# Control, Optimal Transport and Neural Differential Equations in Supervised Learning

arXiv: https://arxiv.org/abs/2503.15105 · HTML: https://mlsanity.com/paper/2503.15105/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 94 references that are in the map.
- First posted on arXiv 19 Mar 2025.

## Builds on (most cited first, 30 of 43)

- [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)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems](https://mlsanity.com/paper/1610.06519/) (2016)
- [Low-Rank Sinkhorn Factorization](https://mlsanity.com/paper/2103.04737/) (2021)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Scalable Unbalanced Optimal Transport using Generative Adversarial Networks](https://mlsanity.com/paper/1810.11447/) (2018)
- [Massively scalable Sinkhorn distances via the Nystr"om method](https://mlsanity.com/paper/1812.05189/) (2018)
- [Continuous-in-Depth Neural Networks](https://mlsanity.com/paper/2008.02389/) (2020)
- [Momentum Residual Neural Networks](https://mlsanity.com/paper/2102.07870/) (2021)
- [Layer-Parallel Training of Deep Residual Neural Networks](https://mlsanity.com/paper/1812.04352/) (2018)
- [Deep learning as optimal control problems: models and numerical methods](https://mlsanity.com/paper/1904.05657/) (2019)
- [Structure preserving deep learning](https://mlsanity.com/paper/2006.03364/) (2020)
- [Unbalanced Optimal Transport through Non-negative Penalized Linear Regression](https://mlsanity.com/paper/2106.04145/) (2021)
- [Faster Unbalanced Optimal Transport: Translation invariant Sinkhorn and 1-D Frank-Wolfe](https://mlsanity.com/paper/2201.00730/) (2022)
- [ResNets Ensemble via the Feynman-Kac Formalism to Improve Natural and Robust Accuracies](https://mlsanity.com/paper/1811.10745/) (2018)

## Built on by (most cited first, 1 of 1)

- [Accelerated Sinkhorn Algorithms for Partial Optimal Transport](https://mlsanity.com/paper/2601.17196/) (2026)

## Abstract (arXiv)

We study the fundamental computational problem of approximating optimal transport (OT) equations using neural differential equations (Neural ODEs). More specifically, we develop a novel framework for approximating unbalanced optimal transport (UOT) in the continuum using Neural ODEs. By generalizing a discrete UOT problem with Pearson divergence, we constructively design vector fields for Neural ODEs that converge to the true UOT dynamics, thereby advancing the mathematical foundations of computational transport and machine learning. To this end, we design a numerical scheme inspired by the Sinkhorn algorithm to solve the corresponding minimization problem and rigorously prove its convergence, providing explicit error estimates. From the obtained numerical solutions, we derive vector fields defining the transport dynamics and construct the corresponding transport equation. Finally, from the numerically obtained transport equation, we construct a neural differential equation whose flow converges to the true transport dynamics in an appropriate limiting regime.