  This paper presents a graph autoencoder architecture capable of performing
projection-based model-order reduction (PMOR) on advection-dominated flows
modeled by unstructured meshes. The autoencoder is coupled with the time
integration scheme from a traditional deep least-squares Petrov-Galerkin
projection and provides the first deployment of a graph autoencoder into a PMOR
framework. The presented graph autoencoder is constructed with a two-part
process that consists of (1) generating a hierarchy of reduced graphs to
emulate the compressive abilities of convolutional neural networks (CNNs) and
(2) training a message passing operation at each step in the hierarchy of
reduced graphs to emulate the filtering process of a CNN. The resulting
framework provides improved flexibility over traditional CNN-based autoencoders
because it is extendable to unstructured meshes. To highlight the capabilities
of the proposed framework, which is named geometric deep least-squares
Petrov-Galerkin (GD-LSPG), we benchmark the method on a one-dimensional
Burgers' equation problem with a structured mesh and demonstrate the
flexibility of GD-LSPG by deploying it to a two-dimensional Euler equations
model that uses an unstructured mesh. The proposed framework provides
considerable improvement in accuracy for very low-dimensional latent spaces in
comparison with traditional affine projections.
