Active learning for structural reliability analysis with multiple limit state functions through variance-enhanced PC-Kriging surrogate models

TL;DR

Efficient probabilistic characterization of multiple limit state functions under a limited budget is addressed by deploying PC-Kriging surrogates with a $LOOCV$-derived variance correction and $U$-function–guided active learning. The framework is extended to multi-LSF settings by strategies that alternate targets or use a convergence metric on the reliability index $\widehat{\beta}_{g_j}$ to balance training across states. Analytical and offshore-wind case studies show that convergence-driven sampling with variance correction yields robust, balanced predictions across all limit states, outperforming single-target approaches. This approach enables accurate and budget-efficient structural reliability assessments for complex systems with multiple performance criteria.

Abstract

Existing active strategies for training surrogate models yield accurate structural reliability estimates by aiming at design space regions in the vicinity of a specified limit state function. In many practical engineering applications, various damage conditions, e.g. repair, failure, should be probabilistically characterized, thus demanding the estimation of multiple performance functions. In this work, we investigate the capability of active learning approaches for efficiently selecting training samples under a limited computational budget while still preserving the accuracy associated with multiple surrogated limit states. Specifically, PC-Kriging-based surrogate models are actively trained considering a variance correction derived from leave-one-out cross-validation error information, whereas the sequential learning scheme relies on U-function-derived metrics. The proposed active learning approaches are tested in a highly nonlinear structural reliability setting, whereas in a more practical application, failure and repair events are stochastically predicted in the aftermath of a ship collision against an offshore wind substructure. The results show that a balanced computational budget administration can be effectively achieved by successively targeting the specified multiple limit state functions within a unified active learning scheme.

Figures (3)

• Figure 1: Active learning evolution corresponding to the tested strategies in terms of the predicted probability associated with both considered events $g_1$ and $g_2$. (Left) U-function score metrics evaluated from PC-Kriging variance predictors. (Right) U-function score metrics calculated from corrected variance predictors (U-LOO).
• Figure 2: Box-plot representation of all investigated active learning approaches in terms of the combined relative error metric, $\varepsilon_{\beta}$. (Left) U-function score metrics evaluated from PC-Kriging variance predictors. (Right) U-function score metrics calculated from corrected PC-Kriging variance predictors (U-LOO).
• Figure 3: Active learning evolution in terms of the predicted probability associated with both considered events, $g_F$ and $g_d$, all relying on U-function metrics estimated from corrected variance predictors (U-LOO).