  Large-scale numerical simulations often come at the expense of daunting
computations. High-Performance Computing has enhanced the process, but adapting
legacy codes to leverage parallel GPU computations remains challenging.
Meanwhile, Machine Learning models can harness GPU computations effectively but
often struggle with generalization and accuracy. Graph Neural Networks (GNNs),
in particular, are great for learning from unstructured data like meshes but
are often limited to small-scale problems. Moreover, the capabilities of the
trained model usually restrict the accuracy of the data-driven solution. To
benefit from both worlds, this paper introduces a novel preconditioner
integrating a GNN model within a multi-level Domain Decomposition framework.
The proposed GNN-based preconditioner is used to enhance the efficiency of a
Krylov method, resulting in a hybrid solver that can converge with any desired
level of accuracy. The efficiency of the Krylov method greatly benefits from
the GNN preconditioner, which is adaptable to meshes of any size and shape, is
executed on GPUs, and features a multi-level approach to enforce the
scalability of the entire process. Several experiments are conducted to
validate the numerical behavior of the hybrid solver, and an in-depth analysis
of its performance is proposed to assess its competitiveness against a C++
legacy solver.
