Graph Neural Network- based nonlocal models: towards application to RANS turbulence modelling

The closure problem of the Reynolds-averaged Navier–Stokes (RANS) equations remains one of the main challenges in turbulence modelling, and no universally accepted solution currently exists. In recent years, machine learning (ML) has emerged as a promising alternative to traditional heuristic modelling approaches. In this thesis, graph neural networks (GNNs) are investigated as a framework to capture non-local interactions in fluid flows, which represent a key limitation of classical turbulence models and many data-driven approaches. The proposed architectures are systematically compared with feed-forward neural networks (FNNs) under comparable training conditions. The analysis is carried out through two numerical experiments. First, a simplified transport framework is considered to study the ability of graph architectures to capture non-local dependencies on computational meshes. Second, the methodology is extended to a RANS based setting using the Periodic Hills benchmark from the Closure Challenge dataset, where neural networks are trained to predict the turbulent eddy viscosity field. Different graph sampling strategies and network configurations are investigated to assess the robustness and predictive capability of the models. The trained architectures are also evaluated on geometries not included in the training set to assess their generalization properties. The results show that graph neural networks can effectively exploit the spatial connectivity of CFD meshes, particularly in problems characterized by strong non-local interactions. However, when only a limited amount of training data is available, feed-forward neural networks can achieve higher prediction accuracy. This suggests that graph-based models may require larger datasets to fully exploit their ability to capture spatial dependencies. These findings provide useful insights toward the development of data-driven turbulence models.