Real-world systems, such as manufacturing systems, can be modeled as network topologies with arcs and nodes. The capacity of each arc has several statuses owing to maintenance or machine failure. Such a system is called a capacitated-flow network (CFN). To learn the performance of the CFN, network reliability, the probability that the CFN can successfully transmit the required demand from the source to the sink, is usually utilized. Based on the minimal path (MP), the network reliability can be calculated by obtaining all the minimal capacity vectors, which denote the minimal required capacity for each arc. Efficient calculation of network reliability for a certain CFN is an NP-hard problem; moreover, different CFN connections need to be considered. Therefore, an artificial neural network (ANN) is adopted herein to overcome the network reliability evaluation for random CFN with different network connections. The generation method of the CFN information with different network connections as well as the related structure and functions are then developed to estimate the network reliability. Random search is used to optimize the hyperparameters of the ANN model. For different CFN connections, the trained model can be implemented with small errors in a short time compared with the MP-based algorithm.