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

What the math gives to ML

This paper provides a principled way to replace the quadratic geometry built into common OT-regularized CNFs with an explicitly chosen p-Wasserstein geometry. The transferable asset is the generalized Benamou–Brenier potential flow: a scalar potential produces a velocity field whose direction and magnitude depend on the dual exponent q=p/(p-1), while a bridge-matching objective can train it without solving a full OT problem at every update. The most promising neural-network use is a p-controlled flow-matching CNF, with p selected according to whether robustness to outliers, sparse displacement, or smooth quadratic transport is desired.