# Reachable Set Estimation and Safety Verification for Piecewise Linear Systems with Neural Network Controllers

arXiv: https://arxiv.org/abs/1802.06981 · HTML: https://mlsanity.com/paper/1802.06981/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2018.

## Built on by (most cited first, 15 of 15)

- [OVERT: An Algorithm for Safety Verification of Neural Network Control Policies for Nonlinear Systems](https://mlsanity.com/paper/2108.01220/) (2021)
- [Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes](https://mlsanity.com/paper/2112.09197/) (2021)
- [Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopes](https://mlsanity.com/paper/2207.02715/) (2022)
- [Reliably-stabilizing piecewise-affine neural network controllers](https://mlsanity.com/paper/2111.07183/) (2021)
- [Neural Abstractions](https://mlsanity.com/paper/2301.11683/) (2023)
- [Efficient Reachability Analysis of Closed-Loop Systems with Neural Network Controllers](https://mlsanity.com/paper/2101.01815/) (2021)
- [SyReNN: A Tool for Analyzing Deep Neural Networks](https://mlsanity.com/paper/2101.03263/) (2021)
- [ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog](https://mlsanity.com/paper/2306.17418/) (2023)
- [BERN-NN: Tight Bound Propagation For Neural Networks Using Bernstein Polynomial Interval Arithmetic](https://mlsanity.com/paper/2211.14438/) (2022)
- [Generating Probabilistic Safety Guarantees for Neural Network Controllers](https://mlsanity.com/paper/2103.01203/) (2021)
- [Hybrid Zonotopes Exactly Represent ReLU Neural Networks](https://mlsanity.com/paper/2304.02755/) (2023)
- [Dual Graphs of Polyhedral Decompositions for the Detection of Adversarial Attacks](https://mlsanity.com/paper/2211.13305/) (2022)
- [Robust Optimization Framework for Training Shallow Neural Networks Using Reachability Method](https://mlsanity.com/paper/2107.12801/) (2021)
- [On the Trade-off Between Efficiency and Precision of Neural Abstraction](https://mlsanity.com/paper/2307.15546/) (2023)
- [Compression Repair for Feedforward Neural Networks Based on Model Equivalence Evaluation](https://mlsanity.com/paper/2402.11737/) (2024)

## Abstract (arXiv)

In this work, the reachable set estimation and safety verification problems for a class of piecewise linear systems equipped with neural network controllers are addressed. The neural network is considered to consist of Rectified Linear Unit (ReLU) activation functions. A layer-by-layer approach is developed for the output reachable set computation of ReLU neural networks. The computation is formulated in the form of a set of manipulations for a union of polytopes. Based on the output reachable set for neural network controllers, the output reachable set for a piecewise linear feedback control system can be estimated iteratively for a given finite-time interval. With the estimated output reachable set, the safety verification for piecewise linear systems with neural network controllers can be performed by checking the existence of intersections of unsafe regions and output reach set. A numerical example is presented to illustrate the effectiveness of our approach.