Multi-Scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators

Leon Migus, Yuan Yin, Jocelyn Ahmed Mazari, Patrick Gallinari1
Proceedings of Topological, Algebraic, and Geometric Learning Workshops 2022, PMLR 196:332-340, 2022.

Abstract

Representing physical signals at different scales is among the most challenging problems in engineering. Several multi-scale modeling tools have been developed to describe physical systems governed by Partial Differential Equations (PDEs). These tools are at the crossroad of principled physical models and numerical schema. Recently, data-driven models have been introduced to speed-up the approximation of PDE solutions compared to numerical solvers. Among these recent data-driven methods, neural integral operators are a class that learn a mapping between function spaces. These functions are discretized on graphs (meshes) which are appropriate for modeling interactions in physical phenomena. In this work, we study three multi-resolution schema with integral kernel operators that can be approximated with Message Passing Graph Neural Networks (MPGNNs). To validate our study, we make extensive MPGNNs experiments with well-chosen metrics considering steady and unsteady PDEs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v196-migus22a, title = {Multi-Scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators}, author = {Migus, Leon and Yin, Yuan and Ahmed Mazari, Jocelyn and Gallinari, Patrick}, booktitle = {Proceedings of Topological, Algebraic, and Geometric Learning Workshops 2022}, pages = {332--340}, year = {2022}, editor = {Cloninger, Alexander and Doster, Timothy and Emerson, Tegan and Kaul, Manohar and Ktena, Ira and Kvinge, Henry and Miolane, Nina and Rieck, Bastian and Tymochko, Sarah and Wolf, Guy}, volume = {196}, series = {Proceedings of Machine Learning Research}, month = {25 Feb--22 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v196/migus22a/migus22a.pdf}, url = {https://proceedings.mlr.press/v196/migus22a.html}, abstract = {Representing physical signals at different scales is among the most challenging problems in engineering. Several multi-scale modeling tools have been developed to describe physical systems governed by Partial Differential Equations (PDEs). These tools are at the crossroad of principled physical models and numerical schema. Recently, data-driven models have been introduced to speed-up the approximation of PDE solutions compared to numerical solvers. Among these recent data-driven methods, neural integral operators are a class that learn a mapping between function spaces. These functions are discretized on graphs (meshes) which are appropriate for modeling interactions in physical phenomena. In this work, we study three multi-resolution schema with integral kernel operators that can be approximated with Message Passing Graph Neural Networks (MPGNNs). To validate our study, we make extensive MPGNNs experiments with well-chosen metrics considering steady and unsteady PDEs.} }
Endnote
%0 Conference Paper %T Multi-Scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators %A Leon Migus %A Yuan Yin %A Jocelyn Ahmed Mazari %A Patrick Gallinari1 %B Proceedings of Topological, Algebraic, and Geometric Learning Workshops 2022 %C Proceedings of Machine Learning Research %D 2022 %E Alexander Cloninger %E Timothy Doster %E Tegan Emerson %E Manohar Kaul %E Ira Ktena %E Henry Kvinge %E Nina Miolane %E Bastian Rieck %E Sarah Tymochko %E Guy Wolf %F pmlr-v196-migus22a %I PMLR %P 332--340 %U https://proceedings.mlr.press/v196/migus22a.html %V 196 %X Representing physical signals at different scales is among the most challenging problems in engineering. Several multi-scale modeling tools have been developed to describe physical systems governed by Partial Differential Equations (PDEs). These tools are at the crossroad of principled physical models and numerical schema. Recently, data-driven models have been introduced to speed-up the approximation of PDE solutions compared to numerical solvers. Among these recent data-driven methods, neural integral operators are a class that learn a mapping between function spaces. These functions are discretized on graphs (meshes) which are appropriate for modeling interactions in physical phenomena. In this work, we study three multi-resolution schema with integral kernel operators that can be approximated with Message Passing Graph Neural Networks (MPGNNs). To validate our study, we make extensive MPGNNs experiments with well-chosen metrics considering steady and unsteady PDEs.
APA
Migus, L., Yin, Y., Ahmed Mazari, J. & Gallinari1, P.. (2022). Multi-Scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators. Proceedings of Topological, Algebraic, and Geometric Learning Workshops 2022, in Proceedings of Machine Learning Research 196:332-340 Available from https://proceedings.mlr.press/v196/migus22a.html.

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