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Aiming for the median solution between cyber-intensive optimal power flow (OPF) solutions and subpar local control, this work advocates deciding inverter injection setpoints using deep neural networks (DNNs). Instead of fitting OPF solutions in a black-box manner, inverter DNNs are naturally integrated with the feeder model and trained to minimize a grid-wide objective subject to inverter and network constraints enforced on the average over uncertain grid conditions. Learning occurs in a quasi-stationary fashion and is posed as a stochastic OPF, handled via stochastic primal-dual updates acting on grid data scenarios. Although trained as a whole, the proposed DNN is operated in a master-slave architecture. Its master part is run at the utility to output a condensed control signal broadcast to all inverters. Its slave parts are implemented by inverters and are driven by the utility signal along with local inverter readings. This novel DNN structure uniquely addresses the small-big data conundrum where utilities collect detailed smart meter readings yet on an hourly basis, while in real time inverters should be driven by local inputs and minimal utility coordination to save on communication. Numerical tests corroborate the efficacy of this physics-aware DNN-based inverter solution over an optimal control policy.