Congestion due to morning and evening traffic peaks has caused economic losses amounting to billions of dollars annually. Thus, accurately predicting the departure time of commuters is of interest to transportation planners, engineers, and elected officials alike. We develop a statistically-informed deep learning approach to improve commuter departure time prediction models. Specifically, we leverage elements of the proportional hazards model, a class of time-to-event prediction approaches, to augment vanilla deep neural network (DNN) architectures. The proposed approach also employs collaborative filtering to segment the commuter population into distinct behavioral classes, allowing tailored predictions for specific commuter profiles. We find that our class of survival analysis-enhanced DNNs outperforms conventional neural networks in predicting trip departure times, while also offering more interpretability through the hazard coefficients.