This work focuses on the persistent monitoring problem, where a set of targets moving based on an unknown model must be monitored by an autonomous mobile robot with a limited sensing range. To keep each target's position estimate as accurate as possible, the robot needs to adaptively plan its path to (re-)visit all the targets and update its belief from measurements collected along the way. In doing so, the main challenge is to strike a balance between exploitation, i.e., re-visiting previously-located targets, and exploration, i.e., finding new targets or re-acquiring lost ones. Encouraged by recent advances in deep reinforcement learning, we introduce an attention-based neural solution to the persistent monitoring problem, where the agent can learn the inter-dependencies between targets, i.e., their spatial and temporal correlations, conditioned on past measurements. This endows the agent with the ability to determine which target, time, and location to attend to across multiple scales, which we show also helps relax the usual limitations of a finite target set. We experimentally demonstrate that our method outperforms other baselines in terms of number of targets visits and average estimation error in complex environments. Finally, we implement and validate our model in a drone-based simulation experiment to monitor mobile ground targets in a high-fidelity simulator.