# Potential Matching Optimal Transport: Continuous Normalizing Flows for Exact p-Wasserstein Dynamics

arXiv: https://arxiv.org/abs/2608.05666 · HTML: https://mlsanity.com/paper/2608.05666/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 17 references that are in the map.
- First posted on arXiv 6 Aug 2026.

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

- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Convergence analysis of OT-Flow for sample generation](https://mlsanity.com/paper/2403.16208/) (2024)

## Abstract (arXiv)

We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general p-cost optimal transport with cp(x,y)=|x-y|p. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou--Brenier form for the chosen exponent p. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou--Brenier optimality system and recovers the corresponding p-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns p-specific maps that agree with the corresponding p-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.