Skip to main content
SpringerLink
Log in

An importance sampling method for structural reliability analysis based on interpretable deep generative network

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

Importance sampling methods are widely used in structural reliability analysis. However, owing to the complex shape of optimal importance sampling densities, it is usually difficult to fit the optimal importance sampling densities and sample from the fitted distributions using conventional importance sampling methods. In this paper, a novel importance sampling method based on interpretable deep generative network (IDGN-IS) is proposed for structural reliability analysis. The proposed IDGN-IS model can be directly trained using the data from original distribution of random variables and efficiently sampling from an arbitrary importance sampling density. The developed interpretable deep generative network consists of a deep generative network and a monotonic network, which enables the network to fit and sample from the target distributions while being interpretable. Using the interpretability of the deep generative network, the IDGN-IS method can sample from an arbitrary conditional probability distribution of the fitted distributions by choosing an appropriate threshold of the input Gaussian distribution samples. When the threshold of the input Gaussian distribution samples is set to a value close to zero, the IDGN-IS method can efficiently sample from the optimal importance sampling density and provide accurate estimation of the failure probability. The calculation efficiency and estimation accuracy of the proposed IDGN-IS method in structural reliability analysis are demonstrated using four examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

Data availability

No additional data are available.

References

  1. Ang AS, Tang W (1975) Probability concepts in engineering planning and design[M]. Wiley

    Google Scholar 

  2. Melchers RE (1999) Structural reliability analysis and prediction, 2nd edn. Wiley, West Sussex

    Google Scholar 

  3. Zhao YG, Ono T (1999) A general procedure for first/second-order reliability method (FORM/SORM)[J]. Struct Saf 21(2):95–112

    Article  Google Scholar 

  4. Moustapha M, Marelli S, Sudret B (2022) Active learning for structural reliability: Survey, general framework and benchmark[J]. Struct Saf. https://doi.org/10.1016/j.strusafe.2021.102174

    Article  Google Scholar 

  5. Guimarães H, Matos JC, Henriques AA (2018) An innovative adaptive sparse response surface method for structural reliability analysis[J]. Struct Saf 73:12–28

    Article  Google Scholar 

  6. Zhou Y, Lu Z, Yun W (2020) Active sparse polynomial chaos expansion for system reliability analysis[J]. Reliab Eng Syst Saf. https://doi.org/10.1016/j.ress.2020.107025

    Article  Google Scholar 

  7. Cheng K, Lu Z (2021) Adaptive Bayesian support vector regression model for structural reliability analysis[J]. Reliab Eng Syst Saf 206:107286

    Article  Google Scholar 

  8. Yang J, Cheng L, Ran L (2021) Research on slope reliability analysis using multi-kernel relevance vector machine and advanced first-order second-moment method[J]. Eng Comput. https://doi.org/10.1007/s00366-021-01331-9

    Article  Google Scholar 

  9. Zhang D, Zhou P, Jiang C et al (2021) A stochastic process discretization method combing active learning Kriging model for efficient time-variant reliability analysis[J]. Comput Methods Appl Mech Eng 384:113990

    Article  MathSciNet  Google Scholar 

  10. Zhou T, Peng Y (2021) Active learning and active subspace enhancement for PDEM-based high-dimensional reliability analysis[J]. Struct Saf 88:102026

    Article  Google Scholar 

  11. Zhang X, Pandey MD, Yu R et al (2021) HALK: A hybrid active-learning Kriging approach and its applications for structural reliability analysis[J]. Eng Comput. https://doi.org/10.1007/s00366-021-01308-8

    Article  Google Scholar 

  12. Zhou C, Xiao NC, Zuo MJ et al (2021) An improved Kriging-based approach for system reliability analysis with multiple failure modes[J]. Eng Comput. https://doi.org/10.1007/s00366-021-01349-z

