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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 labeleddata, i.e., PDE problems together with their solutions. The data generationprocess is exceptionally time-consuming, as it involves solving numeroussystems of linear equations to obtain numerical solutions to the PDEs. Manyexisting methods solve these systems independently without considering theirinherent similarities, resulting in extremely redundant computations. To tacklethis problem, we propose a novel method, namely Sorting Krylov Recycling (SKR),to boost the efficiency of solving these systems, thus significantlyaccelerating data generation for neural operators training. To the best of ourknowledge, SKR is the first attempt to address the time-consuming nature ofdata generation for learning neural operators. The working horse of SKR isKrylov subspace recycling, a powerful technique for solving a series ofinterrelated systems by leveraging their inherent similarities. Specifically,SKR employs a sorting algorithm to arrange these systems in a sequence, whereadjacent systems exhibit high similarities. Then it equips a solver with Krylovsubspace recycling to solve the systems sequentially instead of independently,thus effectively enhancing the solving efficiency. Both theoretical analysisand extensive experiments demonstrate that SKR can significantly accelerateneural operator data generation, achieving a remarkable speedup of up to 13.9times.

Accelerating Data Generation for Neural Operators via Krylov Subspace Recycling