Accelerating PDE Data Generation via Differential Operator Action in Solution Space

Huanshuo Dong, Hong Wang, Haoyang Liu, Jian Luo, Jie Wang
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:11395-11411, 2024.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-dong24d, title = {Accelerating {PDE} Data Generation via Differential Operator Action in Solution Space}, author = {Dong, Huanshuo and Wang, Hong and Liu, Haoyang and Luo, Jian and Wang, Jie}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {11395--11411}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/dong24d/dong24d.pdf}, url = {https://proceedings.mlr.press/v235/dong24d.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Accelerating PDE Data Generation via Differential Operator Action in Solution Space %A Huanshuo Dong %A Hong Wang %A Haoyang Liu %A Jian Luo %A Jie Wang %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-dong24d %I PMLR %P 11395--11411 %U https://proceedings.mlr.press/v235/dong24d.html %V 235 %X 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.
APA
Dong, H., Wang, H., Liu, H., Luo, J. & Wang, J.. (2024). Accelerating PDE Data Generation via Differential Operator Action in Solution Space. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:11395-11411 Available from https://proceedings.mlr.press/v235/dong24d.html.

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