Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models

Congcong Zhu, Xiaoyan Xu, Jiayue Han, Jingrun Chen
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:80223-80258, 2025.

Abstract

Auto-regressive partial differential equation (PDE) foundation models have shown great potential in handling time-dependent data. However, these models suffer from error accumulation caused by the shortcut problem deeply rooted in auto-regressive prediction. The challenge becomes particularly evident for out-of-distribution data, as the pretraining performance may approach random model initialization for downstream tasks with long-term dynamics. To deal with this problem, we propose physics-informed temporal alignment (PITA), a self-supervised learning framework inspired by inverse problem solving. Specifically, PITA aligns the physical dynamics discovered at different time steps on each given PDE trajectory by integrating physics-informed constraints into the self-supervision signal. The alignment is derived from observation data without relying on known physics priors, indicating strong generalization ability to out-of-distribution data. Extensive experiments show that PITA significantly enhances the accuracy and robustness of existing foundation models on diverse time-dependent PDE data. The code is available at https://github.com/SCAILab-USTC/PITA.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-zhu25w, title = {Physics-informed Temporal Alignment for Auto-regressive {PDE} Foundation Models}, author = {Zhu, Congcong and Xu, Xiaoyan and Han, Jiayue and Chen, Jingrun}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {80223--80258}, 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/zhu25w/zhu25w.pdf}, url = {https://proceedings.mlr.press/v267/zhu25w.html}, abstract = {Auto-regressive partial differential equation (PDE) foundation models have shown great potential in handling time-dependent data. However, these models suffer from error accumulation caused by the shortcut problem deeply rooted in auto-regressive prediction. The challenge becomes particularly evident for out-of-distribution data, as the pretraining performance may approach random model initialization for downstream tasks with long-term dynamics. To deal with this problem, we propose physics-informed temporal alignment (PITA), a self-supervised learning framework inspired by inverse problem solving. Specifically, PITA aligns the physical dynamics discovered at different time steps on each given PDE trajectory by integrating physics-informed constraints into the self-supervision signal. The alignment is derived from observation data without relying on known physics priors, indicating strong generalization ability to out-of-distribution data. Extensive experiments show that PITA significantly enhances the accuracy and robustness of existing foundation models on diverse time-dependent PDE data. The code is available at https://github.com/SCAILab-USTC/PITA.} }
Endnote
%0 Conference Paper %T Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models %A Congcong Zhu %A Xiaoyan Xu %A Jiayue Han %A Jingrun Chen %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-zhu25w %I PMLR %P 80223--80258 %U https://proceedings.mlr.press/v267/zhu25w.html %V 267 %X Auto-regressive partial differential equation (PDE) foundation models have shown great potential in handling time-dependent data. However, these models suffer from error accumulation caused by the shortcut problem deeply rooted in auto-regressive prediction. The challenge becomes particularly evident for out-of-distribution data, as the pretraining performance may approach random model initialization for downstream tasks with long-term dynamics. To deal with this problem, we propose physics-informed temporal alignment (PITA), a self-supervised learning framework inspired by inverse problem solving. Specifically, PITA aligns the physical dynamics discovered at different time steps on each given PDE trajectory by integrating physics-informed constraints into the self-supervision signal. The alignment is derived from observation data without relying on known physics priors, indicating strong generalization ability to out-of-distribution data. Extensive experiments show that PITA significantly enhances the accuracy and robustness of existing foundation models on diverse time-dependent PDE data. The code is available at https://github.com/SCAILab-USTC/PITA.
APA
Zhu, C., Xu, X., Han, J. & Chen, J.. (2025). Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:80223-80258 Available from https://proceedings.mlr.press/v267/zhu25w.html.

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