  As an alternative to classical numerical solvers for partial differential
equations (PDEs) subject to boundary value constraints, there has been a surge
of interest in investigating neural networks that can solve such problems
efficiently. In this work, we design a general solution operator for two
different time-independent PDEs using graph neural networks (GNNs) and spectral
graph convolutions. We train the networks on simulated data from a finite
elements solver on a variety of shapes and inhomogeneities. In contrast to
previous works, we focus on the ability of the trained operator to generalize
to previously unseen scenarios. Specifically, we test generalization to meshes
with different shapes and superposition of solutions for a different number of
inhomogeneities. We find that training on a diverse dataset with lots of
variation in the finite element meshes is a key ingredient for achieving good
generalization results in all cases. With this, we believe that GNNs can be
used to learn solution operators that generalize over a range of properties and
produce solutions much faster than a generic solver. Our dataset, which we make
publicly available, can be used and extended to verify the robustness of these
models under varying conditions.
