# Real-Time Safe Control of Neural Network Dynamic Models with Sound Approximation

arXiv: https://arxiv.org/abs/2404.13456 · HTML: https://mlsanity.com/paper/2404.13456/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 24 references that are in the map.
- First posted on arXiv 20 Apr 2024.

## Builds on (most cited first, 18 of 18)

- [When to Trust Your Model: Model-Based Policy Optimization](https://mlsanity.com/paper/1906.08253/) (2019)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Neural Network Dynamics for Model-Based Deep Reinforcement Learning with Model-Free Fine-Tuning](https://mlsanity.com/paper/1708.02596/) (2017)
- [Formal Verification of Piece-Wise Linear Feed-Forward Neural Networks](https://mlsanity.com/paper/1705.01320/) (2017)
- [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)
- [Algorithms for Verifying Deep Neural Networks](https://mlsanity.com/paper/1903.06758/) (2019)
- [Scaling provable adversarial defenses](https://mlsanity.com/paper/1805.12514/) (2018)
- [SoK: Certified Robustness for Deep Neural Networks](https://mlsanity.com/paper/2009.04131/) (2020)
- [POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems](https://mlsanity.com/paper/2106.13867/) (2021)
- [Safe Control with Neural Network Dynamic Models](https://mlsanity.com/paper/2110.01110/) (2021)
- [Safe Control Algorithms Using Energy Functions: A Unified Framework, Benchmark, and New Directions](https://mlsanity.com/paper/1908.01883/) (2019)
- [Model-Based Control with Sparse Neural Dynamics](https://mlsanity.com/paper/2312.12791/) (2023)
- [Safe Control Synthesis via Input Constrained Control Barrier Functions](https://mlsanity.com/paper/2104.01704/) (2021)
- [BERN-NN: Tight Bound Propagation For Neural Networks Using Bernstein Polynomial Interval Arithmetic](https://mlsanity.com/paper/2211.14438/) (2022)
- [DeepBern-Nets: Taming the Complexity of Certifying Neural Networks using Bernstein Polynomial Activations and Precise Bound Propagation](https://mlsanity.com/paper/2305.13508/) (2023)
- [System-level Safety Guard: Safe Tracking Control through Uncertain Neural Network Dynamics Models](https://mlsanity.com/paper/2312.06810/) (2023)

## Built on by (most cited first, 5 of 5)

- [Steering Dialogue Dynamics for Robustness against Multi-turn Jailbreaking Attacks](https://mlsanity.com/paper/2503.00187/) (2025)
- [Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation](https://mlsanity.com/paper/2410.16281/) (2024)
- [Verifiable Safety Q-Filters via Hamilton-Jacobi Reachability and Multiplicative Q-Networks](https://mlsanity.com/paper/2506.15693/) (2025)
- [Bridging Control with Neural Network Verifier alpha-beta-CROWN: A Tutorial](https://mlsanity.com/paper/2605.26577/) (2026)
- [Efficient Verification of Neural Control Barrier Functions with Smooth Nonlinear Activations](https://mlsanity.com/paper/2605.07757/) (2026)

## Abstract (arXiv)

Safe control of neural network dynamic models (NNDMs) is important to robotics and many applications. However, it remains challenging to compute an optimal safe control in real time for NNDM. To enable real-time computation, we propose to use a sound approximation of the NNDM in the control synthesis. In particular, we propose Bernstein over-approximated neural dynamics (BOND) based on the Bernstein polynomial over-approximation (BPO) of ReLU activation functions in NNDM. To mitigate the errors introduced by the approximation and to ensure persistent feasibility of the safe control problems, we synthesize a worst-case safety index using the most unsafe approximated state within the BPO relaxation of NNDM offline. For the online real-time optimization, we formulate the first-order Taylor approximation of the nonlinear worst-case safety constraint as an additional linear layer of NNDM with the l2 bounded bias term for the higher-order remainder. Comprehensive experiments with different neural dynamics and safety constraints show that with safety guaranteed, our NNDMs with sound approximation are 10-100 times faster than the safe control baseline that uses mixed integer programming (MIP), validating the effectiveness of the worst-case safety index and scalability of the proposed BOND in real-time large-scale settings. The code is available at https://github.com/intelligent-control-lab/BOND.