  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.
