Transferability of Learned States in Neural PDE Solvers

Assessing useful reuse in neural PDE solvers is challenging: final accuracy can reflect source learning and target-time computation. Our reuse contract separates solution accuracy, learning contribution, and numerical utility through paired state comparisons, matched target information and budgets, and cost accounting. A literature audit extracts 18 version-specific protocol records from 12 papers, documenting retained states, target-time resources, and reported controls. For a fixed linear system and residual tolerance, we construct two initial guesses with identical solution-error, energy-error, and residual norms, reaching the same solution with different conjugate-gradient (CG) iteration counts. Across 240 core source-training trajectories, two linear PDE families, Fourier neural operators and convolutional networks, a fixed predictor's benefit reverses across correction algorithms. Among pairs with both relative prediction errors $\le 5\%$ on 64 in-distribution tasks ($63\times 63$ interior grids), reductions in all three norms accompany more CG iterations, at mean taskwise rates of 23.5% and 23.9% in two libraries. Work-based selection saves 2.50–3.33 CG iterations on held-out in-distribution tasks; matched adaptation demonstrates finite-budget pretraining value. Independent batches confirm a 0.73% complete online saving for one physics-trained Fourier neural operator against zero-initialized Poisson-preconditioned CG. Reuse requires matched state comparisons and downstream computational evidence.