Learning neural operators for solving partial differential equations (PDEs) has attracted great attention due to its high inference efficiency. However, training such operators requires generating a substantial amount of labeled data, i.e., PDE problems together with their solutions. The data generation process is exceptionally time-consuming, as it involves solving numerous systems of linear equations to obtain numerical solutions to the PDEs. Many existing methods solve these systems independently without considering their inherent similarities, resulting in extremely redundant computations. To tackle this problem, we propose a novel method, namely Sorting Krylov Recycling (SKR), to boost the efficiency of solving these systems, thus significantly accelerating data generation for neural operators training. To the best of our knowledge, SKR is the first attempt to address the time-consuming nature of data generation for learning neural operators. The working horse of SKR is Krylov subspace recycling, a powerful technique for solving a series of interrelated systems by leveraging their inherent similarities. Specifically, SKR employs a sorting algorithm to arrange these systems in a sequence, where adjacent systems exhibit high similarities. Then it equips a solver with Krylov subspace recycling to solve the systems sequentially instead of independently, thus effectively enhancing the solving efficiency. Both theoretical analysis and extensive experiments demonstrate that SKR can significantly accelerate neural operator data generation, achieving a remarkable speedup of up to 13.9 times.

翻译：学习求解偏微分方程（PDEs）的神经算子因其高推理效率而备受关注。然而，训练此类算子需要生成大量标注数据，即PDE问题及其对应的解。这一数据生成过程极为耗时，因为需要求解大量线性方程组以获得PDEs的数值解。现有方法通常独立求解这些方程组，未考虑其内在相似性，导致大量冗余计算。针对这一问题，我们提出了一种新颖方法——排序Krylov循环方法（SKR），以提升方程组求解效率，从而显著加速神经算子训练的数据生成。据我们所知，SKR是首个解决神经算子学习中数据生成耗时问题的尝试。SKR的核心机制是Krylov子空间循环，这是一种通过利用一系列相关系统内在相似性来高效求解的强大技术。具体而言，SKR采用排序算法将这些系统按序列排列，使得相邻系统具有高度相似性，随后借助Krylov子空间循环的求解器依次而非独立地求解这些系统，从而有效提升求解效率。理论分析与大量实验均表明，SKR能显著加速神经算子数据生成，实现高达13.9倍的惊人加速比。