AL-SPCE – Reliability analysis for nondeterministic models using stochastic polynomial chaos expansions and active learning

Authors

A. V. Pires, M. Moustapha, S. Marelli and B. Sudret

Abstract

Reliability analysis traditionally relies on deterministic simulators, where repeated evaluations with identical inputs produce the same outputs. However, many real-world systems exhibit stochastic behavior, leading to non-repeatable outcomes even under identical conditions. To model such systems, stochastic simulators are employed, where the response is a random variable rather than a deterministic quantity. The inherent randomness of these models affects their reliability, and must therefore be accounted for in reliability analysis. While Monte Carlo simulation can be used for this purpose, its computational cost is usually prohibitive. To circumvent this issue, stochastic emulators have been introduced as surrogate models capable to reproduce the random response of the simulator at reduced computational cost. Recent contributions have demonstrated that stochastic emulators can be successfully applied to perform reliability analysis for stochastic simulators. However, these approaches still rely on relatively large training sets to achieve accurate reliability estimates, which may become prohibitive for expensive models. In this work, we propose an active learning framework to further reduce the computational effort required for reliability analysis using stochastic emulators. Focusing on stochastic polynomial chaos expansions (SPCE), we introduce a learning function that identifies relevant regions for reliability estimation in which the emulator exhibits high predictive uncertainty. Additionally, we leverage on the asymptotic normality of the maximum likelihood estimator to quantify the local uncertainty in the predictions of the emulator. The proposed methodology, named active learning stochastic polynomial chaos expansions (AL-SPCE), is validated on three different types of problems. In all cases, the results show that the active learning approach significantly improves efficiency compared to previous surrogate-based approaches and direct Monte Carlo simulation, while maintaining accurate reliability estimates.