Stochastic Sampling with 1D Brownian motion for Flow Matching

Flow Matching (FM) models are commonly sampled with deterministic ordinary differential equations. Stochastic sampling can improve trajectory diversity and sample quality, but conventional stochastic differential equation (SDE) samplers often require a large number of function evaluations (NFE). Building on the classi- cal marginal-preservation identity for the Fokker–Planck equation, we develop an adapted low-rank step-process construction that converts low-dimensional Brown- ian motion into a practical, training-free sampler. The Simplified 1D-Noise-Guided Sampling (SONGS) framework, in its rank-one form, injects a scalar Brownian increment along a history-dependent estimate of the local discretization-error direc- tion. Freezing this direction on each integration interval eliminates the Jacobian and divergence evaluations required by a general state-dependent diffusion and supports an improved small-noise stochastic Runge–Kutta implementation. Experiments on CIFAR-10, FFHQ, and ImageNet demonstrate that SONGS achieves superior sampling performance at low NFE.