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 $\epsilon$. Practical simulations span a working range of $\epsilon$, so a single model covering this range is needed. PDE-specific neural operators are accurate but must be retrained at every $\epsilon$; general neural operators cover $\epsilon$ 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 $\epsilon$ to the weights of this target network amortises the dependence on $\epsilon$ across the working range. A single IADR matches the per-$\epsilon$ specialists on in-distribution data, retains their cross-shape generalisation under both $\epsilon$- and geometry-out-of-distribution shifts, and runs at the per-$\epsilon$ 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.