A graph convolutional autoencoder approach to model order reduction for parametrized PDEs

The present work proposes a framework for nonlinear model order reduction based on a Graph Convolutional Autoencoder (GCA-ROM). In the reduced order modeling (ROM) context, one is interested in obtaining real-time and many-query evaluations of parametric Partial Differential Equations (PDEs). Linear...

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Veröffentlicht in:Journal of computational physics Jg. 501; S. 112762
Hauptverfasser: Pichi, Federico, Moya, Beatriz, Hesthaven, Jan S.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 15.03.2024
Schlagworte:
Convolutional autoencoder
Geometrical parameters
Graph neural networks
Nonlinear reduced order modeling
Parametric PDEs
Nonlinear reduced order modeling
Parametric PDEs
Graph neural networks
Geometrical parameters
Convolutional autoencoder
ISSN:0021-9991, 1090-2716
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Zusammenfassung:The present work proposes a framework for nonlinear model order reduction based on a Graph Convolutional Autoencoder (GCA-ROM). In the reduced order modeling (ROM) context, one is interested in obtaining real-time and many-query evaluations of parametric Partial Differential Equations (PDEs). Linear techniques such as Proper Orthogonal Decomposition (POD) and Greedy algorithms have been analyzed thoroughly, but they are more suitable when dealing with linear and affine models showing a fast decay of the Kolmogorov n-width. On one hand, the autoencoder architecture represents a nonlinear generalization of the POD compression procedure, allowing one to encode the main information in a latent set of variables while extracting their main features. On the other hand, Graph Neural Networks (GNNs) constitute a natural framework for studying PDE solutions defined on unstructured meshes. Here, we develop a non-intrusive and data-driven nonlinear reduction approach, exploiting GNNs to encode the reduced manifold and enable fast evaluations of parametrized PDEs. We show the capabilities of the methodology for several models: linear/nonlinear and scalar/vector problems with fast/slow decay in the physically and geometrically parametrized setting. The main properties of our approach consist of (i) high generalizability in the low-data regime even for complex behaviors, (ii) physical compliance with general unstructured grids, and (iii) exploitation of pooling and un-pooling operations to learn from scattered data. •Nonlinear model order reduction framework exploiting Graph Convolutional Autoencoder.•High generalizability of the methodology in the low-data regime.•Physical compliance with unstructured meshes.•Pooling and un-pooling operations to learn from scattered data.•Tested on several benchmarks with physical and geometrical parameters.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2024.112762