Parameterized Physics-informed Neural Networks for Parameterized PDEs

Woojin Cho, Minju Jo, Haksoo Lim, Kookjin Lee, Dongeun Lee, Sanghyun Hong, Noseong Park
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:8510-8533, 2024.

Abstract

Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Raynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P$^2$INNs). P$^2$INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P$^2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known “failure modes”.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-cho24b, title = {Parameterized Physics-informed Neural Networks for Parameterized {PDE}s}, author = {Cho, Woojin and Jo, Minju and Lim, Haksoo and Lee, Kookjin and Lee, Dongeun and Hong, Sanghyun and Park, Noseong}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {8510--8533}, 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/cho24b/cho24b.pdf}, url = {https://proceedings.mlr.press/v235/cho24b.html}, abstract = {Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Raynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P$^2$INNs). P$^2$INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P$^2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known “failure modes”.} }
Endnote
%0 Conference Paper %T Parameterized Physics-informed Neural Networks for Parameterized PDEs %A Woojin Cho %A Minju Jo %A Haksoo Lim %A Kookjin Lee %A Dongeun Lee %A Sanghyun Hong %A Noseong Park %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-cho24b %I PMLR %P 8510--8533 %U https://proceedings.mlr.press/v235/cho24b.html %V 235 %X Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Raynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P$^2$INNs). P$^2$INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P$^2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known “failure modes”.
APA
Cho, W., Jo, M., Lim, H., Lee, K., Lee, D., Hong, S. & Park, N.. (2024). Parameterized Physics-informed Neural Networks for Parameterized PDEs. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:8510-8533 Available from https://proceedings.mlr.press/v235/cho24b.html.

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