Fatigue life prediction and reliability analysis: Mechanism and data-driven, deterministic and stochastic methods

ZHU Shunpeng; WANG Lanyi; YAN Mengli; LUO Changqi; WANG Qingyuan

doi:10.6052/1000-0992-26-016

Article Contents

Article Navigation > Advances in Mechanics > 2026 > Accepted Manuscript

Zhu S P, Wang L Y, Yan M L, Luo C Q, Wang Q Y. Fatigue life prediction and reliability analysis: Mechanism and data-driven, deterministic and stochastic methods. Advances in Mechanics, in press doi: 10.6052/1000-0992-26-016

Citation:

Zhu S P, Wang L Y, Yan M L, Luo C Q, Wang Q Y. Fatigue life prediction and reliability analysis: Mechanism and data-driven, deterministic and stochastic methods. Advances in Mechanics, in press doi: 10.6052/1000-0992-26-016

Citation:

PDF( 11879 KB)

Fatigue life prediction and reliability analysis: Mechanism and data-driven, deterministic and stochastic methods

doi: 10.6052/1000-0992-26-016 cstr: 32046.14.1000-0992-26-016

1.
School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, 611731
2.
Failure Mechanics and Engineering Disaster Prevention and Mitigation, Key Laboratory of Sichuan Province, Sichuan University, Chengdu, 610065

Funds: Supported by Key Program of the National Natural Science Foundation of China (No. 12232004), Fundamental Research Funds for the Central Universities (No. ZYGX2024Z013).

More Information

Corresponding author: zspeng2007@uestc.edu.cn; wangqy@scu.edu.cn
Received Date: 2026-04-07
Accepted Date: 2026-07-17

Available Online: 2026-07-24

Abstract

Abstract

Fatigue life prediction and reliability analysis are crucial for ensuring the long-term safety of major equipment in service. Traditional research paradigms mainly rely on expert experience, mechanism modeling, and classical statistical analysis methods. However, due to an insufficient understanding for fatigue failure behavior, traditional research paradigms are hard to guarantee accurate predictions when modeling complex fatigue damage evolution. Represented by machine learning (ML), data-driven methods have shown great potential in fatigue life prediction and reliability analysis due to excellent nonlinear fitting capabilities. The limitation of small sample fatigue dataset and the inherent data-driven nature of ML models limit the generalization performance of ML, making them difficult to meet the safety assessment requirements for long-term service. The physics-informed machine learning (PIML) framework offers a novel modeling paradigm for fatigue life prediction and reliability analysis, driven by failure mechanisms and fatigue data. While retaining the powerful fitting capabilities of ML models, it could ensure the physical consistency of the PIML framework. This study comprehensively outlines the mechanism and data-driven methods for fatigue life prediction and reliability analysis, and discusses the extensions and applications from deterministic life prediction to reliability analysis in detail. This study systematically summarizes different modeling paradigms to embed prior physical knowledge into ML, and deeply analyzes the advantages of different modeling strategies. For the application range, the performance and applicable boundaries are clarified, including computational efficiency, interpretability, and generalization performance. Finally, the limitations of current fatigue life prediction and reliability analysis methods are systematically summarized to highlight potential development directions for creating advanced prediction models that ensure the service safety and structural integrity of major engineering equipment.
- fatigue,
- reliability analysis,
- life prediction,
- metallic materials,
- data-driven,
- physics-informed

FullText(HTML)

References(183)

