# Output Reachable Set Estimation and Verification for Multi-Layer Neural Networks

arXiv: https://arxiv.org/abs/1708.03322 · HTML: https://mlsanity.com/paper/1708.03322/

## Key facts

- Cited by 80 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Aug 2017.

## Built on by (most cited first, 30 of 80)

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- [Robust Machine Learning Systems: Challenges, Current Trends, Perspectives, and the Road Ahead](https://mlsanity.com/paper/2101.02559/) (2021)
- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [OVERT: An Algorithm for Safety Verification of Neural Network Control Policies for Nonlinear Systems](https://mlsanity.com/paper/2108.01220/) (2021)
- [A Review of Formal Methods applied to Machine Learning](https://mlsanity.com/paper/2104.02466/) (2021)
- [Efficient Neural Network Analysis with Sum-of-Infeasibilities](https://mlsanity.com/paper/2203.11201/) (2022)
- [DNNV: A Framework for Deep Neural Network Verification](https://mlsanity.com/paper/2105.12841/) (2021)
- [Neural Network Repair with Reachability Analysis](https://mlsanity.com/paper/2108.04214/) (2021)
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- [PRoA: A Probabilistic Robustness Assessment against Functional Perturbations](https://mlsanity.com/paper/2207.02036/) (2022)
- [Convex Bounds on the Softmax Function with Applications to Robustness Verification](https://mlsanity.com/paper/2303.01713/) (2023)
- [Verification-Aided Deep Ensemble Selection](https://mlsanity.com/paper/2202.03898/) (2022)
- [BDD4BNN: A BDD-based Quantitative Analysis Framework for Binarized Neural Networks](https://mlsanity.com/paper/2103.07224/) (2021)
- [CertiFair: A Framework for Certified Global Fairness of Neural Networks](https://mlsanity.com/paper/2205.09927/) (2022)
- [Toward Certified Robustness Against Real-World Distribution Shifts](https://mlsanity.com/paper/2206.03669/) (2022)
- [Neural Network Verification in Control](https://mlsanity.com/paper/2110.01388/) (2021)
- [On Recurrent Neural Networks for learning-based control: recent results and ideas for future developments](https://mlsanity.com/paper/2111.13557/) (2021)
- [Exact Verification of ReLU Neural Control Barrier Functions](https://mlsanity.com/paper/2310.09360/) (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)
- [Verifying Generalization in Deep Learning](https://mlsanity.com/paper/2302.05745/) (2023)
- [Efficient Reachability Analysis of Closed-Loop Systems with Neural Network Controllers](https://mlsanity.com/paper/2101.01815/) (2021)
- [Benchmarking Safe Deep Reinforcement Learning in Aquatic Navigation](https://mlsanity.com/paper/2112.10593/) (2021)
- [A Survey on Verification and Validation, Testing and Evaluations of Neurosymbolic Artificial Intelligence](https://mlsanity.com/paper/2401.03188/) (2024)
- [ProTIP: Probabilistic Robustness Verification on Text-to-Image Diffusion Models against Stochastic Perturbation](https://mlsanity.com/paper/2402.15429/) (2024)
- [Fast BATLLNN: Fast Box Analysis of Two-Level Lattice Neural Networks](https://mlsanity.com/paper/2111.09293/) (2021)
- [Provably Tightest Linear Approximation for Robustness Verification of Sigmoid-like Neural Networks](https://mlsanity.com/paper/2208.09872/) (2022)

## Abstract (arXiv)

In this paper, the output reachable estimation and safety verification problems for multi-layer perceptron neural networks are addressed. First, a conception called maximum sensitivity in introduced and, for a class of multi-layer perceptrons whose activation functions are monotonic functions, the maximum sensitivity can be computed via solving convex optimization problems. Then, using a simulation-based method, the output reachable set estimation problem for neural networks is formulated into a chain of optimization problems. Finally, an automated safety verification is developed based on the output reachable set estimation result. An application to the safety verification for a robotic arm model with two joints is presented to show the effectiveness of proposed approaches.