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The spread of machine learning (ML) techniques in combination with theavailability of high-quality experimental and numerical data boosted in recentyears numerous applications in fluid mechanics. Among those, examples ofclosure models for turbulent flows or data-assimilation based on neuralnetworks (NN) are already numerous. However, it is well known that thesetechniques are prone to over-fit and necessitate an exceedingly large amount ofdata, unless enforcing physical constraints. We address those limitations byapplying graph neural networks (GNN). This architecture is characterized by anet of nodes that can be easily adapted to unstructured meshes. Moreover, it isknown GNN can show remarkable generalization capabilities as compared tostandard network models. Here, we demonstrate the use of GNN by interfacingthem with a finite elements (FEM) solver for the supervised learning ofReynolds-averaged Navier--Stokes equations. We consider as a test-case the flowpast bluff bodies; we train the model using the wake past a cylinder, atdifferent Reynolds numbers 40<Re<150 and resolved on different grids. The GNNmodel successfully predicts the Reynolds stress also in unseen casescharacterized by different geometric configurations and Reynolds numbers.Interestingly, a small data-set is used for achieving these performances, thussuggesting the applications of GNN in replacement of less flexible techniquessuch as convolutional NN.

Operator learning of RANS equations: a Graph Neural Network closure model