References

[1]	甘磊, 吴昊, 仲政. 2025a. 数据驱动的金属疲劳寿命模型研究进展. 力学进展, 55(1): 30-79 (Gan L, Wu H, Zhong Z. 2025a. Advances in data-driven models for fatigue life prediction of metallic materials. Advances in Mechanics, 55(1): 30-79). doi: 10.6052/1000-0992-24-025 Gan L, Wu H, Zhong Z. 2025a. Advances in data-driven models for fatigue life prediction of metallic materials. Advances in Mechanics, 55(1): 30-79. doi: 10.6052/1000-0992-24-025
[2]	廖鼎, 朱顺鹏, 牛晓鹏, 等. 2025. 机械结构概率疲劳研究: 现状及展望. 机械工程学报, 61(8): 47-74 (Liao D, Zhu S P, Niu X P, et al. 2025. Probabilistic fatigue research of mechanical structures: state-of-the-art and future trends. Journal of Mechanical Engineering, 61(8): 47-74). doi: 10.3901/JME.2025.08.047 Liao D, Zhu S P, Niu X P, et al. 2025. Probabilistic fatigue research of mechanical structures: state-of-the-art and future trends. Journal of Mechanical Engineering, 61(8): 47-74. doi: 10.3901/JME.2025.08.047
[3]	刘茜, 胡殿印, 王怡, 等. 2025. 航空发动机结构可靠性设计研究进展与展望. 航空学报, 46(21): 532625 (Liu X, Hu D Y, Wang Y, et al. 2025. Structural reliability design for aero-engines: review and prospects. Acta Aeronautica et Astronautica Sinica, 46(21): 532625). doi: 10.7527/S1000-6893.2025.32625 Liu X, Hu D Y, Wang Y, et al. 2025. Structural reliability design for aero-engines: review and prospects. Acta Aeronautica et Astronautica Sinica, 46(21): 532625. doi: 10.7527/S1000-6893.2025.32625
[4]	刘志状, 吴昊. 2023. 一种基于参数影响的数据驱动下的疲劳寿命预测方法. 机械工程学报, 59(4): 71-79 (Liu Z Z, Wu H. 2023. Data-driven fatigue life prediction method based on the Influence of parameters. Journal of Mechanical Engineering, 59(4): 71-79). doi: 10.3901/JME.2023.04.071 Liu Z Z, Wu H. 2023. Data-driven fatigue life prediction method based on the Influence of parameters. Journal of Mechanical Engineering, 59(4): 71-79. doi: 10.3901/JME.2023.04.071
[5]	孟德彪, 杨恒飞, 杨世源, 等. 2025. 自适应多保真度 Kriging 模型辅助的一阶可靠性分析方法. 机械工程学报, 61(18): 344-365 (Meng D B, Yang H F, Yang S Y, et al. 2025. Adaptive multi-fidelity kriging model-assisted first-order reliability analysis method. Journal of Mechanical Engineering, 61(18): 344-365). Meng D B, Yang H F, Yang S Y, et al. 2025. Adaptive multi-fidelity kriging model-assisted first-order reliability analysis method. Journal of Mechanical Engineering, 61(18): 344-365.
[6]	牛晓鹏, 朱顺鹏, 高杰维, 等. 2022a. 多源不确定性下叶盘结构疲劳可靠性分析与优化设计. 推进技术, 43(2): 200988 (Niu X P, Zhu S P, Gao J W, et al. 2022a. Fatigue reliability analysis and optimization design of turbine blade disks under multi-source uncertainties. Journal of Propulsion Technology, 43(2): 200988). doi: 10.13675/j.cnki.tjjs.200988 Niu X P, Zhu S P, Gao J W, et al. 2022a. Fatigue reliability analysis and optimization design of turbine blade disks under multi-source uncertainties. Journal of Propulsion Technology, 43(2): 200988. doi: 10.13675/j.cnki.tjjs.200988
[7]	王润梓, 廖鼎, 张显程, 等. 2021a. 高温结构蠕变疲劳寿命设计方法: 从材料到结构. 机械工程学报, 57(16): 66-86 (Wang R Z, Liao D, Zhang X C, et al. 2021a. Creep-fatigue Life Design Methods in High-temperature Structures: From Materials to Components. Journal of Mechanical Engineering, 57(16): 66-86). doi: 10.3901/JME.2021.16.066 Wang R Z, Liao D, Zhang X C, et al. 2021a. Creep-fatigue Life Design Methods in High-temperature Structures: From Materials to Components. Journal of Mechanical Engineering, 57(16): 66-86. doi: 10.3901/JME.2021.16.066
