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

## TL;DR

This paper develops an active learning method using PC-Kriging surrogate models with variance correction to efficiently estimate multiple structural reliability limit states, balancing accuracy and computational cost.

## Contribution

It introduces a novel active learning scheme for multiple limit state functions using variance-enhanced PC-Kriging, applicable to complex nonlinear structural reliability problems.

## Key findings

- Effective in predicting failure and repair events after ship collision
- Balances computational resources while maintaining accuracy for multiple limit states
- Validated on practical offshore wind substructure scenario

## 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.