School of Mathematical Sciences, Shanghai Jiao Tong University · Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University
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.