Accelerating PDE Data Generation via Differential Operator Action in Solution Space
This addresses a bottleneck in data-driven PDE solving for researchers and practitioners by making training data generation more efficient, though it is an incremental improvement on existing methods.
The paper tackles the high computational cost of generating large-scale, high-precision training data for neural operators in PDE solving by proposing DiffOAS, a method that accelerates data generation by 300 times for 10,000 instances while maintaining comparable model performance with only 5% of the time.
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.