IADR: Interface-Augmented Neural Operator for Phase-Field Mean-Curvature Flow

The Allen--Cahn equation is one of the standard phase-field models for mean-curvature flow (MCF), parameterised by the diffuse-interface width $\varepsilon$. Practical simulations span a working range of $\varepsilon$, so a single model covering this range is needed. PDE-specific neural operators are accurate but must be retrained at every $\varepsilon$; general neural operators cover $\varepsilon$ in one model but lose the phase-field structure of the target. We introduce Interface-Augmented Diffusion-Reaction (IADR), a neural operator that addresses both shortcomings: a target network with a diffusion-and-reaction structure preserves the phase-field structure of the Allen--Cahn equation, while a hypernetwork that maps $\varepsilon$ to the weights of this target network amortises the dependence on $\varepsilon$ across the working range. A single IADR matches the per-$\varepsilon$ specialists on in-distribution data, retains their cross-shape generalisation under both $\varepsilon$- and geometry-out-of-distribution shifts, and runs at the per-$\varepsilon$ specialist's per-step inference cost. We further establish a $K$-step rollout error bound that aligns with the long-horizon behaviour observed in experiments.