Flow Matching and Optimal Transport: an Illustrated Introduction to Wasserstein–otto Geometry
Year: 2026
Type: preprint
Abstract: We recall and derive the equations underlying Flow Match- ing, making explicit the probability density function of the linearly in- terpolated random variable and relating it to the transporting veloc- ity field through the continuity equation. We visualise how the initial density is transported and compare the resulting path with a geodesic in Wasserstein–Otto geometry. We demonstrate that, for an isotropic Gaussian initial law, the Flow Matching velocity field is the minimum- energy representative of the density variation at each time, among fields satisfying the continuity equation, even though the path is generally not a geodesic under the Wasserstein–Otto metric. We explain the dis- tinction between this local minimisation and the minimisation of the action over paths with fixed endpoints using explicit calculations and illustrations.
Author: Guilbert, Pierre
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225655348