Reliability analysis is one of the methods to consider the safety and stability of an engineering system. It is very important to determine whether a system is safe or not. We need to solve the complex nonlinear and implicit the limit state functions to obtain the reliability index. Traditional reliability analysis methods, First-Order Reliability Method (FORM), Second-Order Reliability Method (SORM), and Monte Carlo simulation (MCS), are not effective and have many limitations. In this paper, at the first step, an artificial neural network was used to model the limit state function. After that, the elite opposition-based learning differential evolution algorithm was selected to solve nonlinear equality constrained optimization problem to find the reliability index and the failure probability of problems in terms of random variables. The proposed method and some reference methods were applied to analyze the test problems in the literature to compare their effectiveness.