  Physics-based deep learning frameworks have shown to be effective in
accurately modeling the dynamics of complex physical systems with
generalization capability across problem inputs. However, time-independent
problems pose the challenge of requiring long-range exchange of information
across the computational domain for obtaining accurate predictions. In the
context of graph neural networks (GNNs), this calls for deeper networks, which,
in turn, may compromise or slow down the training process. In this work, we
present two GNN architectures to overcome this challenge - the Edge Augmented
GNN and the Multi-GNN. We show that both these networks perform significantly
better (by a factor of 1.5 to 2) than baseline methods when applied to
time-independent solid mechanics problems. Furthermore, the proposed
architectures generalize well to unseen domains, boundary conditions, and
materials. Here, the treatment of variable domains is facilitated by a novel
coordinate transformation that enables rotation and translation invariance. By
broadening the range of problems that neural operators based on graph neural
networks can tackle, this paper provides the groundwork for their application
to complex scientific and industrial settings.
