# Verifying Safety of Neural Networks from Topological Perspectives

arXiv: https://arxiv.org/abs/2306.15403 · HTML: https://mlsanity.com/paper/2306.15403/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 60 references that are in the map.
- First posted on arXiv 27 Jun 2023.

## Builds on (most cited first, 30 of 33)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [BARTScore: Evaluating Generated Text as Text Generation](https://mlsanity.com/paper/2106.11520/) (2021)
- [Interpretation of Neural Networks is Fragile](https://mlsanity.com/paper/1710.10547/) (2017)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Safety Verification of Deep Neural Networks](https://mlsanity.com/paper/1610.06940/) (2016)
- [Formal Verification of Piece-Wise Linear Feed-Forward Neural Networks](https://mlsanity.com/paper/1705.01320/) (2017)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Algorithms for Verifying Deep Neural Networks](https://mlsanity.com/paper/1903.06758/) (2019)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Efficient Formal Safety Analysis of Neural Networks](https://mlsanity.com/paper/1809.08098/) (2018)
- [i-RevNet: Deep Invertible Networks](https://mlsanity.com/paper/1802.07088/) (2018)
- [An approach to reachability analysis for feed-forward ReLU neural networks](https://mlsanity.com/paper/1706.07351/) (2017)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Output Reachable Set Estimation and Verification for Multi-Layer Neural Networks](https://mlsanity.com/paper/1708.03322/) (2017)
- [Panoptic 3D Scene Reconstruction From a Single RGB Image](https://mlsanity.com/paper/2111.02444/) (2021)
- [How to Certify Machine Learning Based Safety-critical Systems? A Systematic Literature Review](https://mlsanity.com/paper/2107.12045/) (2021)
- [Improving Neural Network Verification through Spurious Region Guided Refinement](https://mlsanity.com/paper/2010.07722/) (2020)
- [Output Range Analysis for Deep Neural Networks](https://mlsanity.com/paper/1709.09130/) (2017)
- [ReachNN: Reachability Analysis of Neural-Network Controlled Systems](https://mlsanity.com/paper/1906.10654/) (2019)
- [On The Verification of Neural ODEs with Stochastic Guarantees](https://mlsanity.com/paper/2012.08863/) (2020)
- [Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopes](https://mlsanity.com/paper/2207.02715/) (2022)
- [Reachable Set Computation and Safety Verification for Neural Networks with ReLU Activations](https://mlsanity.com/paper/1712.08163/) (2017)
- [Reachability Analysis of a General Class of Neural Ordinary Differential Equations](https://mlsanity.com/paper/2207.06531/) (2022)
- [Reachability Analysis and Safety Verification for Neural Network Control Systems](https://mlsanity.com/paper/1805.09944/) (2018)
- [Testing and verification of neural-network-based safety-critical control software: A systematic literature review](https://mlsanity.com/paper/1910.06715/) (2019)
- [GoTube: Scalable Stochastic Verification of Continuous-Depth Models](https://mlsanity.com/paper/2107.08467/) (2021)

## Built on by (most cited first, 1 of 1)

- [TNODEV: Toolbox for Neural ODE Verification](https://mlsanity.com/paper/2606.16567/) (2026)

## Abstract (arXiv)

Neural networks (NNs) are increasingly applied in safety-critical systems such as autonomous vehicles. However, they are fragile and are often ill-behaved. Consequently, their behaviors should undergo rigorous guarantees before deployment in practice. In this paper, we propose a set-boundary reachability method to investigate the safety verification problem of NNs from a topological perspective. Given an NN with an input set and a safe set, the safety verification problem is to determine whether all outputs of the NN resulting from the input set fall within the safe set. In our method, the homeomorphism property and the open map property of NNs are mainly exploited, which establish rigorous guarantees between the boundaries of the input set and the boundaries of the output set. The exploitation of these two properties facilitates reachability computations via extracting subsets of the input set rather than the entire input set, thus controlling the wrapping effect in reachability analysis and facilitating the reduction of computation burdens for safety verification. The homeomorphism property exists in some widely used NNs such as invertible residual networks (i-ResNets) and Neural ordinary differential equations (Neural ODEs), and the open map is a less strict property and easier to satisfy compared with the homeomorphism property. For NNs establishing either of these properties, our set-boundary reachability method only needs to perform reachability analysis on the boundary of the input set. Moreover, for NNs that do not feature these properties with respect to the input set, we explore subsets of the input set for establishing the local homeomorphism property and then abandon these subsets for reachability computations. Finally, some examples demonstrate the performance of the proposed method.