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

• URL: http://arxiv.org/abs/2608.05666v1
• Date: Thu, 06 Aug 2026 07:04:30 GMT
• Title: Potential Matching Optimal Transport: Continuous Normalizing Flows for Exact $p$-Wasserstein Dynamics
• Abstract summary: Potential Matching Optimal Transport (PMOT) is a potential-flow framework for general $p$-cost optimal transport.<n>It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints.<n>On synthetic benchmarks, PMOT learns $p$-specific maps that agree with the corresponding $p$-matched OT references.
• Score: 7.533486962572339
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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