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One essential task in practice is to quantify and improve the reliability of an infrastructure network in terms of the connectivity of network components (i.e., all-terminal reliability). However, as the number of edges and nodes in the network increases, computing the all-terminal network reliability using exact algorithms becomes prohibitive. This is extremely burdensome in network designs requiring repeated computations. In this paper, we propose a novel machine learning-based framework for evaluating and improving all-terminal network reliability using Deep Neural Networks (DNNs) and Deep Reinforcement Learning (DRL). With the help of DNNs and Stochastic Variational Inference (SVI), we can effectively compute the all-terminal reliability for different network configurations in DRL. Furthermore, the Bayesian nature of the proposed SVI+DNN model allows for quantifying the estimation uncertainty while enforcing regularization and reducing overfitting. Our numerical experiment and case study show that the proposed framework provides an effective tool for infrastructure network reliability improvement.