Transferability of Learned States in Neural PDE Solvers

2026-10-07

AI Summary

This study addresses the challenge of evaluating the reuse value of pretrained states in neural PDE solvers by proposing a “reuse contract” framework that decouples reuse benefits into final accuracy, learning contribution, and numerical utility. Through paired state comparisons and computational cost accounting, a systematic benchmark is constructed using Fourier neural operators, convolutional networks, and conjugate gradient (CG) methods to quantify the practical value of pretrained states within downstream correction algorithms. The research reveals an inversion phenomenon wherein fixed-predictor gains vary with the correction algorithm, establishing matched-state comparison standards. Experiments demonstrate that workload-based selection strategies save 2.5 to 3.3 CG iterations, while online reuse achieves an overall computational saving of 0.73%, thereby validating the effectiveness of pretrained state reuse under limited computational budgets.

Abstract

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 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 less than or equal to 5 percent on 64 in-distribution tasks (63 by 63 interior grids), reductions in all three norms accompany more CG iterations, at mean taskwise rates of 23.5 percent and 23.9 percent 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 percent 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.

Problem

• Neural PDE Solvers
• Transferability
• State Reuse
• Pretraining
• Conjugate Gradient

Innovation

• Neural PDE Solvers
• Transferability
• Conjugate Gradient
• Fourier Neural Operator
• Reuse Contract