Flow-WLS: Regularized Multistep Integration for Fast Sampling in Generative Flow Matching Models

Numerical integration of the probability flow ordinary differential equation (ODE) in diffusion and Flow Matching models is fundamentally constrained by a trade-off: low-order solvers (orders 1–2) require substantial step budgets (30+ NFEs) to avoid oversmoothing, while classical explicit high-order multistep schemes (orders 4+) suffer from Adams-Bashforth instability, Runge's phenomenon, and severe noise amplification. In this work, we analyze the breakdown mechanism of high-order polynomial extrapolation in Flow Matching architectures (such as DiT / Cosmos) and introduce Flow-WLS (Weighted Least Squares / Savitzky-Golay Multistep). Instead of fitting an oscillating interpolating polynomial of degree (L-1) through past velocity evaluations, Flow-WLS constrains the trajectory approximation to a regularized low-degree parabola (d=2) fitted over extended history buffers (L=3 to 8 steps) via weighted least squares with exponential memory decay. This inverts the classic curse of multistep solvers: increasing memory depth suppresses discretization noise rather than amplifying it. Empirical evaluations on text-to-image Flow Matching models demonstrate strictly monotonic metric improvements as history expands from L=3 to L=8, achieving an SSIM of 0.8003 at only 10 NFEs (~11 seconds on a consumer GPU) with zero anatomical distortion.