Transolver: A Fast Transformer Solver for PDEs on General Geometries

Haixu Wu, Huakun Luo, Haowen Wang, Jianmin Wang, Mingsheng Long
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:53681-53705, 2024.

Abstract

Transformers have empowered many milestones across various fields and have recently been applied to solve partial differential equations (PDEs). However, since PDEs are typically discretized into large-scale meshes with complex geometries, it is challenging for Transformers to capture intricate physical correlations directly from massive individual points. Going beyond superficial and unwieldy meshes, we present Transolver based on a more foundational idea, which is learning intrinsic physical states hidden behind discretized geometries. Specifically, we propose a new Physics-Attention to adaptively split the discretized domain into a series of learnable slices of flexible shapes, where mesh points under similar physical states will be ascribed to the same slice. By calculating attention to physics-aware tokens encoded from slices, Transovler can effectively capture intricate physical correlations under complex geometrics, which also empowers the solver with endogenetic geometry-general modeling capacity and can be efficiently computed in linear complexity. Transolver achieves consistent state-of-the-art with 22% relative gain across six standard benchmarks and also excels in large-scale industrial simulations, including car and airfoil designs. Code is available at https://github.com/thuml/Transolver.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-wu24r, title = {Transolver: A Fast Transformer Solver for {PDE}s on General Geometries}, author = {Wu, Haixu and Luo, Huakun and Wang, Haowen and Wang, Jianmin and Long, Mingsheng}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {53681--53705}, 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/wu24r/wu24r.pdf}, url = {https://proceedings.mlr.press/v235/wu24r.html}, abstract = {Transformers have empowered many milestones across various fields and have recently been applied to solve partial differential equations (PDEs). However, since PDEs are typically discretized into large-scale meshes with complex geometries, it is challenging for Transformers to capture intricate physical correlations directly from massive individual points. Going beyond superficial and unwieldy meshes, we present Transolver based on a more foundational idea, which is learning intrinsic physical states hidden behind discretized geometries. Specifically, we propose a new Physics-Attention to adaptively split the discretized domain into a series of learnable slices of flexible shapes, where mesh points under similar physical states will be ascribed to the same slice. By calculating attention to physics-aware tokens encoded from slices, Transovler can effectively capture intricate physical correlations under complex geometrics, which also empowers the solver with endogenetic geometry-general modeling capacity and can be efficiently computed in linear complexity. Transolver achieves consistent state-of-the-art with 22% relative gain across six standard benchmarks and also excels in large-scale industrial simulations, including car and airfoil designs. Code is available at https://github.com/thuml/Transolver.} }
Endnote
%0 Conference Paper %T Transolver: A Fast Transformer Solver for PDEs on General Geometries %A Haixu Wu %A Huakun Luo %A Haowen Wang %A Jianmin Wang %A Mingsheng Long %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-wu24r %I PMLR %P 53681--53705 %U https://proceedings.mlr.press/v235/wu24r.html %V 235 %X Transformers have empowered many milestones across various fields and have recently been applied to solve partial differential equations (PDEs). However, since PDEs are typically discretized into large-scale meshes with complex geometries, it is challenging for Transformers to capture intricate physical correlations directly from massive individual points. Going beyond superficial and unwieldy meshes, we present Transolver based on a more foundational idea, which is learning intrinsic physical states hidden behind discretized geometries. Specifically, we propose a new Physics-Attention to adaptively split the discretized domain into a series of learnable slices of flexible shapes, where mesh points under similar physical states will be ascribed to the same slice. By calculating attention to physics-aware tokens encoded from slices, Transovler can effectively capture intricate physical correlations under complex geometrics, which also empowers the solver with endogenetic geometry-general modeling capacity and can be efficiently computed in linear complexity. Transolver achieves consistent state-of-the-art with 22% relative gain across six standard benchmarks and also excels in large-scale industrial simulations, including car and airfoil designs. Code is available at https://github.com/thuml/Transolver.
APA
Wu, H., Luo, H., Wang, H., Wang, J. & Long, M.. (2024). Transolver: A Fast Transformer Solver for PDEs on General Geometries. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:53681-53705 Available from https://proceedings.mlr.press/v235/wu24r.html.

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