Learning regularized Gauss-Newton methods

We consider variational networks for a class of nonlinear-ill-posed least squares inverse problems. These problems are addressed by regularized Gauss-Newton type optimization algorithms where the regularization is learned by a neural network. Two different data-driven approaches will be investigated. First, we present a learned-regularizer integrated into an unrolled Gauss-Newton network. As an alternative approach, we derive a proximal regularized quasi-Newton (PRQN) method and we unfold the PRQN into a deep network which consists of a cascade of multiple proximal mappings. Therefore the proximal operator is learned directly through a variable metric denoiser network. As a practical application, we show how our methods have been successfully applied to solve the parameter identification problem in elliptic PDEs, such as the nonlinear Electrical Impedance Tomography inverse problem.