# Bridging Neural ODE and ResNet: A Formal Error Bound for Safety Verification

arXiv: https://arxiv.org/abs/2506.03227 · HTML: https://mlsanity.com/paper/2506.03227/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 29 references that are in the map.
- First posted on arXiv 3 Jun 2025.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [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)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Safety Verification of Deep Neural Networks](https://mlsanity.com/paper/1610.06940/) (2016)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2004.05519/) (2020)
- [Do Residual Neural Networks discretize Neural Ordinary Differential Equations?](https://mlsanity.com/paper/2205.14612/) (2022)
- [On The Verification of Neural ODEs with Stochastic Guarantees](https://mlsanity.com/paper/2012.08863/) (2020)
- [Generalization bounds for neural ordinary differential equations and deep residual networks](https://mlsanity.com/paper/2305.06648/) (2023)
- [Implicit regularization of deep residual networks towards neural ODEs](https://mlsanity.com/paper/2309.01213/) (2023)
- [Comprehensive Review of Neural Differential Equations for Time Series Analysis](https://mlsanity.com/paper/2502.09885/) (2025)
- [Reachability Analysis of a General Class of Neural Ordinary Differential Equations](https://mlsanity.com/paper/2207.06531/) (2022)
- [Safety Verification for Neural Networks Based on Set-boundary Analysis](https://mlsanity.com/paper/2210.04175/) (2022)
- [Approximate Bisimulation Relations for Neural Networks and Application to Assured Neural Network Compression](https://mlsanity.com/paper/2202.01214/) (2022)

## 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)
- [TNODEV: Toolbox for Neural ODE Verification](https://mlsanity.com/paper/2606.16567/) (2026)
- [Robustness Verification of an Autonomous Underwater Vehicle-based Plankton Classifier](https://mlsanity.com/paper/2607.04453/) (2026)

## Abstract (arXiv)

A neural ordinary differential equation (neural ODE) is a machine learning model that is commonly described as a continuous-depth generalization of a residual network (ResNet) with a single residual block, or conversely, the ResNet can be seen as the Euler discretization of the neural ODE. These two models are therefore strongly related in a way that the behaviors of either model are considered to be an approximation of the behaviors of the other. In this work, we establish a more formal relationship between these two models by bounding the approximation error between two such related models. The obtained error bound then allows us to use one of the models as a verification proxy for the other, without running the verification tools twice: if the reachable output set expanded by the error bound satisfies a safety property on one of the models, this safety property is then guaranteed to be also satisfied on the other model. This feature is fully reversible, and the initial safety verification can be run indifferently on either of the two models. This novel approach is illustrated on a numerical example of a fixed-point attractor system modeled as a neural ODE.