  In this work, we consider the non-invasive medical imaging modality of
Electrical Impedance Tomography, where the problem is to recover the
conductivity in a medium from a set of data that arises out of a
current-to-voltage map (Neumann-to-Dirichlet operator) defined on the boundary
of the medium. We formulate this inverse problem as an operator-learning
problem where the goal is to learn the implicitly defined operator-to-function
map between the space of Neumann-to-Dirichlet operators to the space of
admissible conductivities. Subsequently, we use an operator-learning
architecture, popularly called DeepONets, to learn this operator-to-function
map. Thus far, most of the operator learning architectures have been
implemented to learn operators between function spaces. In this work, we
generalize the earlier works and use a DeepONet to actually {learn an
operator-to-function} map. We provide a Universal Approximation Theorem type
result which guarantees that this implicitly defined operator-to-function map
between the space of Neumann-to-Dirichlet operator to the space of conductivity
function can be approximated to an arbitrary degree using such a DeepONet.
Furthermore, we provide a computational implementation of our proposed approach
and compare it against a standard baseline. We show that the proposed approach
achieves good reconstructions and outperforms the baseline method in our
experiments.
