  Mesh-based simulations play a key role when modeling complex physical systems
that, in many disciplines across science and engineering, require the solution
of parametrized time-dependent nonlinear partial differential equations (PDEs).
In this context, full order models (FOMs), such as those relying on the finite
element method, can reach high levels of accuracy, however often yielding
intensive simulations to run. For this reason, surrogate models are developed
to replace computationally expensive solvers with more efficient ones, which
can strike favorable trade-offs between accuracy and efficiency. This work
explores the potential usage of graph neural networks (GNNs) for the simulation
of time-dependent PDEs in the presence of geometrical variability. In
particular, we propose a systematic strategy to build surrogate models based on
a data-driven time-stepping scheme where a GNN architecture is used to
efficiently evolve the system. With respect to the majority of surrogate
models, the proposed approach stands out for its ability of tackling problems
with parameter dependent spatial domains, while simultaneously generalizing to
different geometries and mesh resolutions. We assess the effectiveness of the
proposed approach through a series of numerical experiments, involving both
two- and three-dimensional problems, showing that GNNs can provide a valid
alternative to traditional surrogate models in terms of computational
efficiency and generalization to new scenarios. We also assess, from a
numerical standpoint, the importance of using GNNs, rather than classical dense
deep neural networks, for the proposed framework.
