Multi-point active learning probability density evolution method

Probability density evolution method has been efficiently adapted for structural reliability analysis, owing to it rooting in the principle of preservation of probability. Despite achieving significant progress in the past decades, there remains a critical need to enhance its theoretical foundations and improve computational efficiency. In this paper, we develop a multi-point active learning probability density evolution method distinguished by the following four key features: (i) Quantification. An explicit formulation of failure probability is proposed for probability density evolution method by combining the finite difference scheme and the Dirac sequence scheme. Then, an epistemic uncertainty measure of Kriging-based failure probability estimation is quantified. (ii) Reduction. A multi-point learning function is deduced in closed form, aiming to select a batch of new samples to optimally reduce such epistemic uncertainty measure. (iii) Maximization. The multi-point enrichment process is directly conducted based on stepwise maximization of learning function, eliminating the traditional practice of combining a single-point learning function with some additional batch selection procedures. (iv) Termination. The termination of multi-point enrichment process is checked from the actual reduction of epistemic uncertainty of failure probability. The proposed method is tested on four examples and compared against several existing ones in the literature. The results indicate that the proposed method comes with high accuracy of failure probability estimate, whilst gaining favorable savings of the number of iterations and the total computational time, particularly when tackling with complex dynamic reliability problems.