Active learning Kriging with functional dimension reduction for reliability analysis of stochastic dynamical systems

In engineering applications, it is important to evaluate the reliability of dynamical systems under various uncertainties arising from materials, manufacturing processes, and external excitations. Surrogate models are widely employed to enable efficient reliability analysis in complex, computationally intensive engineering problems. However, building high-accuracy surrogate models for estimating dynamic systems’ reliability with limited computational resources remains a significant challenge. This paper presents a novel active learning Kriging method based on functional dimension reduction (AKFDR) for the efficient estimation of first-passage failure probabilities of stochastic dynamical systems. In this approach, Kriging surrogate models are constructed in a latent functional space obtained through functional dimension reduction, enabling probabilistic predictions of dynamic responses. By leveraging the prediction uncertainty and the concept of trajectory misclassification probability (TMP), a new learning function incorporating a weighted correlation criterion is then developed to guide the selection of the best next sample for model enhancement. Furthermore, an error-based stopping criterion is proposed to judge the convergence of the active learning process. The final surrogate model is then used to estimate the first-passage failure probability via Monte Carlo simulation. Through three numerical examples of varying dimensionality and complexity, it is shown that the proposed method is efficient and accurate for first-passage probability evaluation of stochastic dynamical systems.