  The majority of model-based learned image reconstruction methods in medical
imaging have been limited to uniform domains, such as pixelated images. If the
underlying model is solved on nonuniform meshes, arising from a finite element
method typical for nonlinear inverse problems, interpolation and embeddings are
needed. To overcome this, we present a flexible framework to extend model-based
learning directly to nonuniform meshes, by interpreting the mesh as a graph and
formulating our network architectures using graph convolutional neural
networks. This gives rise to the proposed iterative Graph Convolutional
Newton's Method (GCNM), which directly includes the forward model into the
solution of the inverse problem, while all updates are directly computed by the
network on the problem specific mesh. We present results for Electrical
Impedance Tomography, a severely ill-posed nonlinear inverse problem that is
frequently solved via optimization-based methods, where the forward problem is
solved by finite element methods. Results for absolute EIT imaging are compared
to standard iterative methods as well as a graph residual network. We show that
the GCNM has strong generalizability to different domain shapes, out of
distribution data as well as experimental data, from purely simulated training
data.
