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The proliferation of wireless communications networks over the past decades, combined with the scarcity of the wireless spectrum, have motivated a significant effort towards increasing the throughput of wireless networks. One of the major factors which limits the throughput in wireless communications networks is the accuracy of the time synchronization between the nodes in the network, as a higher throughput requires higher synchronization accuracy. Existing time synchronization schemes, and particularly, methods based on pulse-coupled oscillators (PCOs), which are the focus of the current work, have the advantage of simple implementation and achieve high accuracy when the nodes are closely located, yet tend to achieve poor synchronization performance for distant nodes. In this study, we propose a robust PCO-based time synchronization algorithm which retains the simple structure of existing approaches while operating reliably and converging quickly for both distant and closely located nodes. This is achieved by augmenting PCO-based synchronization with deep learning tools that are trainable in a distributed manner, thus allowing the nodes to train their neural network component of the synchronization algorithm without requiring additional exchange of information or central coordination. The numerical results show that our proposed deep learning-aided scheme is notably robust to propagation delays resulting from deployments over large areas, and to relative clock frequency offsets. It is also shown that the proposed approach rapidly attains full (i.e., clock frequency and phase) synchronization for all nodes in the wireless network, while the classic model-based implementation does not.