PINNtegrate: PINN-based Integral-Learning for Variational and Interface Problems

Frank Ehebrecht, Toni Scharle, Martin Atzmueller
Proceedings of the 1st ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications", PMLR 255:1-16, 2024.

Abstract

Physics Informed Neural Networks (PINNs) feature applications to various partial differential equations (PDEs) in physics and engineering. Many real-world problems contain interfaces, i.e., discontinuities in some model parameter, and have to be included in any relevant PDE solver toolkit. These problems do not necessarily admit smooth solutions. Therefore, interfaces cannot be naturally included into classical PINNs, since their learning algorithm uses the strong formulation of the PDE and does not include solutions in the weak sense. The interface information can be incorporated either by an additional flux condition on the interface or by a variational formulation, thus also allowing weak solutions. This paper proposes new approaches to combine either the weak or energy functional formulation with the piece-wise strong formulation, to be able to tackle interface problems. Our new method PINNtegrate can incorporate integrals into the neural network learning algorithm. This novel method cannot only be applied to interface problems but also to other problems that contain an integrand as an optimization objective. We demonstrate PINNtegrate on variational minimal surface and interface problems of linear elliptic PDEs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v255-ehebrecht24a, title = {PINNtegrate: PINN-based Integral-Learning for Variational and Interface Problems}, author = {Ehebrecht, Frank and Scharle, Toni and Atzmueller, Martin}, booktitle = {Proceedings of the 1st ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications"}, pages = {1--16}, year = {2024}, editor = {Coelho, Cecı́lia and Zimmering, Bernd and Costa, M. Fernanda P. and Ferrás, Luı́s L. and Niggemann, Oliver}, volume = {255}, series = {Proceedings of Machine Learning Research}, month = {20 Oct}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v255/main/assets/ehebrecht24a/ehebrecht24a.pdf}, url = {https://proceedings.mlr.press/v255/ehebrecht24a.html}, abstract = {Physics Informed Neural Networks (PINNs) feature applications to various partial differential equations (PDEs) in physics and engineering. Many real-world problems contain interfaces, i.e., discontinuities in some model parameter, and have to be included in any relevant PDE solver toolkit. These problems do not necessarily admit smooth solutions. Therefore, interfaces cannot be naturally included into classical PINNs, since their learning algorithm uses the strong formulation of the PDE and does not include solutions in the weak sense. The interface information can be incorporated either by an additional flux condition on the interface or by a variational formulation, thus also allowing weak solutions. This paper proposes new approaches to combine either the weak or energy functional formulation with the piece-wise strong formulation, to be able to tackle interface problems. Our new method PINNtegrate can incorporate integrals into the neural network learning algorithm. This novel method cannot only be applied to interface problems but also to other problems that contain an integrand as an optimization objective. We demonstrate PINNtegrate on variational minimal surface and interface problems of linear elliptic PDEs.} }
Endnote
%0 Conference Paper %T PINNtegrate: PINN-based Integral-Learning for Variational and Interface Problems %A Frank Ehebrecht %A Toni Scharle %A Martin Atzmueller %B Proceedings of the 1st ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications" %C Proceedings of Machine Learning Research %D 2024 %E Cecı́lia Coelho %E Bernd Zimmering %E M. Fernanda P. Costa %E Luı́s L. Ferrás %E Oliver Niggemann %F pmlr-v255-ehebrecht24a %I PMLR %P 1--16 %U https://proceedings.mlr.press/v255/ehebrecht24a.html %V 255 %X Physics Informed Neural Networks (PINNs) feature applications to various partial differential equations (PDEs) in physics and engineering. Many real-world problems contain interfaces, i.e., discontinuities in some model parameter, and have to be included in any relevant PDE solver toolkit. These problems do not necessarily admit smooth solutions. Therefore, interfaces cannot be naturally included into classical PINNs, since their learning algorithm uses the strong formulation of the PDE and does not include solutions in the weak sense. The interface information can be incorporated either by an additional flux condition on the interface or by a variational formulation, thus also allowing weak solutions. This paper proposes new approaches to combine either the weak or energy functional formulation with the piece-wise strong formulation, to be able to tackle interface problems. Our new method PINNtegrate can incorporate integrals into the neural network learning algorithm. This novel method cannot only be applied to interface problems but also to other problems that contain an integrand as an optimization objective. We demonstrate PINNtegrate on variational minimal surface and interface problems of linear elliptic PDEs.
APA
Ehebrecht, F., Scharle, T. & Atzmueller, M.. (2024). PINNtegrate: PINN-based Integral-Learning for Variational and Interface Problems. Proceedings of the 1st ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications", in Proceedings of Machine Learning Research 255:1-16 Available from https://proceedings.mlr.press/v255/ehebrecht24a.html.

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