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An active learning Kriging model with adaptive parameters for reliability analysis

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Abstract

The prevalence of highly nonlinear and implicit performance functions in structural reliability analysis has increased the computational effort significantly. To solve this problem, an efficiently active learning function, named parameter adaptive expected feasibility function (PAEFF) is proposed using the prediction variance and joint probability density. The PAEFF function first uses the harmonic mean of prediction variances of Kriging model to judge the iteration degree of the current surrogate model, to realize the scaling of the variance in the expected feasibility function. Second, to improve the prediction accuracy of the Kriging model, the joint probability densities are applied to ensure that the sample points to be updated have a higher probability of occurrence. Finally, a new failure probability-based stopping criterion with wider applicability is proposed. Theoretically, the stopping criterion proposed is applicable to all active learning functions. The effectiveness and accuracy of the proposed PAEFF are verified by two mathematical calculations and three engineering examples.

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References

  1. Metropolis N, Ulam S (1949) The Monte Carlo method [J]. J Am Stat Assoc 44(247):335–341

    Article  MATH  Google Scholar 

  2. Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format [J]. J Eng Mech Div 100(1):111–121

    Article  Google Scholar 

  3. Tvedt L (1990) Distribution of quadratic forms in normal space—application to structural reliability [J]. J Eng Mech 116(6):1183–1197

    Article  Google Scholar 

  4. Gaspar B, Teixeira AP, Soares CG (2017) Adaptive surrogate model with active refinement combining Kriging and a trust region method [J]. Reliab Eng Syst Saf 165:277–291

    Article  Google Scholar 

  5. Marelli S, Sudret B (2018) An active-learning algorithm that combines sparse polynomial chaos expansions and bootstrap for structural reliability analysis [J]. Struct Saf 75:67–74

    Article  Google Scholar 

  6. Hariri-Ardebili MA, Pourkamali-Anaraki F (2018) Support vector machine based reliability analysis of concrete dams [J]. Soil Dyn Earthq Eng 104:276–295

    Article  Google Scholar 

  7. Cardoso JB, de Almeida JR, Dias JM et al (2008) Structural reliability analysis using Monte Carlo simulation and neural networks [J]. Adv Eng Softw 39(6):505–513

    Article  Google Scholar 

  8. 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 

  9. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions [J]. J Glob Optim 13(4):455–492

    Article  MathSciNet  MATH  Google Scholar 

  10. Bichon BJ, Eldred MS, Swiler LP et al (2008) Efficient global reliability analysis for nonlinear implicit performance functions [J]. AIAA J 46(10):2459–2468

    Article  Google Scholar 

  11. Echard B, Gayton N, Lemaire M (2011) AK-MCS: an active learning reliability method combining Kriging and Monte Carlo simulation [J]. Struct Saf 33(2):145–154

    Article  Google Scholar 

  12. Meng Z, Zhang Z, Li G et al (2020) An active weight learning method for efficient reliability assessment with small failure probability [J]. Struct Multidiscip Optim 61(3):1157–1170

    Article  MathSciNet  Google Scholar 

  13. Zheng PJ, Wang CM, Zong ZH et al (2017) A new active learning method based on the learning function U of the AK-MCS reliability analysis method [J]. Eng Struct 148:185–194

    Article  Google Scholar 

  14. Zhang X, Wang L, Sørensen JD (2019) REIF: a novel active-learning function toward adaptive Kriging surrogate models for structural reliability analysis [J]. Reliab Eng Syst Saf 185:440–454

    Article  Google Scholar 

  15. Sun Z, Wang J, Li R et al (2017) LIF: a new Kriging based learning function and its application to structural reliability analysis [J]. Reliab Eng Syst Saf 157:152–165

    Article  Google Scholar 

  16. Hong L, Li H, Peng K (2021) A combined radial basis function and adaptive sequential sampling method for structural reliability analysis [J]. Appl Math Model 90:375–393

    Article  MathSciNet  MATH  Google Scholar 

  17. Xiao NC, Zuo MJ, Zhou C (2018) A new adaptive sequential sampling method to construct surrogate models for efficient reliability analysis [J]. Reliab Eng Syst Saf 169:330–338

