We introduce a novel machine learning-based approach for data assimilation applied in the context of fluid mechanics. We consider as baseline the Reynolds Averaged Navier-Stokes (RANS) equations, a set of equations where the unknown is represented by the meanflow and a closure model based on the Reynolds-stress tensor is required for correctly computing the solution. We consider an adjoint-based optimization method augmented by the introduction of Graph Neural Networks (GNNs). To this end, we first train a model for the closure term based on a GNN. Second, the GNN model is introduced in the end-to-end training process of data assimilation, where the RANS equations are part of the architecture and act as a constraint for a physically consistent prediction. We show our results using direct numerical simulations based on a Finite Element Method (FEM) solver; a two-fold interface between the GNN model and the solver allows the GNN’s predictions to be incorporated into post-processing steps of the FEM analysis.