Dimensional compression and reconstruction for unstructured finite volume meshes via geometric deep learning

The finite volume method (FVM) is an attractive approach to simulate complex physical phenomena by solving integral forms of the governing physical equations. The FVM often employs an unstructured mesh (UM) to spatially discretize the domain into a mesh of cells that are not created in the form of a structured grid. This lack of a natural grid structure for an UM makes the direct application of convolutional neural networks for the purpose of model reduction currently untenable. The present work aims to overcome this limitation by incorporating graph neural networks (GNNs) to perform dimensional compression intuitively upon an UM in a machine learning framework. GNNs are a class of machine learning methods selected for this application due to their ability to represent and extract information from relational data, as is seen in an UM. A GNN-based approach to perform dimensional compression and reconstruction upon the FVM employing an UM will be presented, and the method will be tested on a problem with a Kelvin-Helmholtz instability.