# Constructive approximate transport maps with normalizing flows

arXiv: https://arxiv.org/abs/2412.19366 · HTML: https://mlsanity.com/paper/2412.19366/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 42 references that are in the map.
- First posted on arXiv 26 Dec 2024.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Improving Variational Auto-Encoders using Householder Flow](https://mlsanity.com/paper/1611.09630/) (2016)
- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [Deep neural networks, generic universal interpolation, and controlled ODEs](https://mlsanity.com/paper/1908.07838/) (2019)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Interpolation, Approximation and Controllability of Deep Neural Networks](https://mlsanity.com/paper/2309.06015/) (2023)
- [Neural ODE Control for Trajectory Approximation of Continuity Equation](https://mlsanity.com/paper/2205.09241/) (2022)
- [Control of neural transport for normalizing flows](https://mlsanity.com/paper/2307.07817/) (2023)
- [Interplay between depth and width for interpolation in neural ODEs](https://mlsanity.com/paper/2401.09902/) (2024)
- [Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems](https://mlsanity.com/paper/2110.12393/) (2021)
- [Approximate and exact controllability of the continuity equation with a localized vector field](https://mlsanity.com/paper/1710.09287/) (2017)
- [Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs](https://mlsanity.com/paper/2311.01404/) (2023)
- [Some Remarks on Controllability of the Liouville Equation](https://mlsanity.com/paper/2404.14683/) (2024)

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

- [Continuous Temporal Learning of Probability Distributions via Neural ODEs with Applications in Continuous Glucose Monitoring Data](https://mlsanity.com/paper/2505.08698/) (2025)
- [Machine Learning and Control: Foundations, Advances, and Perspectives](https://mlsanity.com/paper/2510.03303/) (2025)
- [Convergence, design and training of continuous-time dropout as a random batch method](https://mlsanity.com/paper/2510.13134/) (2025)
- [Constructive interpolation and generalization rates for neural ODEs: a control perspective](https://mlsanity.com/paper/2606.00469/) (2026)

## Abstract (arXiv)

We study an approximate controllability problem for the continuity equation and its application to constructing transport maps with normalizing flows. Specifically, we construct time-dependent controls θ=(w, a, b) in the vector field xmapsto w(a^top x + b)_+ to approximately transport a known base density ρB to a target density ρ_*. The approximation error is measured in relative entropy, and θ are constructed piecewise constant, with bounds on the number of switches being provided. Our main result relies on an assumption on the relative tail decay of ρ_* and ρB, and provides hints on characterizing the reachable space of the continuity equation in relative entropy.