    Article  Google Scholar 

  18. Zhou C, Xiao NC, Zuo MJ et al (2020) AK-PDF: An active learning method combining Kriging and probability density function for efficient reliability analysis [J]. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 234(3):536–549

    Google Scholar 

  19. Wen Z, Pei H, Liu H et al (2016) A sequential Kriging reliability analysis method with characteristics of adaptive sampling regions and parallelizability [J]. Reliab Eng Syst Saf 153:170–179

    Article  Google Scholar 

  20. Lv Z, Lu Z, Wang P (2015) A new learning function for Kriging and its applications to solve reliability problems in engineering [J]. Comput Math Appl 70(5):1182–1197

    MathSciNet  MATH  Google Scholar 

  21. Shi Y, Lu Z, He R et al (2020) A novel learning function based on Kriging for reliability analysis [J]. Reliab Eng Syst Saf 198:106857

    Article  Google Scholar 

  22. Wang Z, Shafieezadeh A (2019) REAK: reliability analysis through error rate-based adaptive Kriging [J]. Reliab Eng Syst Saf 182:33–45

    Article  Google Scholar 

  23. Echard B, Gayton N, Lemaire M et al (2013) A combined importance sampling and Kriging reliability method for small failure probabilities with time-demanding numerical models [J]. Reliab Eng Syst Saf 111:232–240

    Article  Google Scholar 

  24. Huang X, Chen J, Zhu H (2016) Assessing small failure probabilities by AK–SS: an active learning method combining Kriging and subset simulation [J]. Struct Saf 59:86–95

    Article  Google Scholar 

  25. Fauriat W, Gayton N (2014) AK-SYS: an adaptation of the AK-MCS method for system reliability [J]. Reliab Eng Syst Saf 123:137–144

    Article  Google Scholar 

  26. Bichon BJ, McFarland JM, Mahadevan S (2011) Efficient surrogate models for reliability analysis of systems with multiple failure modes [J]. Reliab Eng Syst Saf 96(10):1386–1395

    Article  Google Scholar 

  27. Yun W, Lu Z, Zhou Y et al (2019) AK-SYSi: an improved adaptive Kriging model for system reliability analysis with multiple failure modes by a refined U learning function [J]. Struct Multidiscip Optim 59(1):263–278

    Article  MathSciNet  Google Scholar 

  28. Xing J, Luo Y, Gao Z (2020) A global optimization strategy based on the Kriging surrogate model and parallel computing [J]. Struct Multidiscip Optim 62:405-417

  29. Krige DG (1951) A statistical approach to some basic mine valuation problems on the Witwatersrand [J]. J South Afr Inst Min Metall 52(6):119–139

    Google Scholar 

  30. Sacks J, Schiller SB, Welch WJ (1989) Design for computer experiment [J]. Technometrics 31(1):41–47

    Article  MathSciNet  Google Scholar 

  31. Koetler JR, Owen AB (1996) Computer experiments [J]. Handb Stat 13(13):261–308

    Article  MathSciNet  Google Scholar 

  32. Lophaven SN, Nielsen HB, Søndergaard J (2002) DACE, A Matlab Kriging toolbox, version 2.0 [J]. Tech. rep

  33. Zhang X, Pandey M D, Yu R, et al. HALK: A hybrid active-learning Kriging approach and its applications for structural reliability analysis [J]. Engineering with Computers, 2021: 1–17.

  34. Guo J, Du X (2009) Reliability sensitivity analysis with random and interval variables [J]. Int J Numer Meth Eng 78(13):1585–1617

    Article  MathSciNet  MATH  Google Scholar 

  35. Roussouly N, Petitjean F, Salaun M (2013) A new adaptive response surface method for reliability analysis [J]. Probab Eng Mech 32:103–115

    Article  Google Scholar 

Download references

Acknowledgements

This research is supported by the National Natural Science Foundation of China under the Contract no. 51975106.

