Probability density evolution method for structural reliability analysis: A parallel Bayesian active learning perspective

While probability density evolution method (PDEM) paired with active learning shows strong promise for structural reliability analysis, its broader adoption is hindered by unresolved theoretical limitations and computational inefficiencies. In this work, we present the first attempt at casting a Bayesian inference perspective for evaluating failure probability in PDEM. First, it quantifies epistemic uncertainty through a posterior mean and a provable upper bound of variance (UBV) of failure probability, overcoming limitations of the traditional frequentist approaches. Then, three critical ingredients of parallel active learning paradigm are designed: (i) A multi-point learning function called the upper bound of variance reduction (UBVR) is analytically deduced to quantify the impact of adding $k(\ge 1)$ new samples. (ii) Batch enrichment process is achieved via a principled stepwise maximization strategy of UBVR, eliminating the need for those goal-inconsistent batch selection strategies. (iii) A hybrid convergence criterion is defined by continuously monitoring the instantaneous value of UBV. The proposed method offers a comprehensive framework for fusing Bayesian inference of failure probability and parallel active learning in PDEM. It is tested on five examples and compared against several existing parallel active learning reliability methods. Results indicate that the proposed approach matches similar accuracy to state-of-the-art methods with great computational cost savings.