  We consider ill-posed inverse problems where the forward operator $T$ is
unknown, and instead we have access to training data consisting of functions
$f_i$ and their noisy images $Tf_i$. This is a practically relevant and
challenging problem which current methods are able to solve only under strong
assumptions on the training set. Here we propose a new method that requires
minimal assumptions on the data, and prove reconstruction rates that depend on
the number of training points and the noise level. We show that, in the regime
of "many" training data, the method is minimax optimal. The proposed method
employs a type of convolutional neural networks (U-nets) and empirical risk
minimization in order to "fit" the unknown operator. In a nutshell, our
approach is based on two ideas: the first is to relate U-nets to multiscale
decompositions such as wavelets, thereby linking them to the existing theory,
and the second is to use the hierarchical structure of U-nets and the low
number of parameters of convolutional neural nets to prove entropy bounds that
are practically useful. A significant difference with the existing works on
neural networks in nonparametric statistics is that we use them to approximate
operators and not functions, which we argue is mathematically more natural and
technically more convenient.
