We develop an energy-efficient indoor positioning system based on Artificial Intelligence (AI). In our system, first, at the positioning layer, a Multi-Layer Perceptron (MLP) estimates the current indoor position of an IoT device based on positioning indicators obtained from the anchors. Second, at the forecasting layer, a pair of MLPs estimate the future positions of the device based on the past position estimates obtained when the device woke up as well as the forecast positions of the device during the sleep periods. Third, the device is awakened to send a positioning beacon at intervals over which a significant displacement is predicted to occur by the forecasting layer. Our results demonstrate that our indoor positioning system saves significant energy via adaptive sleep cycles whose duration is determined by the  prediction of a significant displacement. This work establishes a foundation for indoor positioning that utilizes AI-based positioning and trajectory forecasting.