# Safety Verification for Neural Networks Based on Set-boundary Analysis

arXiv: https://arxiv.org/abs/2210.04175 · HTML: https://mlsanity.com/paper/2210.04175/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 48 references that are in the map.
- First posted on arXiv 9 Oct 2022.

## Builds on (most cited first, 28 of 28)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [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)
- [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)
- [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)
- [GoTube: Scalable Stochastic Verification of Continuous-Depth Models](https://mlsanity.com/paper/2107.08467/) (2021)
- [Lagrangian Reachtubes: The Next Generation](https://mlsanity.com/paper/2012.07458/) (2020)
- [Grounding Spatio-Temporal Language with Transformers](https://mlsanity.com/paper/2106.08858/) (2021)
- [Over- and Under-Approximating Reachable Sets for Perturbed Delay Differential Equations](https://mlsanity.com/paper/1812.11718/) (2018)

## Built on by (most cited first, 5 of 5)

- [Repairing Deep Neural Networks Based on Behavior Imitation](https://mlsanity.com/paper/2305.03365/) (2023)
- [Bridging Neural ODE and ResNet: A Formal Error Bound for Safety Verification](https://mlsanity.com/paper/2506.03227/) (2025)
- [Mixed Monotonicity Reachability Analysis of Neural ODE: A Trade-Off Between Tightness and Efficiency](https://mlsanity.com/paper/2510.17859/) (2025)
- [An Automata-Theoretic Approach to Synthesizing Binarized Neural Networks](https://mlsanity.com/paper/2307.15907/) (2023)
- [UR4NNV: Neural Network Verification, Under-approximation Reachability Works!](https://mlsanity.com/paper/2401.12550/) (2024)

## 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 of NNs is mainly exploited, which establishes a relationship mapping boundaries to boundaries. The exploitation of this property 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 NNs. Notable representations are invertible residual networks (i-ResNets) and Neural ordinary differential equations (Neural ODEs). For these NNs, our set-boundary reachability method only needs to perform reachability analysis on the boundary of the input set. For NNs which do not feature this property 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.