    Article  Google Scholar 

  13. Bao Y, Xiang Z, Li H (2021) Adaptive subset searching-based deep neural network method for structural reliability analysis[J]. Reliab Eng Syst Saf 213:107778

    Article  Google Scholar 

  14. Xiang Z, Bao Y, Tang Z et al (2020) Deep reinforcement learning-based sampling method for structural reliability assessment[J]. Reliab Eng Syst Saf. https://doi.org/10.1016/j.ress.2020.106901

    Article  Google Scholar 

  15. Ren C, Aoues Y, Lemosse D et al (2022) Ensemble of surrogates combining kriging and artificial neural networks for reliability analysis with local goodness measurement[J]. Struct Saf 96:102186

    Article  Google Scholar 

  16. Rubinstein RY, Kroese DP (2016) Simulation and the monte carlo method[M]. Wiley

    Book  Google Scholar 

  17. Yuan X, Liu S, Faes M et al (2021) An efficient importance sampling approach for reliability analysis of time-variant structures subject to time-dependent stochastic load[J]. Mech Syst Signal Process 159:107699

    Article  Google Scholar 

  18. Guo H, Dong Y, Gardoni P (2022) Efficient subset simulation for rare-event integrating point-evolution kernel density and adaptive polynomial chaos kriging[J]. Mech Syst Signal Process 169:108762

    Article  Google Scholar 

  19. Abdollahi A, Azhdary Moghaddam M, Hashemi Monfared SA et al (2021) Subset simulation method including fitness-based seed selection for reliability analysis[J]. Eng Comput 37(4):2689–2705

    Article  Google Scholar 

  20. Zhang X, Lu Z, Cheng K (2021) AK-DS: An adaptive Kriging-based directional sampling method for reliability analysis[J]. Mech Syst Signal Process 156:107610

    Article  Google Scholar 

  21. Papaioannou I, Straub D (2021) Combination line sampling for structural reliability analysis[J]. Struct Saf 88:102025

    Article  Google Scholar 

  22. Au SK, Beck JL (1999) A new adaptive importance sampling scheme for reliability calculations[J]. Struct Saf 21(2):135–158

    Article  Google Scholar 

  23. Wang Z, Song J (2016) Cross-entropy-based adaptive importance sampling using von Mises-Fisher mixture for high dimensional reliability analysis[J]. Struct Saf 59:42–52

    Article  Google Scholar 

  24. Papaioannou I, Geyer S, Straub D (2019) Improved cross entropy-based importance sampling with a flexible mixture model[J]. Reliab Eng Syst Saf 191:106564

    Article  Google Scholar 

  25. Kanjilal O, Papaioannou I, Straub D (2022) Series system reliability of uncertain linear structures under gaussian excitation by cross entropy-based importance sampling[J]. J Eng Mech 148(1):04021136

    Article  Google Scholar 

  26. Zhao Z, Lu ZH, Zhao YG (2022) Time-variant reliability analysis using moment-based equivalent Gaussian process and importance sampling[J]. Struct Multidiscip Optim 65(2):1–17

    Article  MathSciNet  Google Scholar 

  27. Zhang X, Wang L, Sørensen JD (2020) AKOIS: an adaptive Kriging oriented importance sampling method for structural system reliability analysis[J]. Struct Saf 82:101876

    Article  Google Scholar 

  28. Yun W, Lu Z, Wang L et al (2021) Error-based stopping criterion for the combined adaptive Kriging and importance sampling method for reliability analysis[J]. Probab Eng Mech 65:103131

    Article  Google Scholar 

  29. Zhang J, Xiao M, Gao L et al (2019) A combined projection-outline-based active learning Kriging and adaptive importance sampling method for hybrid reliability analysis with small failure probabilities[J]. Comput Methods Appl Mech Eng 344:13–33

    Article  MathSciNet  Google Scholar 

  30. Xiao NC, Zhan H, Yuan K (2020) A new reliability method for small failure probability problems by combining the adaptive importance sampling and surrogate models[J]. Comput Methods Appl Mech Eng 372:113336

