A new acquisition function combined with subset simulation for active learning of small and time-dependent failure probability

The time-dependent reliability analysis aims at estimating the probability of failure, occurring within a specified time period, of a structure subjected to stochastic and dynamic loads or stochastic degradation of performance. Development of efficient numerical algorithms with accuracy assurance for solving this problem, although has been investigated with, e.g., Gaussian Process Regression (GPR)-based active learning procedures, keeps being a bottleneck. Inspired by the concept of up-crossing rate used in the first-passage methods, a new acquisition function (also called learning function) is developed with the consideration of the temporal correlation information across each sample trajectory. It measures the (subjective) probability of mis-judging the occurrence of the up-crossing event within each time sub-interval. With this new acquisition function, the classical active learning procedure is improved. Considering the necessity for estimating small failure probability, the proposed active learning method is then combined with the subset simulation for multi-stage learning. With this method, a series of intermediate surrogate failure surface is actively updated with the target of approaching the true failure surface with pre-specified error tolerance. The effectiveness of the proposed methods are demonstrated with numerical and engineering examples.