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

TL;DR

This paper introduces a nonlinear model order reduction method using Graph Convolutional Autoencoders and Graph Neural Networks to efficiently approximate parametrized PDEs on unstructured meshes, outperforming traditional linear techniques.

Contribution

The work develops a novel GCA-ROM framework combining GNNs and autoencoders for nonlinear PDE model reduction, enabling fast, data-driven, and physically compliant evaluations on unstructured grids.

Findings

Effective for linear and nonlinear PDEs with various parametrizations.

High accuracy with limited training data.

Applicable to unstructured mesh geometries.

Abstract

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 regimes, (ii) physical compliance with general unstructured grids, and (iii) exploitation of pooling and un-pooling operations to learn from scattered data.