# A minimax optimal control approach for robust neural ODEs

arXiv: https://arxiv.org/abs/2310.17584 · HTML: https://mlsanity.com/paper/2310.17584/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 30 references that are in the map.
- First posted on arXiv 26 Oct 2023.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [The Space of Transferable Adversarial Examples](https://mlsanity.com/paper/1704.03453/) (2017)
- [Tilted Empirical Risk Minimization](https://mlsanity.com/paper/2007.01162/) (2020)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Deep learning as optimal control problems: models and numerical methods](https://mlsanity.com/paper/1904.05657/) (2019)
- [Understanding Adversarial Training: Increasing Local Stability of Neural Nets through Robust Optimization](https://mlsanity.com/paper/1511.05432/) (2015)
- [Existence and Minimax Theorems for Adversarial Surrogate Risks in Binary Classification](https://mlsanity.com/paper/2206.09098/) (2022)
- [A Measure Theoretical Approach to the Mean-field Maximum Principle for Training NeurODEs](https://mlsanity.com/paper/2107.08707/) (2021)
- [On the regularized risk of distributionally robust learning over deep neural networks](https://mlsanity.com/paper/2109.06294/) (2021)
- [From NeurODEs to AutoencODEs: a mean-field control framework for width-varying Neural Networks](https://mlsanity.com/paper/2307.02279/) (2023)
- [An Abstract Maximum Principle for constrained minimum problems](https://mlsanity.com/paper/2310.09845/) (2023)

## Built on by (most cited first, 4 of 4)

- [Lyapunov Probes for Hallucination Detection in Large Foundation Models](https://mlsanity.com/paper/2603.06081/) (2026)
- [How to beat a Bayesian adversary](https://mlsanity.com/paper/2407.08678/) (2024)
- [Control Disturbance Rejection in Neural ODEs](https://mlsanity.com/paper/2509.18034/) (2025)
- [Time-optimal neural feedback control of nilpotent systems as a binary classification problem](https://mlsanity.com/paper/2503.17581/) (2025)

## Abstract (arXiv)

In this paper, we address the adversarial training of neural ODEs from a robust control perspective. This is an alternative to the classical training via empirical risk minimization, and it is widely used to enforce reliable outcomes for input perturbations. Neural ODEs allow the interpretation of deep neural networks as discretizations of control systems, unlocking powerful tools from control theory for the development and the understanding of machine learning. In this specific case, we formulate the adversarial training with perturbed data as a minimax optimal control problem, for which we derive first order optimality conditions in the form of Pontryagin's Maximum Principle. We provide a novel interpretation of robust training leading to an alternative weighted technique, which we test on a low-dimensional classification task.