An active learning function-on-function nonlinear autoregressive model for evaluating small first-passage probabilities of stochastic structural dynamical systems

In engineering applications, evaluating the reliability of structural dynamical systems subject to uncertainties from materials, manufacturing, and external excitations is critical. Surrogate models are widely used to enable efficient reliability analysis for complex, computationally in-tensive problems. However, constructing high-accuracy surrogates for reliability analysis of stochastic structural dynamical systems under limited computational budgets remains chal-lenging, particularly when estimating small first-passage probabilities. Building on the recently developed F2NARX method for emulating stochastic structural dynamical systems, we propose an efficient scheme that combines active learning with subset simulation to evaluate small first-passage failure probabilities. We first introduce a general framework that integrates active learning into subset simulation, where active learning is used to accurately estimate each in-termediate first-passage probability. We then develop an effective active-learning strategy, in-cluding a misclassification rate-based learning function and a stability-based stopping criterion, to further reduce the number of training time histories required to construct surrogates that accurately estimate the intermediate probabilities. Across several examples of varying dimen-sionality and complexity, the proposed method demonstrates high efficiency and accuracy in estimating small first-passage failure probabilities for stochastic structural dynamical systems.