发表机构
University of Chicago; University of Cambridge; Princeton University; ByteDance Inc.; Amazon（芝加哥大学; 剑桥大学; 普林斯顿大学; 字节跳动公司; 亚马逊公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对神经PDE求解器中学习状态的可迁移性问题，提出复用契约方法，通过配对状态比较等分离解精度等指标，在实验中验证了该方法可节省CG迭代次数并证明有限预算预训练的价值。
AI 中文摘要
评估神经PDE求解器中有用的复用具有挑战性：最终精度可能反映源学习和目标时间计算。我们的复用契约通过配对状态比较、匹配的目标信息与预算以及成本核算，将解精度、学习贡献和数值效用分离开来。文献审查从12篇论文中提取了18个特定版本的协议记录，记录了保留的状态、目标时间资源和报告的控制项。对于固定的线性系统和残差容差，我们构造了两个初始猜测，它们具有相同的解误差、能量误差和残差范数，但达到相同解所需的共轭梯度（CG）迭代次数不同。在240条源训练轨迹、两个线性PDE族、傅里叶神经算子和卷积网络中，固定预测器的益处会在不同校正算法间发生反转。在64个分布内任务（63×63内部网格）上，当两个相对预测误差均不超过5%时，所有三个范数的降低伴随CG迭代次数的增加，在两个库中，任务级平均速率分别为23.5%和23.9%。基于工作的选择在保留的分布内任务上节省了2.50-3.33次CG迭代；匹配的自适应方法证明了有限预算预训练的价值。独立批次证实，与零初始化的泊松预处理CG相比，一个物理训练的傅里叶神经算子可实现0.73%的完整在线节省。复用需要匹配的状态比较和下游计算证据。
英文摘要
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.
CommentsUnder review as a conference paper at ICLR 2027
