/2106.13176/) (2021)
- [Self-triggered Control for Safety Critical Systems using Control Barrier Functions](https://mlsanity.com/paper/1903.03692/) (2019)
- [Decentralized Optimal Merging Control for Connected and Automated Vehicles](https://mlsanity.com/paper/1809.07916/) (2018)

## Built on by (most cited first, 12 of 12)

- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [On the Forward Invariance of Neural ODEs](https://mlsanity.com/paper/2210.04763/) (2022)
- [Differentiable Control Barrier Functions for Vision-based End-to-End Autonomous Driving](https://mlsanity.com/paper/2203.02401/) (2022)
- [Differentiable Safe Controller Design through Control Barrier Functions](https://mlsanity.com/paper/2209.10034/) (2022)
- [Are All Vision Models Created Equal? A Study of the Open-Loop to Closed-Loop Causality Gap](https://mlsanity.com/paper/2210.04303/) (2022)
- [LeTO: Learning Constrained Visuomotor Policy with Differentiable Trajectory Optimization](https://mlsanity.com/paper/2401.17500/) (2024)
- [Reachability Barrier Networks: Learning Hamilton-Jacobi Solutions for Smooth and Flexible Control Barrier Functions](https://mlsanity.com/paper/2505.11755/) (2025)
- [T-CBF: Traversability-based Control Barrier Function to Navigate Vertically Challenging Terrain](https://mlsanity.com/paper/2503.06083/) (2025)
- [Interpreting Neural Policies with Disentangled Tree Representations](https://mlsanity.com/paper/2210.06650/) (2022)
- [GradSTL: Comprehensive Signal Temporal Logic for Neurosymbolic Reasoning and Learning](https://mlsanity.com/paper/2508.04438/) (2025)
- [Fault Tolerant Neural Control Barrier Functions for Robotic Systems under Sensor Faults and Attacks](https://mlsanity.com/paper/2402.18677/) (2024)
- [Safe Reinforcement Learning with Receding-Horizon Safety Critic and Differentiable Action Shields](https://mlsanity.com/paper/2610.20255/) (2026)

## Abstract (arXiv)

This paper introduces differentiable higher-order control barrier functions (CBF) that are end-to-end trainable together with learning systems. CBFs are usually overly conservative, while guaranteeing safety. Here, we address their conservativeness by softening their definitions using environmental dependencies without loosing safety guarantees, and embed them into differentiable quadratic programs. These novel safety layers, termed a BarrierNet, can be used in conjunction with any neural network-based controller, and can be trained by gradient descent. BarrierNet allows the safety constraints of a neural controller be adaptable to changing environments. We evaluate them on a series of control problems such as traffic merging and robot navigations in 2D and 3D space, and demonstrate their effectiveness compared to state-of-the-art approaches.