Physics and Lie symmetry informed Gaussian processes

David Dalton, Dirk Husmeier, Hao Gao
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:9953-9975, 2024.

Abstract

Physics-informed machine learning (PIML) has established itself as a new scientific paradigm which enables the seamless integration of observational data with partial differential equation (PDE) based physics models. A powerful tool for the analysis, reduction and solution of PDEs is the Lie symmetry method. Nevertheless, only recently has the integration of such symmetries into PIML frameworks begun to be explored. The present work adds to this growing literature by introducing an approach for incorporating a Lie symmetry into a physics-informed Gaussian process (GP) model. The symmetry is introduced as a constraint on the GP; either in a soft manner via virtual observations of an induced PDE called the invariant surface condition, or explicitly through the design of the kernel. Experimental results demonstrate that the use of symmetry constraints improves the performance of the GP for both forward and inverse problems, and that our approach offers competitive performance with neural networks in the low-data environment.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-dalton24a, title = {Physics and Lie symmetry informed {G}aussian processes}, author = {Dalton, David and Husmeier, Dirk and Gao, Hao}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {9953--9975}, 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/dalton24a/dalton24a.pdf}, url = {https://proceedings.mlr.press/v235/dalton24a.html}, abstract = {Physics-informed machine learning (PIML) has established itself as a new scientific paradigm which enables the seamless integration of observational data with partial differential equation (PDE) based physics models. A powerful tool for the analysis, reduction and solution of PDEs is the Lie symmetry method. Nevertheless, only recently has the integration of such symmetries into PIML frameworks begun to be explored. The present work adds to this growing literature by introducing an approach for incorporating a Lie symmetry into a physics-informed Gaussian process (GP) model. The symmetry is introduced as a constraint on the GP; either in a soft manner via virtual observations of an induced PDE called the invariant surface condition, or explicitly through the design of the kernel. Experimental results demonstrate that the use of symmetry constraints improves the performance of the GP for both forward and inverse problems, and that our approach offers competitive performance with neural networks in the low-data environment.} }
Endnote
%0 Conference Paper %T Physics and Lie symmetry informed Gaussian processes %A David Dalton %A Dirk Husmeier %A Hao Gao %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-dalton24a %I PMLR %P 9953--9975 %U https://proceedings.mlr.press/v235/dalton24a.html %V 235 %X Physics-informed machine learning (PIML) has established itself as a new scientific paradigm which enables the seamless integration of observational data with partial differential equation (PDE) based physics models. A powerful tool for the analysis, reduction and solution of PDEs is the Lie symmetry method. Nevertheless, only recently has the integration of such symmetries into PIML frameworks begun to be explored. The present work adds to this growing literature by introducing an approach for incorporating a Lie symmetry into a physics-informed Gaussian process (GP) model. The symmetry is introduced as a constraint on the GP; either in a soft manner via virtual observations of an induced PDE called the invariant surface condition, or explicitly through the design of the kernel. Experimental results demonstrate that the use of symmetry constraints improves the performance of the GP for both forward and inverse problems, and that our approach offers competitive performance with neural networks in the low-data environment.
APA
Dalton, D., Husmeier, D. & Gao, H.. (2024). Physics and Lie symmetry informed Gaussian processes. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:9953-9975 Available from https://proceedings.mlr.press/v235/dalton24a.html.

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