    Article  MathSciNet  Google Scholar 

  31. Liu H, He X, Wang P et al (2022) Time-dependent reliability analysis method based on ARBIS and Kriging surrogate model[J]. Eng Comput. https://doi.org/10.1007/s00366-021-01570-w

    Article  Google Scholar 

  32. Tabandeh A, Jia G, Gardoni P (2022) A review and assessment of importance sampling methods for reliability analysis[J]. Struct Saf 97:102216

    Article  Google Scholar 

  33. Wan X, Wei S (2020) Coupling the reduced-order model and the generative model for an importance sampling estimator[J]. J Comput Phys 408:109281

    Article  MathSciNet  Google Scholar 

  34. Dinh L, Sohl-Dickstein J, Bengio S (2016) Density estimation using real nvp[J]. arXiv preprint arXiv:1605.08803

  35. Sill J (1997) Monotonic networks[J]. Advances in neural information processing systems 10. https://proceedings.neurips.cc/paper/1997/hash/83adc9225e4deb67d7ce42d58fe5157c-Abstract.html

  36. Geyer S, Papaioannou I, Straub D (2019) Cross entropy-based importance sampling using Gaussian densities revisited[J]. Struct Saf 76:15–27

    Article  Google Scholar 

  37. Papaioannou I, Papadimitriou C, Straub D (2016) Sequential importance sampling for structural reliability analysis[J]. Struct Saf 62:66–75

    Article  Google Scholar 

  38. Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation[J]. Probab Eng Mech 16(4):263–277

    Article  Google Scholar 

  39. Xiang Z, Chen J, Bao Y et al (2020) An active learning method combining DNN and weighted sampling for structural reliability analysis[J]. Mech Syst Signal Process 140:106684

    Article  Google Scholar 

  40. China, Ministry of Communications (2020) JTG/T 3365-01-2020, Specifications for Design of Highway Cable-stayed Bridge. Beijing: China Communications Press

  41. Su G, Peng L, Hu L (2017) A Gaussian process-based dynamic surrogate model for complex engineering structural reliability analysis[J]. Struct Saf 68:97–109

    Article  Google Scholar 

Download references

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 51925808, U1934209, and 52208223), the Science and Technology Research and Development Program Project of China railway group limited (Major Special Project, No. 2021-Special-04-2), the Tencent Foundation (Xplorer Prize 2021), and China Postdoctoral Science Foundation (No. 2022M713545).

Author information

Authors and Affiliations

  1. Central South University, National Engineering Research Center of High-Speed Railway Construction Technology, Changsha, 410075, China

    Zhengliang Xiang, Xuhui He, Yunfeng Zou & Haiquan Jing

  2. School of Civil Engineering, Central South University, Changsha, 410075, China

    Zhengliang Xiang, Xuhui He, Yunfeng Zou & Haiquan Jing

  3. Hunan Provincial Key Laboratory for Disaster Prevention and Mitigation of Rail Transit Engineering Structures, Changsha, 410075, China

    Zhengliang Xiang, Xuhui He, Yunfeng Zou & Haiquan Jing

Authors
  1. Zhengliang Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuhui He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yunfeng Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Haiquan Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

ZX: Conceptualization, Methodology, Software, Investigation, Writing—Original Draft; XH: Resources, Writing—Review and Editing, Project administration, Funding acquisition, Supervision; Yunfeng Zou: Writing—Review and Editing, Supervision; HJ: Writing—Review and Editing, Supervision.

Corresponding author

Correspondence to Xuhui He.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiang, Z., He, X., Zou, Y. et al. An importance sampling method for structural reliability analysis based on interpretable deep generative network. Engineering with Computers 40, 367–380 (2024). https://doi.org/10.1007/s00366-023-01790-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-023-01790-2

Keywords

Search

Navigation