APIC: Routing Corrections for Neural PDE Rollouts

Inference-time correction is part of a neural PDE simulator's evolution rule, and its numerical structure determines which prediction errors it can address. A shared correction template can therefore impose a restriction even when its residual is equation-dependent. We study equation class as a prior for selecting correction operators. We introduce Analytical Predictor–Inference Correction (APIC), which selects an analytical update according to the equation's numerical structure and combines it with optional bounded learned refinement. This separates the choice of update family from the surrogate architecture while retaining a frozen pretrained predictor; the Navier–Stokes branch requires no predictor evaluation. Across three PDEs and three architectures, fixed class-level routing reduces rollout relative-L2 error by a mean of 83.48% over PhysicsCorrect, within 0.25 percentage points of selecting the best evaluated variant separately for each PDE–architecture pair. An analytical-only policy achieves 81.68%, and the Navier–Stokes branch reduces error by 93.4% without a network forward pass. These results support a coarse structural prior for correction selection on the evaluated benchmark and establish an analytical reference for assessing the contribution of learned refinement.