Estimation of Small Failure Probability Based on Adaptive Subset Simulation and Deep Neural Network

The critical problem of reliability design is how to obtain a more accurate failure probability with a smaller number of evaluations of actual complex and nonlinear performance function. To achieve this objective, an adaptive subset simulation method with a deep neural network (DNN) is proposed for accurate estimation of small failure probability. A determinate criterion for threshold values is developed, and the subset number is adaptively quantified according to the initial estimated value of small failure probability. Therefore, the estimation of small failure probability is converted to estimation problem of multiple large conditional probabilities. An adaptive deep neural network model is constructed in every subset to predict the conditional probability with a smaller number of evaluations of the actual performance function. Furthermore, the sampling points for the next subset can be adaptively selected according to the constructed DNN model, which can decrease the number of invalid sampling points and evaluations of actual performance function, then the computational efficiency for estimating the conditional probability in every subset is increased. The sampling points with high probability density functions are recalculated with actual performance function values to replace the predicted values of the DNN model, which can verify the accuracy of DNN model and increase the estimation accuracy of small failure probability. By analyzing a nonlinear problem, a multiple failure domain problem and two engineering examples, the effectiveness and accuracy of the proposed methodology for estimating small failure probability are verified.