# Neural ODE Control for Trajectory Approximation of Continuity Equation

arXiv: https://arxiv.org/abs/2205.09241 · HTML: https://mlsanity.com/paper/2205.09241/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 29 references that are in the map.
- First posted on arXiv 18 May 2022.

## Builds on (most cited first, 14 of 14)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [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)
- [Momentum Residual Neural Networks](https://mlsanity.com/paper/2102.07870/) (2021)
- [Deep learning as optimal control problems: models and numerical methods](https://mlsanity.com/paper/1904.05657/) (2019)
- [ResNets Ensemble via the Feynman-Kac Formalism to Improve Natural and Robust Accuracies](https://mlsanity.com/paper/1811.10745/) (2018)
- [Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theory](https://mlsanity.com/paper/2007.06007/) (2020)
- [Measure-valued spline curves: an optimal transport viewpoint](https://mlsanity.com/paper/1801.03186/) (2018)
- [Mean-Field Neural ODEs via Relaxed Optimal Control](https://mlsanity.com/paper/1912.05475/) (2019)
- [Optimal Control of Continuity Equations](https://mlsanity.com/paper/1506.08932/) (2015)
- [Approximate and exact controllability of the continuity equation with a localized vector field](https://mlsanity.com/paper/1710.09287/) (2017)

## Built on by (most cited first, 9 of 9)

- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [Interplay between depth and width for interpolation in neural ODEs](https://mlsanity.com/paper/2401.09902/) (2024)
- [Double-well Net for Image Segmentation](https://mlsanity.com/paper/2401.00456/) (2023)
- [Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEs](https://mlsanity.com/paper/2305.08849/) (2023)
- [Constructive approximate transport maps with normalizing flows](https://mlsanity.com/paper/2412.19366/) (2024)
- [Control, Optimal Transport and Neural Differential Equations in Supervised Learning](https://mlsanity.com/paper/2503.15105/) (2025)
- [Noise in the reverse process improves the approximation capabilities of diffusion models](https://mlsanity.com/paper/2312.07851/) (2023)
- [Constructive interpolation and generalization rates for neural ODEs: a control perspective](https://mlsanity.com/paper/2606.00469/) (2026)
- [Deep learning from a control vantage point](https://mlsanity.com/paper/2610.14385/) (2026)

## Abstract (arXiv)

We consider the controllability problem for the continuity equation, corresponding to neural ordinary differential equations (ODEs), which describes how a probability measure is pushedforward by the flow. We show that the controlled continuity equation has very strong controllability properties. Particularly, a given solution of the continuity equation corresponding to a bounded Lipschitz vector field defines a trajectory on the set of probability measures. For this trajectory, we show that there exist piecewise constant training weights for a neural ODE such that the solution of the continuity equation corresponding to the neural ODE is arbitrarily close to it. As a corollary to this result, we establish that the continuity equation of the neural ODE is approximately controllable on the set of compactly supported probability measures that are absolutely continuous with respect to the Lebesgue measure.