Inversion of Integral Models: a Neural Network Approach

Abstract

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.