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Authors and Affiliations

  1. School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Huanwei Xu, Wei Zhang, Naixun Zhou, Lu Xiao & Jingtian Zhang

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  1. Huanwei Xu
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  2. Wei Zhang
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  3. Naixun Zhou
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  4. Lu Xiao
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  5. Jingtian Zhang
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Correspondence to Huanwei Xu.

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Appendix 1: The monotonicity proof of EFF function

Appendix 1: The monotonicity proof of EFF function

The derivative of EFF with respect to prediction variance \(\sigma_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} }} ({\varvec{x}})\) can be given as:

$$\begin{gathered} \frac{{\partial E[\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})]}}{{\partial \sigma_{{\hat{G}}} ({\varvec{x}})}} = 2\left[ {\Phi \left( {\frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) - \Phi \left( {\frac{{ - 2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)} \right] \hfill \\ \, + \left[ {\phi \left( {\frac{{ - 2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) + \phi \left( {\frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) - 2\phi \left( {\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)} \right]. \hfill \\ \end{gathered}$$
(41)

Now prove that the partial derivative of the EFF function for the prediction variance is greater than zero. The last three items of Eq. (41), when the \(\frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}} \le - 1\), that is less than the abscissa of the inflection point of the standard normal distribution − 1. At this time, the probability density function of the standard normal is an up concave function, therefore:

$$\phi \left( {\frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) + \phi \left( {\frac{{ - 2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) - 2\phi \left( {\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) \ge 0.$$
(42)

When \(- 1 < \frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}} \le 1\), ignore the \(\phi \left( {\frac{{ - 2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)\), \(\phi \left( {\frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)\) shows a trend of increasing first and then decreasing, \(2\phi \left( {\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)\) shows a monotonous increasing trend. Because of the nature of concave convex function, the growth rate of \(2\phi \left( {\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)\) is greater than that of \(\phi \left( {\frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)\). Calculate the value of \(\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}\) that satisfies the following formula:

$$\phi \left( {\frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) = 2\phi \left( {\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right).$$
(43)

The calculation result of the above formula is \(\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}} = - \frac{\ln (2)}{2} - 1\), the \(\phi \left( {\frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) = 2\phi \left( {\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) = 0.322\). For the former term of Eq. (41):

$$Z({\varvec{x}}) = 2\left[ {\Phi \left( {\frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right) - \Phi \left( {\frac{{ - 2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)} \right] = {1}{\text{.49}}{.}$$
(44)

The value of Z(x) is much greater than 0.322, therefore, the value of \(\frac{{\partial E[\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})]}}{{\partial \sigma_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} }} ({\varvec{x}})}}\) in interval \(- 1 < \frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}} \le 1 - \frac{\ln (2)}{2}\) is still positive. In the interval of \(1 - \frac{\ln (2)}{2} < \frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}} \le 2\), the value of Z(x) and \(2\phi \left( {\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)\) show a single increasing trend. But the maximum function value of \(2\phi \left( {\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)\) is obtained at \(\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}} = {0}\). The maximum value is \(2\phi \left( {\frac{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}}} \right)_{\max } { = 0}{\text{.80 < 1}}{.49}\). Therefore, it is always true that \(\frac{{\partial E[\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} (x)]}}{{\partial \sigma_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} }} (x)}}\) is greater than zero in the interval of \(- 1 < \frac{{2\sigma_{{\hat{G}}} ({\varvec{x}}) - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})}}{{\sigma_{{\hat{G}}} ({\varvec{x}})}} \le 2\). To sum up, it is known from the symmetry of the standard normal function that \(\frac{{\partial E[\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})]}}{{\partial \sigma_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} }} ({\varvec{x}})}}\) is greater than zero in the whole interval. To sum up, it is known from the symmetry of the standard normal function that \(\frac{{\partial E[\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} ({\varvec{x}})]}}{{\partial \sigma_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} }} ({\varvec{x}})}}\) is greater than zero in the whole interval. Therefore, it can be seen that the EFF function is a monotonic function about the prediction variance.

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Xu, H., Zhang, W., Zhou, N. et al. An active learning Kriging model with adaptive parameters for reliability analysis. Engineering with Computers 39, 3251–3268 (2023). https://doi.org/10.1007/s00366-022-01747-x

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