Transmit antenna muting (TAM) in multiple-user multiple-input multiple-output (MU-MIMO) networks allows reducing the power consumption of the base station (BS) by properly utilizing only a subset of antennas in the BS. In this paper, we consider the downlink transmission of an MU-MIMO network where TAM is formulated to minimize the number of active antennas in the BS while guaranteeing the per-user throughput requirements. To address the computational complexity of the combinatorial optimization problem, we propose an algorithm called neural antenna muting (NAM) with an asymmetric custom loss function. NAM is a classification neural network trained in a supervised manner. The classification error in this scheme leads to either sub-optimal energy consumption or lower quality of service (QoS) for the communication link. We control the classification error probability distribution by designing an asymmetric loss function such that the erroneous classification outputs are more likely to result in fulfilling the QoS requirements. Furthermore, we present three heuristic algorithms and compare them with the NAM. Using a 3GPP compliant system-level simulator, we show that NAM achieves $\sim73\%$ energy saving compared to the full antenna configuration in the BS with $\sim95\%$ reliability in achieving the user throughput requirements while being around $1000\times$ and $24\times$ less computationally intensive than the greedy heuristic algorithm and the fixed column antenna muting algorithm, respectively.