Asymptotic Preservation and Uniform Accuracy of Diffusion and Flow-Matching Samplers

What the math gives to ML

The paper identifies sampler quality as a property of the complete specification—numerical update, time grid, and terminal rule—rather than of the base ODE/SDE solver alone. Its key transferable mechanism is to stop integration at a positive noise scale and analytically complete the final segment, avoiding the logarithmically diverging number of steps required by bounded log-noise stepping. For diffusion models with manifold-like or rank-deficient data, the proposed terminal map exploits the denoiser's normal-mode behavior and has error proportional to the switching scale squared, so choosing the cutoff as a function of base resolution preserves the solver's convergence order. This gives a directly testable way to reduce function evaluations while maintaining endpoint accuracy.