PSI - Issue 52

Wu Zonghui et al. / Procedia Structural Integrity 52 (2024) 203–213 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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bottom at a rate of 0.05 then appears with an upward tendency, it shows a fluctuation in the whole interval. Conversely, the MAPE stabilizes at the calculus value with a low rate (below 0.08) and then shows a slight deviation. As can be seen from Fig.11, the ANNs have similar behaviors in the complex situation during the interval [0,0.10]. The MSE slightly fluctuates after dropping in the wrong rate interval [0,0.05] from the exact solution. Meanwhile, the MSAPE shows a fluctuating downward trend from 0 to 0.1. In contrast, the MAPE is still more robust in the dynamic implicit model, which nearly stays the same. Fig.12 indicates that all these three loss functions have good and similar performances in dealing with tiny vibrations. Meanwhile, they all perform worse in handling the negative equipment error which will increase the failure region. Moreover, Fig.13 and Fig.14 show that the MAPE is the most robust loss function in the two events and the MSAPE performs better than MSE in event 2 while they are similar in event 1. Overall, the MAPE is more robust than the MSAPE and the MSE in dealing with questionable samples. And 10% seems to be the threshold value below which the MAPE can still perform well while others have already deviated from the exact solution significantly. The reason, why the MAPE and the MSAPE perform better than the MSE in the sample set containing incorrect data, has been previously mentioned in 2.1.1. And the best result in the MAPE benefits from its first-moment form, which leads to a lower gradient than the second-moment form during the backward propagation processing of ANN so that it can reduce the influence from questionable data while the large derivatives in the second-moment form generated by incorrect samples will destroy the entire model more easily. ANNs have an acceptable error in dealing with slight deviation samples, and they are sensitive to the expansion of the failure region. 5. Conclusions The following conclusions can be drawn from this study: (1) The UED is an efficient sampling method in small-size sample generation for ANNs. This method can reduce the training to one-third of the samples created by the LHS and performance well, which can cut down the total time cost in complex model reliability analysis. (2) ANNs with different loss functions have similar performances when the training samples are adequate. This proves the capacity of ANNs in resolving dynamic implicit model reliability analysis problems. (3) ANNs have capacities to resist the tiny deviation. And the MAPE is an efficient loss function to deal with the samples containing large deviation data, and the MSE has effective performance in handling correct data. Hence, ANN with MAPE can be taken as a substitution for MSE to deal with complex and questionable data in real engineering. (4) ANNs are more sensitive to the expansion of the failure region in the small probability failure model. (5) The limit of the robustness of the MAPE is around ten percent (wrong sample rate). Although the MAPE shows strong robustness in two cases in this article, it still needs further study to prove its universality in other models. Meanwhile, the MSE is sensitive to incorrect data which has the potential in searching for questionable data in the real project sample sets. References Teixeira R, Nogal M, O'connor A, 2021. Adaptive approaches in metamodel-based reliability analysis: A review. Structural Safety, 89. Gomes H M, Awruch A M, 2004. Comparison of response surface and neural network with other methods for structural reliability analysis. Structural Safety, 26(1): 49-67. Elhewy A H, Mesbahi E, Pu Y, 2006. Reliability analysis of structures using neural network method. Probabilistic Engineering Mechanics, 21(1): 44-53. Dai H Z, Zhang H, Wang W, 2015. A Multiwavelet Neural Network-Based Response Surface Method for Structural Reliability Analysis. Computer Aided Civil and Infrastructure Engineering, 30(2): 151-162. Hurtado J E, Alvarez D A, 2001. Neural-network-based reliability analysis: a comparative study. Computer Methods in Applied Mechanics and Engineering, 191(1-2): 113-132. Mao G J, Niffenegger M, Mao X L, 2022. Probabilistic risk assessment for the piping of a nuclear power plant: Uncertainty and sensitivity analysis by using SINTAP procedure. International Journal of Pressure Vessels and Piping, 200. Thedy J, Liao K W, 2021. Multisphere-based importance sampling for structural reliability. Structural Safety, 91. Rashki M, 2021. Structural reliability reformulation. Structural Safety, 88. Xu C L, Chen W D, Ma J X, et al., 2020. AK-MSS: An adaptation of the AK-MCS method for small failure probabilities. Structural Safety, 86. Chojaczyk A A, Teixeira A P, Neves L C, et al., 2015. Review and application of Artificial Neural Networks models in reliability analysis of steel structures. Structural Safety, 52: 78-89.