New Framework for Optimal Transport with Continuous Normalizing Flows

Optimal Transport (OT) is a mathematical framework for comparing probability distributions, with applications in machine learning like generative models and domain adaptation. This paper introduces Potential Matching Optimal Transport (PMOT), a novel approach for general p-cost optimal transport, where the cost function is based on the p-norm distance between points. PMOT leverages continuous normalizing flows (CNFs), which model transformations between distributions as continuous-time dynamics. The key innovation is parameterizing the CNF's velocity field using a scalar potential, following a generalized Benamou-Brenier formulation. The model is trained using a self-induced matching loss that aligns with straight "bridges" determined by the model's own endpoints, allowing flexible matching of terminal distributions. A significant finding is the establishment of "zero-loss exactness": under certain regularity and uniqueness assumptions, a zero-loss solution guarantees the recovery of the p-optimal transport map and dynamics. Experiments show PMOT accurately learns p-specific maps, performs competitively as a likelihood-based density model, and demonstrates flexible sample-based terminal matching in tasks like color transformation.