Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning

Junfeng Chen, Kailiang Wu
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:7526-7552, 2024.

Abstract

Operator learning for Partial Differential Equations (PDEs) is rapidly emerging as a promising approach for surrogate modeling of intricate systems. Transformers with the self-attention mechanism—a powerful tool originally designed for natural language processing—have recently been adapted for operator learning. However, they confront challenges, including high computational demands and limited interpretability. This raises a critical question: Is there a more efficient attention mechanism for Transformer-based operator learning? This paper proposes the Position-induced Transformer (PiT), built on an innovative position-attention mechanism, which demonstrates significant advantages over the classical self-attention in operator learning. Position-attention draws inspiration from numerical methods for PDEs. Different from self-attention, position-attention is induced by only the spatial interrelations of sampling positions for input functions of the operators, and does not rely on the input function values themselves, thereby greatly boosting efficiency. PiT exhibits superior performance over current state-of-the-art neural operators in a variety of complex operator learning tasks across diverse PDE benchmarks. Additionally, PiT possesses an enhanced discretization convergence feature, compared to the widely-used Fourier neural operator.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-chen24au, title = {Positional Knowledge is All You Need: Position-induced Transformer ({P}i{T}) for Operator Learning}, author = {Chen, Junfeng and Wu, Kailiang}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {7526--7552}, 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/chen24au/chen24au.pdf}, url = {https://proceedings.mlr.press/v235/chen24au.html}, abstract = {Operator learning for Partial Differential Equations (PDEs) is rapidly emerging as a promising approach for surrogate modeling of intricate systems. Transformers with the self-attention mechanism—a powerful tool originally designed for natural language processing—have recently been adapted for operator learning. However, they confront challenges, including high computational demands and limited interpretability. This raises a critical question: Is there a more efficient attention mechanism for Transformer-based operator learning? This paper proposes the Position-induced Transformer (PiT), built on an innovative position-attention mechanism, which demonstrates significant advantages over the classical self-attention in operator learning. Position-attention draws inspiration from numerical methods for PDEs. Different from self-attention, position-attention is induced by only the spatial interrelations of sampling positions for input functions of the operators, and does not rely on the input function values themselves, thereby greatly boosting efficiency. PiT exhibits superior performance over current state-of-the-art neural operators in a variety of complex operator learning tasks across diverse PDE benchmarks. Additionally, PiT possesses an enhanced discretization convergence feature, compared to the widely-used Fourier neural operator.} }
Endnote
%0 Conference Paper %T Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning %A Junfeng Chen %A Kailiang Wu %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-chen24au %I PMLR %P 7526--7552 %U https://proceedings.mlr.press/v235/chen24au.html %V 235 %X Operator learning for Partial Differential Equations (PDEs) is rapidly emerging as a promising approach for surrogate modeling of intricate systems. Transformers with the self-attention mechanism—a powerful tool originally designed for natural language processing—have recently been adapted for operator learning. However, they confront challenges, including high computational demands and limited interpretability. This raises a critical question: Is there a more efficient attention mechanism for Transformer-based operator learning? This paper proposes the Position-induced Transformer (PiT), built on an innovative position-attention mechanism, which demonstrates significant advantages over the classical self-attention in operator learning. Position-attention draws inspiration from numerical methods for PDEs. Different from self-attention, position-attention is induced by only the spatial interrelations of sampling positions for input functions of the operators, and does not rely on the input function values themselves, thereby greatly boosting efficiency. PiT exhibits superior performance over current state-of-the-art neural operators in a variety of complex operator learning tasks across diverse PDE benchmarks. Additionally, PiT possesses an enhanced discretization convergence feature, compared to the widely-used Fourier neural operator.
APA
Chen, J. & Wu, K.. (2024). Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:7526-7552 Available from https://proceedings.mlr.press/v235/chen24au.html.

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