Accelerating PDE Data Generation via Differential Operator Action in
  Solution Space
Huanshuo Dong Hong Wang Haoyang Liu Jian Luo Jie Wang
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.