Mesh-based simulations play a key role when modeling complex physical systems that, in many disciplines across science and engineering, require the solution to parametrized time-dependent nonlinear partial differential equations (PDEs). In this context, full order models (FOMs), such as those relying on the finite element method, can reach high levels of accuracy, however often yielding intensive simulations to run. For this reason, surrogate models are developed to replace computationally expensive solvers with more efficient ones, which can strike favorable trade-offs between accuracy and efficiency. This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs in the presence of geometrical variability. In particular, we propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme where a GNN architecture is used to efficiently evolve the system. With respect to the majority of surrogate models, the proposed approach stands out for its ability of tackling problems with parameter-dependent spatial domains, while simultaneously generalizing to different geometries and mesh resolutions. We assess the effectiveness of the proposed approach through a series of numerical experiments, involving both two- and three-dimensional problems, showing that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios.
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December 2023
Research Article|
December 12 2023
Deep learning-based surrogate models for parametrized PDEs: Handling geometric variability through graph neural networks
Special Collection:
Nonlinear Model Reduction From Equations and Data
Nicola Rares Franco
;
Nicola Rares Franco
a)
(Conceptualization, Formal analysis, Methodology, Software, Supervision, Writing – original draft, Writing – review & editing)
MOX, Department of Mathematics, Politecnico di Milano
, Milan 20133, Italy
a)Author to whom correspondence should be addressed: nicolarares.franco@polimi.it
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Stefania Fresca
;
Stefania Fresca
(Conceptualization, Methodology, Resources, Supervision, Writing – original draft, Writing – review & editing)
MOX, Department of Mathematics, Politecnico di Milano
, Milan 20133, Italy
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Filippo Tombari
;
Filippo Tombari
(Data curation, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft)
MOX, Department of Mathematics, Politecnico di Milano
, Milan 20133, Italy
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Andrea Manzoni
Andrea Manzoni
(Conceptualization, Formal analysis, Funding acquisition, Methodology, Project administration, Resources, Supervision, Validation, Writing – review & editing)
MOX, Department of Mathematics, Politecnico di Milano
, Milan 20133, Italy
Search for other works by this author on:
a)Author to whom correspondence should be addressed: nicolarares.franco@polimi.it
Chaos 33, 123121 (2023)
Article history
Received:
July 31 2023
Accepted:
November 18 2023
Citation
Nicola Rares Franco, Stefania Fresca, Filippo Tombari, Andrea Manzoni; Deep learning-based surrogate models for parametrized PDEs: Handling geometric variability through graph neural networks. Chaos 1 December 2023; 33 (12): 123121. https://doi.org/10.1063/5.0170101
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