Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networks

Shibo Li, Michael Penwarden, Yiming Xu, Conor Tillinghast, Akil Narayan, Mike Kirby, Shandian Zhe
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:19855-19881, 2023.

Abstract

Physics-informed neural networks (PINNs) are emerging as popular mesh-free solvers for partial differential equations (PDEs). Recent extensions decompose the domain, apply different PINNs to solve the problem in each subdomain, and stitch the subdomains at the interface. Thereby, they can further alleviate the problem complexity, reduce the computational cost, and allow parallelization. However, the performance of multi-domain PINNs is sensitive to the choice of the interface conditions. While quite a few conditions have been proposed, there is no suggestion about how to select the conditions according to specific problems. To address this gap, we propose META Learning of Interface Conditions (METALIC), a simple, efficient yet powerful approach to dynamically determine appropriate interface conditions for solving a family of parametric PDEs. Specifically, we develop two contextual multi-arm bandit (MAB) models. The first one applies to the entire training course, and online updates a Gaussian process (GP) reward that given the PDE parameters and interface conditions predicts the performance. We prove a sub-linear regret bound for both UCB and Thompson sampling, which in theory guarantees the effectiveness of our MAB. The second one partitions the training into two stages, one is the stochastic phase and the other deterministic phase; we update a GP reward for each phase to enable different condition selections at the two stages to further bolster the flexibility and performance. We have shown the advantage of METALIC on four bench-mark PDE families.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-li23w, title = {Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networks}, author = {Li, Shibo and Penwarden, Michael and Xu, Yiming and Tillinghast, Conor and Narayan, Akil and Kirby, Mike and Zhe, Shandian}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {19855--19881}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/li23w/li23w.pdf}, url = {https://proceedings.mlr.press/v202/li23w.html}, abstract = {Physics-informed neural networks (PINNs) are emerging as popular mesh-free solvers for partial differential equations (PDEs). Recent extensions decompose the domain, apply different PINNs to solve the problem in each subdomain, and stitch the subdomains at the interface. Thereby, they can further alleviate the problem complexity, reduce the computational cost, and allow parallelization. However, the performance of multi-domain PINNs is sensitive to the choice of the interface conditions. While quite a few conditions have been proposed, there is no suggestion about how to select the conditions according to specific problems. To address this gap, we propose META Learning of Interface Conditions (METALIC), a simple, efficient yet powerful approach to dynamically determine appropriate interface conditions for solving a family of parametric PDEs. Specifically, we develop two contextual multi-arm bandit (MAB) models. The first one applies to the entire training course, and online updates a Gaussian process (GP) reward that given the PDE parameters and interface conditions predicts the performance. We prove a sub-linear regret bound for both UCB and Thompson sampling, which in theory guarantees the effectiveness of our MAB. The second one partitions the training into two stages, one is the stochastic phase and the other deterministic phase; we update a GP reward for each phase to enable different condition selections at the two stages to further bolster the flexibility and performance. We have shown the advantage of METALIC on four bench-mark PDE families.} }
Endnote
%0 Conference Paper %T Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networks %A Shibo Li %A Michael Penwarden %A Yiming Xu %A Conor Tillinghast %A Akil Narayan %A Mike Kirby %A Shandian Zhe %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-li23w %I PMLR %P 19855--19881 %U https://proceedings.mlr.press/v202/li23w.html %V 202 %X Physics-informed neural networks (PINNs) are emerging as popular mesh-free solvers for partial differential equations (PDEs). Recent extensions decompose the domain, apply different PINNs to solve the problem in each subdomain, and stitch the subdomains at the interface. Thereby, they can further alleviate the problem complexity, reduce the computational cost, and allow parallelization. However, the performance of multi-domain PINNs is sensitive to the choice of the interface conditions. While quite a few conditions have been proposed, there is no suggestion about how to select the conditions according to specific problems. To address this gap, we propose META Learning of Interface Conditions (METALIC), a simple, efficient yet powerful approach to dynamically determine appropriate interface conditions for solving a family of parametric PDEs. Specifically, we develop two contextual multi-arm bandit (MAB) models. The first one applies to the entire training course, and online updates a Gaussian process (GP) reward that given the PDE parameters and interface conditions predicts the performance. We prove a sub-linear regret bound for both UCB and Thompson sampling, which in theory guarantees the effectiveness of our MAB. The second one partitions the training into two stages, one is the stochastic phase and the other deterministic phase; we update a GP reward for each phase to enable different condition selections at the two stages to further bolster the flexibility and performance. We have shown the advantage of METALIC on four bench-mark PDE families.
APA
Li, S., Penwarden, M., Xu, Y., Tillinghast, C., Narayan, A., Kirby, M. & Zhe, S.. (2023). Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networks. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:19855-19881 Available from https://proceedings.mlr.press/v202/li23w.html.

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