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This work combines a Bee Recurrent Neural Network (BRNN) optimized by the Artificial Bee Colony (ABC) algorithm with Monte Carlo Simulation (MCS) to generate a novel approximate model for predicting network reliability. We utilize the Wavelet Transform (WT)-based Latin Hypercube Sampling (LHS) (WLHS) to select input training data, and open the black box of neural networks by constructing a limited space reliability function from neural network parameters. Furthermore, the proposed method compares favorably with existing methods in literature based on experimental results for a benchmark example. The result reveals that the novel WLHS-MCS based on BRNN (WLHS-BRNN-MCS for short) is an excellent estimator of the reliability function.