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

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.