We present DeepIA, a deep neural network (DNN) framework for enabling fast and reliable initial access for AI-driven beyond 5G and 6G millimeter (mmWave) networks. DeepIA reduces the beam sweep time compared to a conventional exhaustive search-based IA process by utilizing only a subset of the available beams. DeepIA maps received signal strengths (RSSs) obtained from a subset of beams to the beam that is best oriented to the receiver. In both line of sight (LoS) and non-line of sight (NLoS) conditions, DeepIA reduces the IA time and outperforms the conventional IAs beam prediction accuracy. We show that the beam prediction accuracy of DeepIA saturates with the number of beams used for IA and depends on the particular selection of the beams. In LoS conditions, the selection of the beams is consequential and improves the accuracy by up to 70%. In NLoS situations, it improves accuracy by up to 35%. We find that, averaging multiple RSS snapshots further reduces the number of beams needed and achieves more than 95% accuracy in both LoS and NLoS conditions. Finally, we evaluate the beam prediction time of DeepIA through embedded hardware implementation and show the improvement over the conventional beam sweeping.