# Control of neural transport for normalizing flows

arXiv: https://arxiv.org/abs/2307.07817 · HTML: https://mlsanity.com/paper/2307.07817/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jul 2023.

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

- [Interplay between depth and width for interpolation in neural ODEs](https://mlsanity.com/paper/2401.09902/) (2024)
- [Cluster-based classification with neural ODEs via control](https://mlsanity.com/paper/2312.13807/) (2023)
- [Constructive approximate transport maps with normalizing flows](https://mlsanity.com/paper/2412.19366/) (2024)
- [Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics](https://mlsanity.com/paper/2412.01619/) (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)
- [Machine Learning and Control: Foundations, Advances, and Perspectives](https://mlsanity.com/paper/2510.03303/) (2025)
- [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)

Inspired by normalizing flows, we analyze the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfill, at any time instance, a simple neural network ansatz. The L^1 approximate controllability property is proved, showing that any probability density can be driven arbitrarily close to any other one in any time horizon. The control vector fields are built explicitly and inductively and this provides quantitative estimates on their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.