# Efficient Reachability Analysis of Closed-Loop Systems with Neural Network Controllers

arXiv: https://arxiv.org/abs/2101.01815 · HTML: https://mlsanity.com/paper/2101.01815/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 39 references that are in the map.
- First posted on arXiv 5 Jan 2021.

## Builds on (most cited first, 21 of 21)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [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)
- [Certified Defenses against Adversarial Examples](https://mlsanity.com/paper/1801.09344/) (2018)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Safety Verification of Deep Neural Networks](https://mlsanity.com/paper/1610.06940/) (2016)
- [Towards Fast Computation of Certified Robustness for ReLU Networks](https://mlsanity.com/paper/1804.09699/) (2018)
- [Formal Verification of Piece-Wise Linear Feed-Forward Neural Networks](https://mlsanity.com/paper/1705.01320/) (2017)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [An approach to reachability analysis for feed-forward ReLU neural networks](https://mlsanity.com/paper/1706.07351/) (2017)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Graduated Non-Convexity for Robust Spatial Perception: From Non-Minimal Solvers to Global Outlier Rejection](https://mlsanity.com/paper/1909.08605/) (2019)
- [Verisig: verifying safety properties of hybrid systems with neural network controllers](https://mlsanity.com/paper/1811.01828/) (2018)
- [Output Reachable Set Estimation and Verification for Multi-Layer Neural Networks](https://mlsanity.com/paper/1708.03322/) (2017)
- [Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming](https://mlsanity.com/paper/2004.07876/) (2020)
- [Reachable Set Estimation for Neural Network Control Systems: A Simulation-Guided Approach](https://mlsanity.com/paper/2004.12273/) (2020)
- [ReachNN: Reachability Analysis of Neural-Network Controlled Systems](https://mlsanity.com/paper/1906.10654/) (2019)
- [Reachable Set Estimation and Safety Verification for Piecewise Linear Systems with Neural Network Controllers](https://mlsanity.com/paper/1802.06981/) (2018)

## Built on by (most cited first, 10 of 10)

- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [Neural Network Verification in Control](https://mlsanity.com/paper/2110.01388/) (2021)
- [Learning Density Distribution of Reachable States for Autonomous Systems](https://mlsanity.com/paper/2109.06728/) (2021)
- [Verification of Neural Network Control Systems using Symbolic Zonotopes and Polynotopes](https://mlsanity.com/paper/2306.14619/) (2023)
- [Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis](https://mlsanity.com/paper/2406.04476/) (2024)
- [Case Studies for Computing Density of Reachable States for Safe Autonomous Motion Planning](https://mlsanity.com/paper/2209.08073/) (2022)
- [Refining Perception Contracts: Case Studies in Vision-based Safe Auto-landing](https://mlsanity.com/paper/2311.08652/) (2023)
- [How Safe Will I Be Given What I Saw? Calibrated Prediction of Safety Chances for Image-Controlled Autonomy](https://mlsanity.com/paper/2508.09346/) (2025)
- [Delay Independent Safe Control with Neural Networks: Positive Lur'e Certificates for Risk Aware Autonomy](https://mlsanity.com/paper/2510.06661/) (2025)
- [Robust Verification of Controllers under State Uncertainty via Hamilton-Jacobi Reachability Analysis](https://mlsanity.com/paper/2511.14755/) (2025)

## Abstract (arXiv)

Neural Networks (NNs) can provide major empirical performance improvements for robotic systems, but they also introduce challenges in formally analyzing those systems' safety properties. In particular, this work focuses on estimating the forward reachable set of closed-loop systems with NN controllers. Recent work provides bounds on these reachable sets, yet the computationally efficient approaches provide overly conservative bounds (thus cannot be used to verify useful properties), whereas tighter methods are too intensive for online computation. This work bridges the gap by formulating a convex optimization problem for reachability analysis for closed-loop systems with NN controllers. While the solutions are less tight than prior semidefinite program-based methods, they are substantially faster to compute, and some of the available computation time can be used to refine the bounds through input set partitioning, which more than overcomes the tightness gap. The proposed framework further considers systems with measurement and process noise, thus being applicable to realistic systems with uncertainty. Finally, numerical comparisons show 10× reduction in conservatism in frac12 of the computation time compared to the state-of-the-art, and the ability to handle various sources of uncertainty is highlighted on a quadrotor model.