Discretization of parameter identification in PDEs using neural networks

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