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State-of-the-art drone technologies have severe flight time limitations due to weight constraints, which inevitably lead to a relatively small amount of available energy. Therefore, frequent battery replacement or recharging is necessary in applications such as delivery, exploration, or support to the wireless infrastructure. Mobile charging stations (i.e., mobile stations with charging equipment) for outdoor ad-hoc battery charging is one of the feasible solutions to address this issue. However, the ability of these platforms to charge the drones is limited in terms of the number and charging time. This paper designs an auction-based mechanism to control the charging schedule in multi-drone setting. In this paper, charging time slots are auctioned, and their assignment is determined by a bidding process. The main challenge in developing this framework is the lack of prior knowledge on the distribution of the number of drones participating in the auction. Based on optimal second-price-auction, the proposed formulation, then, relies on deep learning algorithms to learn such distribution online. Numerical results from extensive simulations show that the proposed deep learning-based approach provides effective battery charging control in multi-drone scenarios.