Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations

Ze Cheng, Zhongkai Hao, Xiaoqiang Wang, Jianing Huang, Youjia Wu, Xudan Liu, Yiru Zhao, Songming Liu, Hang Su
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:8060-8076, 2024.

Abstract

For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a sufficiently accurate neural operator. However, for many industrial applications, e.g., engineering design optimization, it can be prohibitive to satisfy the requirement since even a single simulation may take hours or days of computation. To address this issue, we propose reference neural operators (RNO), a novel way of implementing neural operators, i.e., to learn the smooth dependence of solutions on geometric deformations. Specifically, given a reference solution, RNO can predict solutions corresponding to arbitrary deformations of the referred geometry. This approach turns out to be much more data efficient. Through extensive experiments, we show that RNO can learn the dependence across various types and different numbers of geometry objects with relatively small datasets. RNO outperforms baseline models in accuracy by a large lead and achieves up to 80% error reduction.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-cheng24c, title = {Reference Neural Operators: Learning the Smooth Dependence of Solutions of {PDE}s on Geometric Deformations}, author = {Cheng, Ze and Hao, Zhongkai and Wang, Xiaoqiang and Huang, Jianing and Wu, Youjia and Liu, Xudan and Zhao, Yiru and Liu, Songming and Su, Hang}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {8060--8076}, 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/cheng24c/cheng24c.pdf}, url = {https://proceedings.mlr.press/v235/cheng24c.html}, abstract = {For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a sufficiently accurate neural operator. However, for many industrial applications, e.g., engineering design optimization, it can be prohibitive to satisfy the requirement since even a single simulation may take hours or days of computation. To address this issue, we propose reference neural operators (RNO), a novel way of implementing neural operators, i.e., to learn the smooth dependence of solutions on geometric deformations. Specifically, given a reference solution, RNO can predict solutions corresponding to arbitrary deformations of the referred geometry. This approach turns out to be much more data efficient. Through extensive experiments, we show that RNO can learn the dependence across various types and different numbers of geometry objects with relatively small datasets. RNO outperforms baseline models in accuracy by a large lead and achieves up to 80% error reduction.} }
Endnote
%0 Conference Paper %T Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations %A Ze Cheng %A Zhongkai Hao %A Xiaoqiang Wang %A Jianing Huang %A Youjia Wu %A Xudan Liu %A Yiru Zhao %A Songming Liu %A Hang Su %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-cheng24c %I PMLR %P 8060--8076 %U https://proceedings.mlr.press/v235/cheng24c.html %V 235 %X For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a sufficiently accurate neural operator. However, for many industrial applications, e.g., engineering design optimization, it can be prohibitive to satisfy the requirement since even a single simulation may take hours or days of computation. To address this issue, we propose reference neural operators (RNO), a novel way of implementing neural operators, i.e., to learn the smooth dependence of solutions on geometric deformations. Specifically, given a reference solution, RNO can predict solutions corresponding to arbitrary deformations of the referred geometry. This approach turns out to be much more data efficient. Through extensive experiments, we show that RNO can learn the dependence across various types and different numbers of geometry objects with relatively small datasets. RNO outperforms baseline models in accuracy by a large lead and achieves up to 80% error reduction.
APA
Cheng, Z., Hao, Z., Wang, X., Huang, J., Wu, Y., Liu, X., Zhao, Y., Liu, S. & Su, H.. (2024). Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:8060-8076 Available from https://proceedings.mlr.press/v235/cheng24c.html.

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