A Quantitative Approximation Framework for Flow Distillation in Diffusion Models

We develop a quantitative framework for diffusion distillation by viewing few step sampling as approximation through compositions of learned flow maps. For trajectory distillation of the probability flow ODE, we show that low noise multimodal regimes separate score approximability from dynamical stability: the score remains efficiently approximable, while small local errors may be strongly amplified by stiff flow dynamics. In a Gaussian mixture Ornstein--Uhlenbeck model, we prove time uniform (Lp(pt)) score approximation by ReLU and ReQU networks with explicit polylogarithmic complexity, and derive a computable Lipschitz bound (L(t)) for the flow velocity. The stability factor (expbigl(intst L(u)mathrm dubigr)) can grow exponentially as noise decreases and mixture separation increases. Comparing this certificate with a certified local Lipschitz budget for one step students identifies regimes of direct distillation difficulty, without implying an approximation lower bound. We also show that deep residual compositions control global transport error through propagated local errors, and that equalizing cumulative stability yields an optimal nonuniform segmentation. With eight segments, this grid reduces final mean relative MSE by up to (51.9%) versus uniform grids.