2024-10-06-ehebrecht24a.md

title	abstract	layout	series	publisher	issn	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	container-title	volume	genre	issued	pdf	extras
PINNtegrate: PINN-based Integral-Learning for Variational and Interface Problems	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.	inproceedings	Proceedings of Machine Learning Research	PMLR	2640-3498	ehebrecht24a	0	PINNtegrate: PINN-based Integral-Learning for Variational and Interface Problems	1	16	1-16	1	false	Ehebrecht, Frank and Scharle, Toni and Atzmueller, Martin	given family Frank Ehebrecht given family Toni Scharle given family Martin Atzmueller	2024-10-06		Proceedings of the 1st ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications"	255	inproceedings	date-parts 2024 10 6	https://raw.githubusercontent.com/mlresearch/v255/main/assets/ehebrecht24a/ehebrecht24a.pdf