  We consider the ill-posed inverse problem of identifying a nonlinearity in a
time-dependent PDE model. The nonlinearity is approximated by a neural network,
and needs to be determined alongside other unknown physical parameters and the
unknown state. Hence, it is not possible to construct input-output data pairs
to perform a supervised training process. Proposing an all-at-once approach, we
bypass the need for training data and recover all the unknowns simultaneously.
In the general case, the approximation via a neural network can be realized as
a discretization scheme, and the training with noisy data can be viewed as an
ill-posed inverse problem. Therefore, we study discretization of regularization
in terms of Tikhonov and projected Landweber methods for discretization of
inverse problems, and prove convergence when the discretization error (network
approximation error) and the noise level tend to zero.
