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M. Chenaud, F. Magoules, J. Alves, "Physics-Informed Graph Convolutional Networks: Towards a generalized framework for complex geometries", in P. Iványi, J. Logo, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Soft Computing, Machine Learning and Optimisation in Civil, Structural and Environmental Engineering", Civil-Comp Press, Edinburgh, UK, Online volume: CCC 5, Paper 4.2, 2023, doi:10.4203/ccc.5.4.2

Keywords: partial differential equations, finite element method, scientific machine learning, graph neural networks, physics-informed neural networks.

Abstract

Since the seminal work of M. Raissi, P. Perdikaris and G. E. Karniadakis [1] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for instance the extension of such models to complex three-dimensional geometries, and a study on how such approaches could be combined to classical numerical solvers. In this work, we justify the use of graph neural networks for these problems, based on the similarity between these architectures and the meshes used in traditional numerical techniques for solving partial differential equations. After proving an issue with the Physics- Informed framework for complex geometries, during the computation of PDE residuals, an alternative procedure is proposed, by combining classical numerical solvers and the Physics-Informed framework. Finally, we propose an implementation of this approach, that we test on a three-dimensional problem on an irregular geometry.

References

[1] M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, 2019, doi: 10.1016/j.jcp.2018.10.045

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Front Page Browse CCC CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Conferences ISSN 2753-3239 CCC: 5 PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON SOFT COMPUTING, MACHINE LEARNING AND OPTIMISATION IN CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING Edited by: P. Iványi, J. Logo and B.H.V. Topping Paper 4.2 Physics-Informed Graph Convolutional Networks: Towards a generalized framework for complex geometries M. Chenaud^1,2, F. Magoules^1,3 and J. Alves² ¹MICS, CentraleSupelec Universite Paris Saclay, Gif-sur-Yvette, France ²Transvalor S.A. E-Golf Park, Biot, France ³Faculty of Engineering and Information Technology, University of Pecs, Hungary doi:10.4203/ccc.5.4.2 Full Bibliographic Reference for this paper M. Chenaud, F. Magoules, J. Alves, "Physics-Informed Graph Convolutional Networks: Towards a generalized framework for complex geometries", in P. Iványi, J. Logo, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Soft Computing, Machine Learning and Optimisation in Civil, Structural and Environmental Engineering", Civil-Comp Press, Edinburgh, UK, Online volume: CCC 5, Paper 4.2, 2023, doi:10.4203/ccc.5.4.2 Keywords: partial differential equations, finite element method, scientific machine learning, graph neural networks, physics-informed neural networks. Abstract Since the seminal work of M. Raissi, P. Perdikaris and G. E. Karniadakis [1] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for instance the extension of such models to complex three-dimensional geometries, and a study on how such approaches could be combined to classical numerical solvers. In this work, we justify the use of graph neural networks for these problems, based on the similarity between these architectures and the meshes used in traditional numerical techniques for solving partial differential equations. After proving an issue with the Physics- Informed framework for complex geometries, during the computation of PDE residuals, an alternative procedure is proposed, by combining classical numerical solvers and the Physics-Informed framework. Finally, we propose an implementation of this approach, that we test on a three-dimensional problem on an irregular geometry. References [1] M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, 2019, doi: 10.1016/j.jcp.2018.10.045 download the full-text of this paper (PDF, 0 pages, 431 Kb) go to the previous paper go to the next paper return to the table of contents return to the volume description
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