We present DeepMend, a novel approach to reconstruct restorations to fractured shapes using learned occupancy functions. Existing shape repair approaches predict low-resolution voxelized restorations or smooth restorations, or require symmetries or access to a pre-existing complete oracle. We represent the occupancy of a fractured shape as the conjunction of the occupancy of an underlying complete shape and a break surface, which we model as functions of latent codes using neural networks. Given occupancy samples from a fractured shape, we estimate latent codes using an inference loss augmented with novel penalties to avoid empty or voluminous restorations. We use the estimated codes to reconstruct a restoration shape. We show results with simulated fractures on synthetic and real-world scanned objects, and with scanned real fractured mugs. Compared to existing approaches and to two baseline methods, our work shows state-of-the-art results in accuracy and avoiding restoration artifacts over non-fracture regions of the fractured shape. We share our code at: https://github.com/Terascale-All-sensing-Research-Studio/DeepMend