# Continuum-marginal optimal transport: a mesh-free kernel method

arXiv: https://arxiv.org/abs/2604.24226 · HTML: https://mlsanity.com/paper/2604.24226/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 27 references that are in the map.
- First posted on arXiv 27 Apr 2026.

## Builds on (most cited first, 12 of 12)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Multimarginal generative modeling with stochastic interpolants](https://mlsanity.com/paper/2310.03695/) (2023)
- [A kernel-based method for Schr"odinger bridges](https://mlsanity.com/paper/2310.14522/) (2023)
- [A deep learning approach to multi-marginal optimal transport via Hilbert space embeddings of probability measures](https://mlsanity.com/paper/2507.09206/) (2025)

## Abstract (arXiv)

In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the continuum limit of the classical two-marginal Benamou--Brenier formulation, and also the deterministic limit of the Nelson problem of stochastic optimal transport. We propose a practical mesh-free solver for this problem. The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency. The same computational framework also applies to the Nelson problem.