  In this paper we consider inverse problems that are mathematically ill-posed.
That is, given some (noisy) data, there is more than one solution that
approximately fits the data. In recent years, deep neural techniques that find
the most appropriate solution, in the sense that it contains a-priori
information, were developed. However, they suffer from several shortcomings.
First, most techniques cannot guarantee that the solution fits the data at
inference. Second, while the derivation of the techniques is inspired by the
existence of a valid scalar regularization function, such techniques do not in
practice rely on such a function, and therefore veer away from classical
variational techniques. In this work we introduce a new family of neural
regularizers for the solution of inverse problems. These regularizers are based
on a variational formulation and are guaranteed to fit the data. We demonstrate
their use on a number of highly ill-posed problems, from image deblurring to
limited angle tomography.
