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

Abstract

Structural reliability analysis of multi-physics engineering systems can be efficiently implemented through actively trained surrogate models, which often profit from the probabilistic advantageous attributes offered by Gaussian processes. Existing active learning approaches can be categorized according to four distinctive features: surrogate model choice, failure probability estimation technique, learning enhancement metric, and stopping criteria. To substantially minimize computational efforts throughout the process, a formulated learning function yields a proxy for smartly allocating samples with the objective to reduce prediction uncertainty around a specified limit state boundary until a pre-defined threshold value is reached. Derived from the originally proposed Expected Feasibility Function (EFF) and/or U-function, sophisticated active learning schemes have been recently investigated and widely reported in the literature.

While the aforementioned active learning approaches are able to efficiently yield accurate estimates associated with a specific event, e.g., failure probability, the surrogate model and training process mainly focus on regions near the limit state border and might render inaccurate predictions on other relevant regions within the design space. In inspection and maintenance planning applications, for example, a limit state associated with a repair event can be formulated in conjunction with a failure limit state, and in some cases, global probabilistic indicators, e.g., quantity of interest distribution, directly drive inspection and/or repair decisions.

In this work, we investigate the capability of active learning approaches to efficiently identify training points able to yield accurate estimates associated with multiple events. In particular, PC-Kriging-based surrogate models are actively trained considering a variance correction derived from leave-one-out cross validation error information, whereas the learning scheme relies on sequentially generated Monte Carlo samples along with metrics computed via EFF or U-function. Additionally, combinations of typical learning functions able to effectively balance the generation of training points around multiple limit states are further proposed.

All investigated active learning methods are thoroughly tested, in terms of accuracy and sample efficiency, in a typical nonlinear multimodal structural reliability setting and a more practical case study, in which maintenance decisions are planned in the aftermath of a ship collision against an offshore wind substructure. The results reveal that training points selected according to a learning function that specifically focuses on a particular limit state boundary might logically provide inaccurate estimates on other regions of interest. Also, we showcase the benefits of relying on enhanced Gaussian process variance metrics throughout the overall training process.