Active learning and active subspace enhancement for PDEM-based high-dimensional reliability analysis

Abstract An efficient reliability method, termed as the AL-AS-GPR-PDEM, is proposed in this study, which effectively combines the probability density evolution method (PDEM) and the Gaussian process regression (GPR) surrogate model enhanced by both the active subspace (AS)-based dimension reduction and the active learning (AL)-based sampling strategy. In this method, the AS-GPR model is developed, where the AS and the GPR are jointly calibrated so as to bypass the tricky prerequisite of gradient information to discover the latent AS of the original parameter space; then, the AL-based sampling strategy is employed to construct a satisfactorily accurate AS-GPR model using as fewer training samples as possible. This treatment allows that the newly-added training sample at each iteration can be simultaneously used to both update the discovered low-dimensional AS and refine the constructed GPR model within the AS. Lastly, the finalized AS-GPR model is employed to predict the equivalent extreme values (EEVs) of structural responses at the representative point set, thereby the PDEM-based reliability analysis can be readily performed based on the estimated EEVs. To testify the effectiveness of the proposed method, four numerical examples are addressed, involving both surrogate modelling of analytical functions with different characteristics and dynamic reliability analysis of linear/nonlinear engineering structures under seismic excitations. It is demonstrated that the proposed method is of satisfactory accuracy and efficiency for structural reliability analysis in high dimensions.