# Regularity of Solutions to Beckmann's Parametric Optimal Transport

arXiv: https://arxiv.org/abs/2603.19755 · HTML: https://mlsanity.com/paper/2603.19755/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 29 references that are in the map.
- First posted on arXiv 20 Mar 2026.

## Builds on (most cited first, 13 of 13)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [On minimax density estimation via measure transport](https://mlsanity.com/paper/2207.10231/) (2022)
- [Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans](https://mlsanity.com/paper/2501.16839/) (2025)
- [Rates of convergence for density estimation with generative adversarial networks](https://mlsanity.com/paper/2102.00199/) (2021)
- [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024)
- [A Convenient Infinite Dimensional Framework for Generative Adversarial Learning](https://mlsanity.com/paper/2011.12087/) (2020)
- [Convergence of the Time Discrete Metamorphosis Model on Hadamard Manifolds](https://mlsanity.com/paper/1902.10930/) (2019)
- [Numerical and statistical analysis of NeuralODE with Runge-Kutta time integration](https://mlsanity.com/paper/2503.10729/) (2025)
- [Learning Brenier Potentials with Convex Generative Adversarial Neural Networks](https://mlsanity.com/paper/2504.19779/) (2025)

## Abstract (arXiv)

Beckmann's problem in optimal transport minimizes the total squared flux in a continuous transport problem from a source to a target distribution. In this article, the regularity theory for solutions to Beckmann's problem in optimal transport is developed utilizing an unconstrained Lagrangian formulation and solving the variational first order optimality conditions. It turns out that the Lagrangian multiplier that enforces Beckmann's divergence constraint fulfills a Poisson equation and the flux vector field is obtained as the potential's gradient. Utilizing Schauder estimates from elliptic regularity theory, the exact H"older regularity of the potential, the flux and the flow generating is derived on the basis of H"older regularity of source and target densities on a bounded, regular domain. If the target distribution depends on parameters, as is the case in conditional (``promptable'') generative learning, we provide sufficient conditions for separate and joint H"older continuity of the resulting vector field in the parameter and the data dimension. Following a recent result by Belomnestny et al., one can thus approximate such vector fields with deep ReQu neural networks in C^(k,alpha)-H"older norm. We also show that this approach generalizes to other probability paths, like Fisher-Rao gradient flows.