A single-loop active learning kriging method for failure probability upper bound function estimation

In engineering practice, both random and interval variables are commonly involved, making the failure probability upper bound (FPUB) a key indicator for assessing structural safety. To minimize the FPUB of an engineering structure, the distribution information of the random variables must be properly designed. As a result, FPUB becomes a function of the design parameters, referred to as the failure probability upper bound function (FPUBF). Existing methods for FPUBF estimation suffer from excessive computational cost. This paper proposes a single-loop active learning Kriging (ALK) method for the estimation of FPUBF. This paper first improves the existing improved cross-entropy importance sampling (ICE-IS) method. Based on the constructed Kriging model that maps all variables to the response, the improved ICE-IS is used to obtain importance sampling (IS) random samples in the augmented space. Then, to improve the computational efficiency of ALK, this paper proposes a Kullback-Leibler (KL) divergence-based learning function. This learning function aims to maximize the KL divergence between the correct prediction probability of IS random samples and the expected correct prediction probability. Once the Kriging model converges, the FPUBF is evaluated using the IS random samples. The proposed method is validated using four numerical examples and two engineering examples. The results demonstrate that the method achieves high accuracy and efficiency in FPUBF estimation.