The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space

The paper presents a unified geometric framework for understanding diffusion models and Flow Matching within the context of Wasserstein space, specifically $\mathcal{P}_2(\mathbb{R}^d)$. It establishes that diffusion models can be viewed as gradient flows of free energy, utilizing the Fokker-Planck equation and the JKO scheme, while Flow Matching operates along geodesics defined by the Benamou-Brenier formula. This integration of both approaches on a single Riemannian manifold highlights their relationship, allowing for more efficient sampling in generative processes by treating them as deterministic ODEs along optimal paths.