Recent advancements in data-driven approaches, such as Neural Operator (NO), have demonstrated their effectiveness in reducing the solving time of Partial Differential Equations (PDEs). However, one major challenge faced by these approaches is the requirement for a large amount of high-precision training data, which needs significant computational costs during the generation process. To address this challenge, we propose a novel PDE dataset generation algorithm, namely Differential Operator Action in Solution space (DiffOAS), which speeds up the data generation process and enhances the precision of the generated data simultaneously. Specifically, DiffOAS obtains a few basic PDE solutions and then combines them to get solutions. It applies differential operators on these solutions, a process we call 'operator action', to efficiently generate precise PDE data points. Theoretical analysis shows that the time complexity of DiffOAS method is one order lower than the existing generation method. Experimental results show that DiffOAS accelerates the generation of large-scale datasets with 10,000 instances by 300 times. Even with just 5% of the generation time, NO trained on the data generated by DiffOAS exhibits comparable performance to that using the existing generation method, which highlights the efficiency of DiffOAS.

通过提出名为解空间微分算子作用的新型偏微分方程（PDE）数据集生成算法（DiffOAS），加速数据生成过程并同时提高准确性，实验证明DiffOAS方法的时间复杂度比现有方法低一个数量级。实验结果表明，DiffOAS将大规模数据集的生成速度加快了300倍，甚至只用到了5%的生成时间，使用DiffOAS生成的数据训练的神经运算符（NO）表现出与现有方法相当的性能，突出了DiffOAS的效率。

在解空间中通过微分算子作用加速PDE数据生成