In this paper, an operation model for distribution systems with energy storage (ES) is proposed and solved with the aid of machine learning. The model considers ES applications with uncertainty realizations. It also considers ES applications for economy and security purposes. Considering the special features of ES operations under day-ahead decision mechanisms of distribution systems, an ES operation scheme is designed for transferring uncertainties to later hours through ES to ensure the secure operation of distribution system. As a result, uncertainties from different time intervals are assembled and may counteract each other, thereby alleviating the uncertainties. As different ES applications rely on ES flexibility (in terms of charging and discharging) and interact with each other, by coordinating different ES applications, the proposed operation model achieves efficient exploit of ES flexibility. To shorten the computation time, a long short-term memory recurrent neural network is used to determine the binary variables corresponding to ES status. The proposed operation model then becomes a convex optimization problem and is solved precisely. Thus, the solving efficiency is greatly improved while ensuring the satisfactory use of ES flexibility in distribution system operation.