NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Data

Songming Liu, Zhongkai Hao, Chengyang Ying, Hang Su, Ze Cheng, Jun Zhu
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:21658-21671, 2023.

Abstract

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO .

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-liu23o, title = {{NUNO}: A General Framework for Learning Parametric {PDE}s with Non-Uniform Data}, author = {Liu, Songming and Hao, Zhongkai and Ying, Chengyang and Su, Hang and Cheng, Ze and Zhu, Jun}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {21658--21671}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/liu23o/liu23o.pdf}, url = {https://proceedings.mlr.press/v202/liu23o.html}, abstract = {The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO .} }
Endnote
%0 Conference Paper %T NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Data %A Songming Liu %A Zhongkai Hao %A Chengyang Ying %A Hang Su %A Ze Cheng %A Jun Zhu %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-liu23o %I PMLR %P 21658--21671 %U https://proceedings.mlr.press/v202/liu23o.html %V 202 %X The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO .
APA
Liu, S., Hao, Z., Ying, C., Su, H., Cheng, Z. & Zhu, J.. (2023). NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Data. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:21658-21671 Available from https://proceedings.mlr.press/v202/liu23o.html.

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