# Reachability analysis of neural networks using mixed monotonicity

arXiv: https://arxiv.org/abs/2111.07683 · HTML: https://mlsanity.com/paper/2111.07683/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 23 references that are in the map.
- First posted on arXiv 15 Nov 2021.

## Builds on (most cited first, 12 of 12)

- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Certified Defenses against Adversarial Examples](https://mlsanity.com/paper/1801.09344/) (2018)
- [Algorithms for Verifying Deep Neural Networks](https://mlsanity.com/paper/1903.06758/) (2019)
- [Formal Security Analysis of Neural Networks using Symbolic Intervals](https://mlsanity.com/paper/1804.10829/) (2018)
- [Efficient Formal Safety Analysis of Neural Networks](https://mlsanity.com/paper/1809.08098/) (2018)
- [NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2004.05519/) (2020)
- [A Dual Approach to Scalable Verification of Deep Networks](https://mlsanity.com/paper/1803.06567/) (2018)
- [A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks](https://mlsanity.com/paper/1902.08722/) (2019)
- [The Second International Verification of Neural Networks Competition (VNN-COMP 2021): Summary and Results](https://mlsanity.com/paper/2109.00498/) (2021)
- [Reachable Set Estimation for Neural Network Control Systems: A Simulation-Guided Approach](https://mlsanity.com/paper/2004.12273/) (2020)
- [Verification for Machine Learning, Autonomy, and Neural Networks Survey](https://mlsanity.com/paper/1810.01989/) (2018)
- [A Programmatic and Semantic Approach to Explaining and DebuggingNeural Network Based Object Detectors](https://mlsanity.com/paper/1912.00289/) (2019)

## Built on by (most cited first, 3 of 3)

- [Mixed Monotonicity Reachability Analysis of Neural ODE: A Trade-Off Between Tightness and Efficiency](https://mlsanity.com/paper/2510.17859/) (2025)
- [Reachability Analysis of Neural Networks with Uncertain Parameters](https://mlsanity.com/paper/2303.07917/) (2023)
- [Certifying Robustness of Graph Convolutional Networks for Node Perturbation with Polyhedra Abstract Interpretation](https://mlsanity.com/paper/2405.08645/) (2024)

## Abstract (arXiv)

This paper presents a new reachability analysis approach to compute interval over-approximations of the output set of feedforward neural networks with input uncertainty. We adapt to neural networks an existing mixed-monotonicity method for the reachability analysis of dynamical systems and apply it to each partial network within the main network. This ensures that the intersection of the obtained results is the tightest interval over-approximation of the output of each layer that can be obtained using mixed-monotonicity on any partial network decomposition. Unlike other tools in the literature focusing on small classes of piecewise-affine or monotone activation functions, the main strength of our approach is its generality: it can handle neural networks with any Lipschitz-continuous activation function. In addition, the simplicity of our framework allows users to very easily add unimplemented activation functions, by simply providing the function, its derivative and the global argmin and argmax of the derivative. Our algorithm is compared to five other interval-based tools (Interval Bound Propagation, ReluVal, Neurify, VeriNet, CROWN) on both existing benchmarks and two sets of small and large randomly generated networks for four activation functions (ReLU, TanH, ELU, SiLU).