Estimating first-passage probabilities of stochastic dynamical systems: A POD–GPR based approximate Bayesian active learning approach

Evaluating the first-passage probabilities of stochastic dynamical systems is often required for assessing the safety and performance of modern engineered systems. However, despite significant progress, this task remains rather challenging both analytically and numerically. To address the challenge, this work presents a novel method—proper orthogonal decomposition (POD) and Gaussian process regression (GPR) based approximate Bayesian active learning (ABAL), referred to as POD–GPR–ABAL. First, POD is employed to obtain an optimal low-rank representation of the stochastic dynamical response of interest using a compact set of basis functions and coefficients, and the POD coefficients are subsequently approximated by independent GPRs. Second, the integral of the first-passage probability is interpreted as a Bayesian inference problem by leveraging the Bayesian formulation of the combined POD–GPR model. Further, an approximate Bayesian inference scheme is developed, which avoids the computational intractability of the exact Bayesian inference. In this context, the mean of an approximate posterior first-passage probability is derived, which can be used a Bayesian estimator for the first-passage probability. In addition, we derive an upper bound on the standard deviation of the approximate posterior first-passage probability, which provides a uncertainty measure for the estimator. Third, based on the posterior statistics, a new stopping criterion is proposed to determine when the active learning process should terminate and a novel learning function is also developed to identify the best next evaluation point when the stopping criterion is not satisfied. Three numerical examples are presented to demonstrate the effectiveness of the proposed POD–GPR–ABAL method for evaluating the first-passage probabilities in stochastic dynamical systems. The results indicate that the proposed method achieves accurate first-passage probability estimates using only a small number of time-history analyses.