  Reconstruction of tomographic images from boundary measurements requires
flexibility with respect to target domains. For instance, when the system
equations are modeled by partial differential equations the reconstruction is
usually done on finite element (FE) meshes, allowing for flexible geometries.
Thus, any processing of the obtained reconstructions should be ideally done on
the FE mesh as well. For this purpose, we extend the hugely successful U-Net
architecture that is limited to rectangular pixel or voxel domains to an
equivalent that works flexibly on FE meshes. To achieve this, the FE mesh is
converted into a graph and we formulate a graph U-Net with a new cluster
pooling and unpooling on the graph that mimics the classic neighborhood based
max-pooling. We demonstrate effectiveness and flexibility of the graph U-Net
for improving reconstructions from electrical impedance tomographic (EIT)
measurements, a nonlinear and highly ill-posed inverse problem. The performance
is evaluated for simulated data and from three measurement devices with
different measurement geometries and instrumentations. We successfully show
that such networks can be trained with a simple two-dimensional simulated
training set and generalize to very different domains, including measurements
from a three-dimensional device and subsequent 3D reconstructions.
