Segmented Physics-Consistent Diffusion Anchors with Deterministic Rollouts for Allen–Cahn Dynamics

We propose a segmented diffusion–Euler method for conditional coarse-to-fine reconstruction of Allen–Cahn dynamics. Residual-centered diffusion reconstructs sparse high-resolution anchors from coarse observations, and deterministic fine-grid Euler steps update each segment interior. An Allen–Cahn grid residual provides physical guidance, with the ReMD multigrid correction kept as an optional refinement. For any fixed finite spatial dimension, we derive a rollout bound that separates discretization bias, anchor risk, and segment-local error amplification. Controlled experiments show that periodic re-anchoring substantially improves rollout accuracy over unsegmented original ReMD. The Allen–Cahn-aware variant achieves comparable field accuracy with lower mean energy mismatch, and independent training runs show low variability. These results identify anchor reconstruction as the main accuracy bottleneck and quantify the tradeoffs between learned reconstruction and deterministic dynamics.