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To correct for respiratory motion in PET imaging, an interpretable and unsupervised deep learning technique, FlowNet-PET, was constructed. The network was trained to predict the optical flow between two PET frames from different breathing amplitude ranges. The trained model aligns different retrospectively-gated PET images, providing a final image with similar counting statistics as a non-gated image, but without the blurring effects. FlowNet-PET was applied to anthropomorphic digital phantom data, which provided the possibility to design robust metrics to quantify the corrections. When comparing the predicted optical flows to the ground truths, the median absolute error was found to be smaller than the pixel and slice widths. The improvements were illustrated by comparing against images without motion and computing the intersection over union (IoU) of the tumors as well as the enclosed activity and coefficient of variation (CoV) within the no-motion tumor volume before and after the corrections were applied. The average relative improvements provided by the network were 64%, 89%, and 75% for the IoU, total activity, and CoV, respectively. FlowNet-PET achieved similar results as the conventional retrospective phase binning approach, but only required one sixth of the scan duration. The code and data have been made publicly available (https://github.com/teaghan/FlowNet_PET).