Two-Step Diffusion: Fast Sampling and Reliable Prediction for 3D Keller--Segel and KPP Equations in Fluid Flows

Abstract

We study fast and reliable generative transport for the 3D KS (Keller-Segel) and KPP (Kolmogorov-Petrovsky-Piskunov) equations in the presence of fluid flows with the goal to approximate the map between initial and terminal distributions for a range of physical parameters $\sigma$ under the Wasserstein metric. To minimize the inaccuracy of direct Wasserstein solver, we propose a two-stage pipeline that retains one-step efficiency while reinstating an explicit $W_2$ objective where it is tractable. In Stage I, a Meanflow-style regressor yields a deterministic, one-step global transport that moves particles close to their terminal states. In Stage II, we freeze this initializer and train a near-identity corrector (Deep Particle, DP) that directly minimizes a mini-batch $W_2$ objective using warm-started optimal transport couplings computed on the Meanflow outputs. Crucially, after the one-step transport (from Stage I) concentrating mass on the approximated correct support, the induced geometry stabilizes high-dimensional $W_2$ computation of the direct Wasserstein solver. We validate our construction in the 3D KS and KPP equations subject to fluid flows with ordered and chaotic streamlines.