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In this paper, we consider a power allocation optimization technique for a time-varying fading broadcast channel in energy harvesting communication systems, in which a transmitter with a rechargeable battery transmits messages to receivers using the harvested energy. We first prove that the optimal online power allocation policy for the sum-rate maximization of the transmitter is a monotonically increasing function of harvested energy, remaining battery, and the channel gain of each user. We then construct a lightweight neural network architecture to take advantage of the monotonicity of the optimal policy. This two-step approach, which relies on effective function approximation to provide a mathematical guideline for neural network design, can prevent us from wasting the representational capacity of neural networks. The tailored neural network architectures enable stable learning and eliminate the heuristic network design. For performance assessment, the proposed approach is compared with the closed-form optimal policy for a partially observable Markov problem. Through additional experiments, it is observed that our online solution achieves a performance close to the theoretical upper bound of the performance in a time-varying fading broadcast channel.