# A Measure Theoretical Approach to the Mean-field Maximum Principle for Training NeurODEs

arXiv: https://arxiv.org/abs/2107.08707 · HTML: https://mlsanity.com/paper/2107.08707/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2021.

## Built on by (most cited first, 11 of 11)

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- [Variational formulations of ODE-Net as a mean-field optimal control problem and existence results](https://mlsanity.com/paper/2303.05924/) (2023)
- [From NeurODEs to AutoencODEs: a mean-field control framework for width-varying Neural Networks](https://mlsanity.com/paper/2307.02279/) (2023)
- [Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs](https://mlsanity.com/paper/2311.01404/) (2023)
- [Geometric Foundations of Tuning without Forgetting in Neural ODEs](https://mlsanity.com/paper/2509.03474/) (2025)
- [Training-Induced Escape from Token Clustering in a Mean-Field Formulation of Transformers](https://mlsanity.com/paper/2605.07772/) (2026)
- [Taming Epilepsy: Mean Field Control of Whole-Brain Dynamics](https://mlsanity.com/paper/2603.18035/) (2026)
- [Dropout and Random Gradient Masking Are Asymptotically Equivalent in Large ResNets](https://mlsanity.com/paper/2607.16761/) (2026)

## Abstract (arXiv)

In this paper we consider a measure-theoretical formulation of the training of NeurODEs in the form of a mean-field optimal control with L2-regularization of the control. We derive first order optimality conditions for the NeurODE training problem in the form of a mean-field maximum principle, and show that it admits a unique control solution, which is Lipschitz continuous in time. As a consequence of this uniqueness property, the mean-field maximum principle also provides a strong quantitative generalization error for finite sample approximations. Our derivation of the mean-field maximum principle is much simpler than the ones currently available in the literature for mean-field optimal control problems, and is based on a generalized Lagrange multiplier theorem on convex sets of spaces of measures. The latter is also new, and can be considered as a result of independent interest.