# Certified Training with Branch-and-Bound for Lyapunov-stable Neural Control

arXiv: https://arxiv.org/abs/2411.18235 · HTML: https://mlsanity.com/paper/2411.18235/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 70 references that are in the map.
- First posted on arXiv 27 Nov 2024.

## Builds on (most cited first, 30 of 50)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification](https://mlsanity.com/paper/2103.06624/) (2021)
- [Towards Stable and Efficient Training of Verifiably Robust Neural Networks](https://mlsanity.com/paper/1906.06316/) (2019)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Branch and Bound for Piecewise Linear Neural Network Verification](https://mlsanity.com/paper/1909.06588/) (2019)
- [NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2004.05519/) (2020)
- [A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks](https://mlsanity.com/paper/1902.08722/) (2019)
- [General Cutting Planes for Bound-Propagation-Based Neural Network Verification](https://mlsanity.com/paper/2208.05740/) (2022)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [Lyapunov-stable neural-network control](https://mlsanity.com/paper/2109.14152/) (2021)
- [Enforcing Hard Constraints with Soft Barriers: Safe Reinforcement Learning in Unknown Stochastic Environments](https://mlsanity.com/paper/2209.15090/) (2022)
- [Certified Training: Small Boxes are All You Need](https://mlsanity.com/paper/2210.04871/) (2022)
- [Learning Layout and Style Reconfigurable GANs for Controllable Image Synthesis](https://mlsanity.com/paper/2003.11571/) (2020)
- [Neural Network Verification with Branch-and-Bound for General Nonlinearities](https://mlsanity.com/paper/2405.21063/) (2024)
- [Fast Certified Robust Training with Short Warmup](https://mlsanity.com/paper/2103.17268/) (2021)
- [Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming](https://mlsanity.com/paper/2004.07876/) (2020)
- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems](https://mlsanity.com/paper/2106.13867/) (2021)
- [Neural Certificates for Safe Control Policies](https://mlsanity.com/paper/2006.08465/) (2020)
- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes](https://mlsanity.com/paper/2112.09197/) (2021)
- [Planning with Learned Dynamics: Probabilistic Guarantees on Safety and Reachability via Lipschitz Constants](https://mlsanity.com/paper/2010.08993/) (2020)
- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)

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

- [Parallel Differentiable Reachability for Learning and Planning with Certified Neural Dynamics and Controllers](https://mlsanity.com/paper/2605.25346/) (2026)
- [Bridging Control with Neural Network Verifier alpha-beta-CROWN: A Tutorial](https://mlsanity.com/paper/2605.26577/) (2026)
- [Worst-Case Distance-Aware Error Bounds for Neural Networks](https://mlsanity.com/paper/2510.22021/) (2025)

## Abstract (arXiv)

We study the problem of learning verifiably Lyapunov-stable neural controllers that provably satisfy the Lyapunov asymptotic stability condition within a region-of-attraction (ROA). Unlike previous works that adopted counterexample-guided training without considering the computation of verification in training, we introduce Certified Training with Branch-and-Bound (CT-BaB), a new certified training framework that optimizes certified bounds, thereby reducing the discrepancy between training and test-time verification that also computes certified bounds. To achieve a relatively global guarantee on an entire input region-of-interest, we propose a training-time BaB technique that maintains a dynamic training dataset and adaptively splits hard input subregions into smaller ones, to tighten certified bounds and ease the training. Meanwhile, subregions created by the training-time BaB also inform test-time verification, for a more efficient training-aware verification. We demonstrate that CT-BaB yields verification-friendly models that can be more efficiently verified at test time while achieving stronger verifiable guarantees with larger ROA. On the largest output-feedback 2D Quadrotor system experimented, CT-BaB reduces verification time by over 11X relative to the previous state-of-the-art baseline using Counterexample Guided Inductive Synthesis (CEGIS), while achieving 164X larger ROA. Code is available at https://github.com/shizhouxing/CT-BaB.