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

Abstract:We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general $p$-cost optimal transport with $c_p(x,y)=\|x-y\|^p$. 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.

Machine Learning (cs.LG); Numerical Analysis (math.NA)

arXiv:2608.05666 [cs.LG]

(or arXiv:2608.05666v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.05666 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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