Score Accuracy Along the Forward Diffusion Does Not Certify Numerical Stability in Diffusion Sampling

What the math gives to ML

This paper exposes a serious mismatch between standard score-matching objectives and the behavior of discretized reverse diffusion: a score can be highly accurate under forward marginals while producing rare numerical trajectories with exploding moments. The transferable asset is a constructive stability mechanism, not merely the counterexample: when the clean-data support lies in a known bounded closed convex set, projecting the denoiser onto that set preserves pointwise accuracy while preventing unbounded denoiser outputs. The most direct neural-network intervention is therefore to project the denoiser output at every reverse-sampling step, then convert the projected denoiser back into a score before applying the reverse SDE or ODE update.