# Mixed Monotonicity Reachability Analysis of Neural ODE: A Trade-Off Between Tightness and Efficiency

arXiv: https://arxiv.org/abs/2510.17859 · HTML: https://mlsanity.com/paper/2510.17859/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 15 Oct 2025.

## Builds on (most cited first, 11 of 11)

- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2004.05519/) (2020)
- [On The Verification of Neural ODEs with Stochastic Guarantees](https://mlsanity.com/paper/2012.08863/) (2020)
- [Reachability Analysis of a General Class of Neural Ordinary Differential Equations](https://mlsanity.com/paper/2207.06531/) (2022)
- [TIRA: Toolbox for Interval Reachability Analysis](https://mlsanity.com/paper/1902.05204/) (2019)
- [A Toolbox for Fast Interval Arithmetic in numpy with an Application to Formal Verification of Neural Network Controlled Systems](https://mlsanity.com/paper/2306.15340/) (2023)
- [Safety Verification for Neural Networks Based on Set-boundary Analysis](https://mlsanity.com/paper/2210.04175/) (2022)
- [Over- and Under-Approximating Reachable Sets for Perturbed Delay Differential Equations](https://mlsanity.com/paper/1812.11718/) (2018)
- [Reachability analysis of neural networks using mixed monotonicity](https://mlsanity.com/paper/2111.07683/) (2021)
- [Bridging Neural ODE and ResNet: A Formal Error Bound for Safety Verification](https://mlsanity.com/paper/2506.03227/) (2025)
- [On sufficient conditions for mixed monotonicity](https://mlsanity.com/paper/1803.04528/) (2018)

## Built on by (most cited first, 2 of 2)

- [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)

Neural ordinary differential equations (neural ODE) are powerful continuous-time machine learning models for depicting the behavior of complex dynamical systems, but their verification remains challenging due to limited reachability analysis tools adapted to them. We propose a novel interval-based reachability method that leverages continuous-time mixed monotonicity techniques for dynamical systems to compute an over-approximation for the neural ODE reachable sets. By exploiting the geometric structure of full initial sets and their boundaries via the homeomorphism property, our approach ensures efficient bound propagation. By embedding neural ODE dynamics into a mixed monotone system, our interval-based reachability approach, implemented in TIRA with single-step, incremental, and boundary-based approaches, provides sound and computationally efficient over-approximations compared with CORA's zonotopes and NNV2.0 star set representations, while trading tightness for efficiency. This trade-off makes our method particularly suited for high-dimensional, real-time, and safety-critical applications. Applying mixed monotonicity to neural ODE reachability analysis paves the way for lightweight formal analysis by leveraging the symmetric structure of monotone embeddings and the geometric simplicity of interval boxes, opening new avenues for scalable verification. This novel approach is illustrated on two numerical examples of a spiral system and a fixed-point attractor system modeled as a neural ODE.