  We study the use of novel techniques arising in machine learning for inverse
problems. Our approach replaces the complex forward model by a neural network,
which is trained simultaneously in a one-shot sense when estimating the unknown
parameters from data, i.e. the neural network is trained only for the unknown
parameter. By establishing a link to the Bayesian approach to inverse problems,
an algorithmic framework is developed which ensures the feasibility of the
parameter estimate w.r. to the forward model. We propose an efficient,
derivative-free optimization method based on variants of the ensemble Kalman
inversion. Numerical experiments show that the ensemble Kalman filter for
neural network based one-shot inversion is a promising direction combining
optimization and machine learning techniques for inverse problems.
