Directional filter combined with active learning for rare failure events

Estimating the small failure probability is a crucial task in structural engineering, and directional sampling has long been recognized as one of the most promising stochastic simulation method for problems with multiple disconnected failure domains. However, when applied to problems with expensive-to-evaluate and highly nonlinear limit state functions, it is still less satisfactory in terms of numerical accuracy and efficiency. To fill this gap, this paper develops a directional filter equipped active learning (DirFAL) algorithm. It integrates three key components: (1) an active learning directional filter scheme, which plays as a role for adaptively prioritizing the directional samples that dominantly contribute to the failure probability; (2) a tailored closed-form expression of acquisition function, which is newly defined and incorporated into DirFAL to effectively select new training data for enriching a Gaussian process regression surrogate model; (3) an active learning stratification strategy for tackling the performance degradation issue when applied to problems with relatively high dimension and extremely small failure probability. The performance of the proposed methods is ultimately demonstrated through two numerical examples and two engineering problems, and results show that they are of superiority over other parallel methods in terms of numerical efficiency given required accuracy.