Retrospective motion correction techniques have the potential to improve clinical imaging without altering the workflow or acquisition sequence. Yet, they suffer from long reconstruction times and poor conditioning. To address these problems, we developed a Network Accelerated Motion Estimation and Reduction method (NAMER) within a data-consistency based forward model approach to motion parameter estimation. The neural net accelerates convergence up to 15-fold as well as improving final image quality. The ML+MR physics motion correction method combines the speedup provided by fast convolutional neural networks with the robustness of a forward model-based data-consistency reconstruction.