High-fidelity large-eddy simulations (LES) of high Reynolds number flows are essential to design low-carbon footprint energy conversion devices. The two-level Taylor-Galerkin (TTGC) finite-element method (FEM) has remained the workhorse of modern industrial-scale combustion LES. In this work, we propose an improved FEM termed ML-TTGC that introduces locally tunable parameters in the TTGC scheme, whose values are provided by a graph neural network (GNN). We show that ML-TTGC outperforms TTGC in solving the convection problem in both irregular and regular meshes over a wide-range of initial conditions. We train the GNN using parameter values that (i) minimize a weighted loss function of the dispersion and dissipation error and (ii) enforce them to be numerically stable. As a result no additional ad-hoc dissipation is necessary for numerical stability or to damp spurious waves amortizing the additional cost of running the GNN.