# Verisig: verifying safety properties of hybrid systems with neural network controllers

arXiv: https://arxiv.org/abs/1811.01828 · HTML: https://mlsanity.com/paper/1811.01828/

## Key facts

- Cited by 84 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2018.

## Built on by (most cited first, 30 of 84)

- [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)
- [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)
- [Stability Verification in Stochastic Control Systems via Neural Network Supermartingales](https://mlsanity.com/paper/2112.09495/) (2021)
- [Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopes](https://mlsanity.com/paper/2207.02715/) (2022)
- [Backward Reachability Analysis of Neural Feedback Loops: Techniques for Linear and Nonlinear Systems](https://mlsanity.com/paper/2209.14076/) (2022)
- [Closed-loop Analysis of Vision-based Autonomous Systems: A Case Study](https://mlsanity.com/paper/2302.04634/) (2023)
- [CertiFair: A Framework for Certified Global Fairness of Neural Networks](https://mlsanity.com/paper/2205.09927/) (2022)
- [Neural Termination Analysis](https://mlsanity.com/paper/2102.03824/) (2021)
- [Toward Certified Robustness Against Real-World Distribution Shifts](https://mlsanity.com/paper/2206.03669/) (2022)
- [Dynamic Model Predictive Shielding for Provably Safe Reinforcement Learning](https://mlsanity.com/paper/2405.13863/) (2024)
- [Neural Network Verification in Control](https://mlsanity.com/paper/2110.01388/) (2021)
- [Vehicle: Bridging the Embedding Gap in the Verification of Neuro-Symbolic Programs](https://mlsanity.com/paper/2401.06379/) (2024)
- [Infinite Time Horizon Safety of Bayesian Neural Networks](https://mlsanity.com/paper/2111.03165/) (2021)
- [Reachability Analysis of a General Class of Neural Ordinary Differential Equations](https://mlsanity.com/paper/2207.06531/) (2022)
- [Conformal Quantitative Predictive Monitoring of STL Requirements for Stochastic Processes](https://mlsanity.com/paper/2211.02375/) (2022)
- [Risk-averse autonomous systems: A brief history and recent developments from the perspective of optimal control](https://mlsanity.com/paper/2109.08947/) (2021)
- [A Simple and Efficient Sampling-based Algorithm for General Reachability Analysis](https://mlsanity.com/paper/2112.05745/) (2021)
- [POLAR-Express: Efficient and Precise Formal Reachability Analysis of Neural-Network Controlled Systems](https://mlsanity.com/paper/2304.01218/) (2023)
- [Neural Network Verification is a Programming Language Challenge](https://mlsanity.com/paper/2501.05867/) (2025)
- [Learning Density Distribution of Reachable States for Autonomous Systems](https://mlsanity.com/paper/2109.06728/) (2021)
- [Efficient Reachability Analysis of Closed-Loop Systems with Neural Network Controllers](https://mlsanity.com/paper/2101.01815/) (2021)
- [Safe Neurosymbolic Learning with Differentiable Symbolic Execution](https://mlsanity.com/paper/2203.07671/) (2022)
- [Backward Reachability Analysis for Neural Feedback Loops](https://mlsanity.com/paper/2204.08319/) (2022)
- [Interval Reachability of Nonlinear Dynamical Systems with Neural Network Controllers](https://mlsanity.com/paper/2301.07912/) (2023)
- [Fast BATLLNN: Fast Box Analysis of Two-Level Lattice Neural Networks](https://mlsanity.com/paper/2111.09293/) (2021)
- [Neural Network Compression of ACAS Xu Early Prototype is Unsafe: Closed-Loop Verification through Quantized State Backreachability](https://mlsanity.com/paper/2201.06626/) (2022)
- [Reachability Analysis of Neural Network Control Systems](https://mlsanity.com/paper/2301.12100/) (2023)
- [Neural Predictive Monitoring under Partial Observability](https://mlsanity.com/paper/2108.07134/) (2021)

## Abstract (arXiv)

This paper presents Verisig, a hybrid system approach to verifying safety properties of closed-loop systems using neural networks as controllers. Although techniques exist for verifying input/output properties of the neural network itself, these methods cannot be used to verify properties of the closed-loop system (since they work with piecewise-linear constraints that do not capture non-linear plant dynamics). To overcome this challenge, we focus on sigmoid-based networks and exploit the fact that the sigmoid is the solution to a quadratic differential equation, which allows us to transform the neural network into an equivalent hybrid system. By composing the network's hybrid system with the plant's, we transform the problem into a hybrid system verification problem which can be solved using state-of-the-art reachability tools. We show that reachability is decidable for networks with one hidden layer and decidable for general networks if Schanuel's conjecture is true. We evaluate the applicability and scalability of Verisig in two case studies, one from reinforcement learning and one in which the neural network is used to approximate a model predictive controller.