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An active learning-oriented error-based stopping criterion for efficient structural reliability analysis

PUBLISHED May 24, 2023 (DOI: https://doi.org/10.54985/peeref.2305p9520990)



Jinsheng Wang1 , Guoji Xu1 , Lei Yu2
  1. Southwest Jiaotong University
  2. China United Engineering Corporation Limited

Conference / event

Ninth International Conference on Engineering Failure Analysis (ICEFA2022), July 2022 (Virtual)

Poster summary

Structural reliability analysis aims to estimate the probability of failure, considering various sources of uncertainty. Among the existing reliability analysis methods, surrogate modeling-based active learning methods are gaining increasing popularity due to their excellent trade-off between accuracy and efficiency. This work focuses on developing an efficient algorithm to terminate the active learning process at an appropriate stage. Specifically, we derive an error-based stopping criterion (ESC) using Chebyshev’s inequality. This criterion allows for the easy calculation of the upper bound of the estimation error for failure probability, without making any assumptions or resorting to bootstrap resampling analysis (BESC). Furthermore, we develop a hybrid stopping criterion that considers both the estimation error of failure probability and its stabilization property at the converged stage. This criterion aims to enhance the computational efficiency of active learning methods. We demonstrate the applicability and effectiveness of the proposed approach through several numerical examples with varying complexity.


Structural reliabiliy analysis, Active learning, Error-based stopping criterion, Chebyshev’s inequality, Surrogate modelling

Research areas

Civil Engineering, Statistics, Computer and Information Science


  1. Cheng K, Lu Z. Active learning Bayesian support vector regression model for global approximation. Information Sciences. 2021, 544:549-63.
  2. Wang J, Li C, Xu G, Li Y, Kareem A. Efficient structural reliability analysis based on adaptive Bayesian support vector regression. Computer Methods in Applied Mechanics and Engineering. 2021, 387:114172.
  3. Wang Z, Shafieezadeh A. ESC: an efficient error-based stopping criterion for kriging-based reliability analysis methods. Structural and Multidisciplinary Optimization. 2019, 59(5):1621-37.
  4. Yi J, Zhou Q, Cheng Y, Liu J. Efficient adaptive Kriging-based reliability analysis combining new learning function and error-based stopping criterion. Structural and Multidisciplinary Optimization. 2020, 62(5):2517-36.
  5. Wang J, Xu G, Mitoulis SA, Li C, Kareem A. Structural reliability analysis using Bayesian support vector regression and subset-assisted importance sampling with active learning. Available at SSRN 4372629. 2023 Feb 28.


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Additional information

Competing interests
No competing interests were disclosed.
Data availability statement
The datasets generated during and / or analyzed during the current study are available from the corresponding author on reasonable request.
Creative Commons license
Copyright © 2023 Wang et al. This is an open access work distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Wang, J., Xu, G., Yu, L. An active learning-oriented error-based stopping criterion for efficient structural reliability analysis [not peer reviewed]. Peeref 2023 (poster).
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