Accelerating high resolution brain imaging at 7T is needed to reach clinically feasible scanning times. Deep learning applies multi-layered neural networks as universal function approximators and is able to find its own compression implicitly. We propose a Recurrent Inference Machine (RIM) that is designed to be a general inverse problem solver. Its recurrent architecture can acquire great network depth, while still retaining a low number of parameters. The RIM outperforms compressed sensing in reconstructing 0.7mm brain data. On the reconstructed phase images, Quantitative Susceptibility Mapping can be performed.