Learn or Refine? Predicting When Neural PDE Correction Is Worth Training

Learned error correctors repair the output of a cheap classical PDE solver, but training one costs high-fidelity data and optimization. Before paying that cost, one would like to know whether the corrector will beat the classical alternative of refining the grid. Accuracy against a fixed reference cannot answer this, because refinement spends the same resource the corrector spends on training data. We formulate the question as a prospective matched-cost decision: a corrector is worth training only if its achievable error beats both a ridge control and the best refinement affordable within its own budget. The hard part is that the achievable learned error is unknown before training. We show that the pilot-to-full remaining learning gain is strongly family-dependent, so a fixed pilot-to-full calibration does not transfer reliably across families; but the gain is predictable from the pilot's early learning progress. This yields a two-feature estimator and a Learn-or-Refine rule costing about $15\%$ of full training. In leave-one-family-out evaluation over 74 prior regime$\times$architecture cases it has the lowest cross-family decision regret among competing estimators; after freezing the estimator it decides correctly on 9 of 10 regimes of an unseen Allen–Cahn family with zero decision regret. The method decides whether a specified corrector recipe is worth completing relative to matched-cost classical refinement.