Curvature-aware Graph Attention for PDEs on Manifolds

Yunfeng Liao, Jiawen Guan, Xiucheng Li
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:37476-37494, 2025.

Abstract

Deep models have recently achieved remarkable performances in solving partial differential equations (PDEs). The previous methods are mostly focused on PDEs arising in Euclidean spaces with less emphasis on the general manifolds with rich geometry. Several proposals attempt to account for the geometry by exploiting the spatial coordinates but overlook the underlying intrinsic geometry of manifolds. In this paper, we propose a Curvature-aware Graph Attention for PDEs on manifolds by exploring the important intrinsic geometric quantities such as curvature and discrete gradient operator. It is realized via parallel transport and tensor field on manifolds. To accelerate computation, we present three curvature-oriented graph embedding approaches and derive closed-form parallel transport equations, and a subtree partition method is also developed to promote parameter-sharing. Our proposed curvature-aware attention can be used as a replacement for vanilla attention, and experiments show that it significantly improves the performance of the existing methods for solving PDEs on manifolds. Our code is available at https://github.com/Supradax/CurvGT.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-liao25b, title = {Curvature-aware Graph Attention for {PDE}s on Manifolds}, author = {Liao, Yunfeng and Guan, Jiawen and Li, Xiucheng}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {37476--37494}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/liao25b/liao25b.pdf}, url = {https://proceedings.mlr.press/v267/liao25b.html}, abstract = {Deep models have recently achieved remarkable performances in solving partial differential equations (PDEs). The previous methods are mostly focused on PDEs arising in Euclidean spaces with less emphasis on the general manifolds with rich geometry. Several proposals attempt to account for the geometry by exploiting the spatial coordinates but overlook the underlying intrinsic geometry of manifolds. In this paper, we propose a Curvature-aware Graph Attention for PDEs on manifolds by exploring the important intrinsic geometric quantities such as curvature and discrete gradient operator. It is realized via parallel transport and tensor field on manifolds. To accelerate computation, we present three curvature-oriented graph embedding approaches and derive closed-form parallel transport equations, and a subtree partition method is also developed to promote parameter-sharing. Our proposed curvature-aware attention can be used as a replacement for vanilla attention, and experiments show that it significantly improves the performance of the existing methods for solving PDEs on manifolds. Our code is available at https://github.com/Supradax/CurvGT.} }
Endnote
%0 Conference Paper %T Curvature-aware Graph Attention for PDEs on Manifolds %A Yunfeng Liao %A Jiawen Guan %A Xiucheng Li %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-liao25b %I PMLR %P 37476--37494 %U https://proceedings.mlr.press/v267/liao25b.html %V 267 %X Deep models have recently achieved remarkable performances in solving partial differential equations (PDEs). The previous methods are mostly focused on PDEs arising in Euclidean spaces with less emphasis on the general manifolds with rich geometry. Several proposals attempt to account for the geometry by exploiting the spatial coordinates but overlook the underlying intrinsic geometry of manifolds. In this paper, we propose a Curvature-aware Graph Attention for PDEs on manifolds by exploring the important intrinsic geometric quantities such as curvature and discrete gradient operator. It is realized via parallel transport and tensor field on manifolds. To accelerate computation, we present three curvature-oriented graph embedding approaches and derive closed-form parallel transport equations, and a subtree partition method is also developed to promote parameter-sharing. Our proposed curvature-aware attention can be used as a replacement for vanilla attention, and experiments show that it significantly improves the performance of the existing methods for solving PDEs on manifolds. Our code is available at https://github.com/Supradax/CurvGT.
APA
Liao, Y., Guan, J. & Li, X.. (2025). Curvature-aware Graph Attention for PDEs on Manifolds. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:37476-37494 Available from https://proceedings.mlr.press/v267/liao25b.html.

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