# Approximate Bisimulation Relations for Neural Networks and Application to Assured Neural Network Compression

arXiv: https://arxiv.org/abs/2202.01214 · HTML: https://mlsanity.com/paper/2202.01214/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 16 references that are in the map.
- First posted on arXiv 2 Feb 2022.

## Builds on (most cited first, 9 of 9)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [A Survey on Deep Learning in Medical Image Analysis](https://mlsanity.com/paper/1702.05747/) (2017)
- [End to End Learning for Self-Driving Cars](https://mlsanity.com/paper/1604.07316/) (2016)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2004.05519/) (2020)
- [Output Reachable Set Estimation and Verification for Multi-Layer Neural Networks](https://mlsanity.com/paper/1708.03322/) (2017)
- [Reachable Set Estimation for Neural Network Control Systems: A Simulation-Guided Approach](https://mlsanity.com/paper/2004.12273/) (2020)
- [DeepCABAC: A Universal Compression Algorithm for Deep Neural Networks](https://mlsanity.com/paper/1907.11900/) (2019)

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

- [Bridging Dimensions: Confident Reachability for High-Dimensional Controllers](https://mlsanity.com/paper/2311.04843/) (2023)
- [Bridging Neural ODE and ResNet: A Formal Error Bound for Safety Verification](https://mlsanity.com/paper/2506.03227/) (2025)
- [Compression Repair for Feedforward Neural Networks Based on Model Equivalence Evaluation](https://mlsanity.com/paper/2402.11737/) (2024)

## Abstract (arXiv)

In this paper, we propose a concept of approximate bisimulation relation for feedforward neural networks. In the framework of approximate bisimulation relation, a novel neural network merging method is developed to compute the approximate bisimulation error between two neural networks based on reachability analysis of neural networks. The developed method is able to quantitatively measure the distance between the outputs of two neural networks with the same inputs. Then, we apply the approximate bisimulation relation results to perform neural networks model reduction and compute the compression precision, i.e., assured neural networks compression. At last, using the assured neural network compression, we accelerate the verification processes of ACAS Xu neural networks to illustrate the effectiveness and advantages of our proposed approximate bisimulation approach.