Y-net: A Physics-constrained and Semi-supervised Learning Approach to the Phase Problem in Computational Electron Imaging

The phase problem in diffraction physics is one of the oldest inverse problems in all of science. The central difficulty that any approach to solving this inverse problem must overcome is that half of the information, namely the phase of the diffracted beam, is always missing. In the context of electron microscopy, the phase problem is generally non-linear and solutions provided by phase-retrieval techniques are known to be poor approximations to the physics of electrons interacting with matter. Here, we show that a semi-supervised learning approach can effectively solve the phase problem in electron microscopy/scattering. In particular, we introduce a new Deep Neural Network (DNN), Y-net, which simultaneously learns a reconstruction algorithm via supervised training in addition to learning a physics-based regularization via unsupervised training. We demonstrate that this constrained, semi-supervised approach is an order of magnitude more data-efficient and accurate than the same model trained in a purely supervised fashion. In addition, the architecture of the Y-net model provides for a straightforward evaluation of the consistency of the model's prediction during inference and is generally applicable to the phase problem in other settings.