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A key challenge in inventory management is to identify policies that optimally replenish inventory from multiple suppliers. To solve such optimization problems, inventory managers need to decide what quantities to order from each supplier, given the net inventory and outstanding orders, so that the expected backlogging, holding, and sourcing costs are jointly minimized. Inventory management problems have been studied extensively for over 60 years, and yet even basic dual-sourcing problems, in which orders from an expensive supplier arrive faster than orders from a regular supplier, remain intractable in their general form. In addition, there is an emerging need to develop proactive, scalable optimization algorithms that can adjust their recommendations to dynamic demand shifts in a timely fashion. In this work, we approach dual sourcing from a neural network--based optimization lens and incorporate information on inventory dynamics and its replenishment (i.e., control) policies into the design of recurrent neural networks. We show that the proposed neural network controllers (NNCs) are able to learn near-optimal policies of commonly used instances within a few minutes of CPU time on a regular personal computer. To demonstrate the versatility of NNCs, we also show that they can control inventory dynamics with empirical, non-stationary demand distributions that are challenging to tackle effectively using alternative, state-of-the-art approaches. Our work shows that high-quality solutions of complex inventory management problems with non-stationary demand can be obtained with deep neural-network optimization approaches that directly account for inventory dynamics in their optimization process. As such, our research opens up new ways of efficiently managing complex, high-dimensional inventory dynamics.