  We introduce a neural network architecture to solve inverse problems linked
to a one-dimensional integral operator. This architecture is built by unfolding
a forward-backward algorithm derived from the minimization of an objective
function which consists of the sum of a data-fidelity function and a
Tikhonov-type regularization function. The robustness of this inversion method
with respect to a perturbation of the input is theoretically analyzed. Ensuring
robustness is consistent with inverse problem theory since it guarantees both
the continuity of the inversion method and its insensitivity to small noise.
The latter is a critical property as deep neural networks have been shown to be
vulnerable to adversarial perturbations. One of the main novelties of our work
is to show that the proposed network is also robust to perturbations of its
bias. In our architecture, the bias accounts for the observed data in the
inverse problem. We apply our method to the inversion of Abel integral
operators, which define a fractional integration involved in wide range of
physical processes. The neural network is numerically implemented and tested to
illustrate the efficiency of the method. Lipschitz constants after training are
computed to measure the robustness of the neural networks.
