  The present paper deals with the data-driven design of regularizers in the
form of artificial neural networks, for solving certain inverse problems
formulated as optimal control problems. These regularizers aim at improving
accuracy, wellposedness or compensating uncertainties for a given class of
optimal control problems (inner-problems). Parameterized as neural networks,
their weights are chosen in order to reduce a misfit between data and
observations of the state solution of the inner-optimal control problems.
Learning these weights constitutes the outer-problem. Based on necessary
first-order optimality conditions for the inner-problems, a relaxation approach
is proposed in order to implement efficient solving of these inner-problems,
namely the forward operator of the outer-problem. Optimality conditions are
derived for the latter, and are implemented in numerical illustrations dealing
with the inverse conductivity problem. The numerical tests show the feasibility
of the relaxation approach, first for rediscovering standard
$L^2$-regularizers, and next for designing regularizers that compensate unknown
noise on the observed state of the inner-problem.
