# Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs

arXiv: https://arxiv.org/abs/2311.01404 · HTML: https://mlsanity.com/paper/2311.01404/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 51 references that are in the map.
- First posted on arXiv 2 Nov 2023.

## Builds on (most cited first, 25 of 25)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Learning normalizing flows from Entropy-Kantorovich potentials](https://mlsanity.com/paper/2006.06033/) (2020)
- [Quadratically regularized optimal transport](https://mlsanity.com/paper/1903.01112/) (2019)
- [Large-time asymptotics in deep learning](https://mlsanity.com/paper/2008.02491/) (2020)
- [A Measure Theoretical Approach to the Mean-field Maximum Principle for Training NeurODEs](https://mlsanity.com/paper/2107.08707/) (2021)
- [Control in the spaces of ensembles of points](https://mlsanity.com/paper/1907.00905/) (2019)
- [Non-asymptotic convergence bounds for Wasserstein approximation using point clouds](https://mlsanity.com/paper/2106.07911/) (2021)
- [Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems](https://mlsanity.com/paper/2110.12393/) (2021)
- [A gradient flow equation for optimal control problems with end-point cost](https://mlsanity.com/paper/2107.00556/) (2021)
- [Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows](https://mlsanity.com/paper/2209.10873/) (2022)
- [Shape deformation analysis from the optimal control viewpoint](https://mlsanity.com/paper/1401.0661/) (2014)
- [Computing Kantorovich-Wasserstein Distances on d-dimensional histograms using (d+1)-partite graphs](https://mlsanity.com/paper/1805.07416/) (2018)
- [Optimal Control of ensembles of dynamical systems](https://mlsanity.com/paper/2203.16485/) (2022)
- [From NeurODEs to AutoencODEs: a mean-field control framework for width-varying Neural Networks](https://mlsanity.com/paper/2307.02279/) (2023)
- [Embedding Capabilities of Neural ODEs](https://mlsanity.com/paper/2308.01213/) (2023)
- [Approximation Theory, Computing, and Deep Learning on the Wasserstein Space](https://mlsanity.com/paper/2310.19548/) (2023)
- [On the Structure of Optimal Transportation Plans between Discrete Measures](https://mlsanity.com/paper/2104.13106/) (2021)

## Built on by (most cited first, 3 of 3)

- [Constructive approximate transport maps with normalizing flows](https://mlsanity.com/paper/2412.19366/) (2024)
- [Complex normalizing flows can almost be information K"ahler-Ricci flows](https://mlsanity.com/paper/2604.17954/) (2026)
- [Deep learning from a control vantage point](https://mlsanity.com/paper/2610.14385/) (2026)

## Abstract (arXiv)

In this paper, we consider the problem of recovering the W2-optimal transport map T between absolutely continuous measures μ,nu∈P(ℝn) as the flow of a linear-control neural ODE, where the control depends only on the time variable and takes values in a finite-dimensional space. We first show that, under suitable assumptions on μ,nu and on the controlled vector fields governing the neural ODE, the optimal transport map is contained in the C0c-closure of the flows generated by the system. Then, we tackle the problem under the assumption that only discrete approximations of μN,nuN of the original measures μ,nu are available: we formulate approximated optimal control problems, and we show that their solutions give flows that approximate the original optimal transport map T. In the framework of generative models, the approximating flow constructed here can be seen as a `Normalizing Flow', which usually refers to the task of providing invertible transport maps between probability measures by means of deep neural networks. We propose an iterative numerical scheme based on the Pontryagin Maximum Principle for the resolution of the optimal control problem, resulting in a method for the practical computation of the approximated optimal transport map, and we test it on a two-dimensional example.