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

PMOT parameterizes the velocity field of a continuous normalizing flow (CNF) using a scalar potential in the generalized Benamou-Brenier form for p-costs c_p(x,y)=||x-y||^p. It trains the potential gradient via a self-induced matching loss along straight bridges determined by the model's own endpoints, avoiding the need for precomputed optimal transport plans. This method generalizes prior work that was limited to p=2, enabling exact Wasserstein dynamics for arbitrary p. The paper claims this leads to more flexible terminal conditions and better handling of non-Euclidean geometry. Implications include improved OT-based generative models and new tools for analyzing Wasserstein gradient flows.