SetPINNs: Set-based Physics-informed Neural Networks

Mayank Nagda, Phil Ostheimer, Thomas Specht, Frank Rhein, Fabian Jirasek, Stephan Mandt, Marius Kloft, Sophie Fellenz
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:3061-3069, 2026.

Abstract

Physics-Informed Neural Networks (PINNs) solve partial differential equations using deep learning. However, conventional PINNs perform pointwise predictions that neglect dependencies within a domain, which may result in suboptimal solutions. We introduce SetPINNs, a framework that effectively captures local dependencies. With a finite element-inspired sampling scheme, we partition the domain into sets to model local dependencies while simultaneously enforcing physical laws. We provide a rigorous theoretical analysis showing that SetPINNs yield unbiased, lower-variance estimates of residual energy and its gradients, ensuring improved domain coverage and reduced residual error. Extensive experiments on synthetic and real-world tasks show improved accuracy, efficiency, and robustness.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-nagda26a, title = { SetPINNs: Set-based Physics-informed Neural Networks }, author = {Nagda, Mayank and Ostheimer, Phil and Specht, Thomas and Rhein, Frank and Jirasek, Fabian and Mandt, Stephan and Kloft, Marius and Fellenz, Sophie}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {3061--3069}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/nagda26a/nagda26a.pdf}, url = {https://proceedings.mlr.press/v300/nagda26a.html}, abstract = { Physics-Informed Neural Networks (PINNs) solve partial differential equations using deep learning. However, conventional PINNs perform pointwise predictions that neglect dependencies within a domain, which may result in suboptimal solutions. We introduce SetPINNs, a framework that effectively captures local dependencies. With a finite element-inspired sampling scheme, we partition the domain into sets to model local dependencies while simultaneously enforcing physical laws. We provide a rigorous theoretical analysis showing that SetPINNs yield unbiased, lower-variance estimates of residual energy and its gradients, ensuring improved domain coverage and reduced residual error. Extensive experiments on synthetic and real-world tasks show improved accuracy, efficiency, and robustness. } }
Endnote
%0 Conference Paper %T SetPINNs: Set-based Physics-informed Neural Networks %A Mayank Nagda %A Phil Ostheimer %A Thomas Specht %A Frank Rhein %A Fabian Jirasek %A Stephan Mandt %A Marius Kloft %A Sophie Fellenz %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-nagda26a %I PMLR %P 3061--3069 %U https://proceedings.mlr.press/v300/nagda26a.html %V 300 %X Physics-Informed Neural Networks (PINNs) solve partial differential equations using deep learning. However, conventional PINNs perform pointwise predictions that neglect dependencies within a domain, which may result in suboptimal solutions. We introduce SetPINNs, a framework that effectively captures local dependencies. With a finite element-inspired sampling scheme, we partition the domain into sets to model local dependencies while simultaneously enforcing physical laws. We provide a rigorous theoretical analysis showing that SetPINNs yield unbiased, lower-variance estimates of residual energy and its gradients, ensuring improved domain coverage and reduced residual error. Extensive experiments on synthetic and real-world tasks show improved accuracy, efficiency, and robustness.
APA
Nagda, M., Ostheimer, P., Specht, T., Rhein, F., Jirasek, F., Mandt, S., Kloft, M. & Fellenz, S.. (2026). SetPINNs: Set-based Physics-informed Neural Networks . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:3061-3069 Available from https://proceedings.mlr.press/v300/nagda26a.html.

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