Estimation of Failure Probability Function under Imprecise Probabilities by Active Learning–Augmented Probabilistic Integration

Imprecise probabilities have gained increasing popularity for quantitatively modeling uncertainty under incomplete information in various fields. However, it is still a computationally challenging task to propagate imprecise probabilities because a double-loop procedure is usually involved. In this contribution, a fully decoupled method, termed as active learning–augmented probabilistic integration (ALAPI), is developed to efficiently estimate the failure probability function (FPF) in the presence of imprecise probabilities. Specially, the parameterized probability-box models are of specific concern. By interpreting the failure probability integral from a Bayesian probabilistic integration perspective, the discretization error can be regarded as a kind of epistemic uncertainty, allowing it to be properly quantified and propagated through computational pipelines. Accordingly, an active learning probabilistic integration (ALPI) method is developed for failure probability estimation, in which a new learning function and a new stopping criterion associated with the upper bound of the posterior variance and coefficient of variation are proposed. Based on the idea of constructing an augmented uncertainty space, an imprecise augmented stochastic simulation (IASS) method is devised by using the random sampling high-dimensional representation model (RS-HDMR) for estimating the FPF in a pointwise stochastic simulation manner. To further improve the efficiency of IASS, the ALAPI is formed by an elegant combination of the ALPI and IASS, allowing the RS-HDMR component functions of the FPF to be properly inferred. Three benchmark examples are investigated to demonstrate the accuracy and efficiency of the proposed method.