[8]	谢里阳, 任俊刚, 吴宁祥, 等. 2015. 复杂结构部件概率疲劳寿命预测方法与模型. 航空学报, 36(8): 2688-2695 (Xie L Y, Ren J G, Wu N X, et al. 2015. Probabilistic fatigue life prediction method and modeling for complex structural parts. Acta Aeronautica et Astronautica Sinica, 36(8): 2688-2695). Xie L Y, Ren J G, Wu N X, et al. 2015. Probabilistic fatigue life prediction method and modeling for complex structural parts. Acta Aeronautica et Astronautica Sinica, 36(8): 2688-2695.
[9]	徐燊, 朱顺鹏, 郝永振, 等. 2018. 基于临界面−损伤参量法的高压涡轮盘多轴疲劳寿命预测. 航空学报, 39(9): 221930 (Xu S, Zhu S P, Hao Y Z, et al. 2018. Multiaxial fatique life prediction of an HPT disc based on critical plane-damage parameter. Acta Aeronautica et Astronautica Sinica, 39(9): 221930). doi: 10.7527/S1000-6893.2018.21930 Xu S, Zhu S P, Hao Y Z, et al. 2018. Multiaxial fatique life prediction of an HPT disc based on critical plane-damage parameter. Acta Aeronautica et Astronautica Sinica, 39(9): 221930. doi: 10.7527/S1000-6893.2018.21930
[10]	赵丙峰, 廖鼎, 朱顺鹏, 等. 2021. 机械结构概率疲劳寿命预测研究进展. 机械工程学报, 57(16): 173-184 (Zhao B F, Liao D, Zhu S P, et al. 2021. Probabilistic fatigue life prediction of mechanical structures: state of the art. Journal of Mechanical Engineering, 57(16): 173-184). doi: 10.3901/JME.2021.16.173 Zhao B F, Liao D, Zhu S P, et al. 2021. Probabilistic fatigue life prediction of mechanical structures: state of the art. Journal of Mechanical Engineering, 57(16): 173-184. doi: 10.3901/JME.2021.16.173
[11]	郑新前, 王钧莹, 黄维娜, 等. 2023. 航空发动机不确定性设计体系探讨. 航空学报, 44(7): 027099 (Zheng X Q, Wang J Y, Huang W N, et al. 2023. Uncertainty-based design system for aeroengines. Acta Aeronautica et Astronautica Sinica, 44(7): 027099). Zheng X Q, Wang J Y, Huang W N, et al. 2023. Uncertainty-based design system for aeroengines. Acta Aeronautica et Astronautica Sinica, 44(7): 027099.
[12]	Abiria I, Wang C, Zhang Q, et al. 2025. High-cycle and very-high-cycle fatigue life prediction in additive manufacturing using hybrid physics-informed neural networks. Engineering Fracture Mechanics, 319: 111026. doi: 10.1016/j.engfracmech.2025.111026
[13]	Alibrandi U, Alani A M, Ricciardi G. 2015. A new sampling strategy for SVM-based response surface for structural reliability analysis. Probabilistic Engineering Mechanics, 41: 1-12. doi: 10.1016/j.probengmech.2015.04.001
[14]	Bai Z, Song S. 2023. Structural reliability analysis based on neural networks with physics-informed training samples. Engineering Applications of Artificial Intelligence, 126: 107157. doi: 10.1016/j.engappai.2023.107157
[15]	Bao H, Wu S, Wu Z, et al. 2021. A machine-learning fatigue life prediction approach of additively manufactured metals. Engineering Fracture Mechanics, 242: 107508. doi: 10.1016/j.engfracmech.2020.107508
[16]	Bao Y, Sun H, Guan X, et al. 2024. An active learning method using deep adversarial autoencoder-based sufficient dimension reduction neural network for high-dimensional reliability analysis. Reliability Engineering & System Safety, 247: 110140. doi: 10.1016/j.ress.2024.110140
[17]	Bichon B J, Eldred M S, Swiler L P, et al. 2008. Efficient Global Reliability Analysis for Nonlinear Implicit Performance Functions. AIAA Journal, 46(10): 2459-2468. doi: 10.2514/1.34321
[18]	Bourinet J M, Deheeger F, Lemaire M. 2011. Assessing small failure probabilities by combined subset simulation and Support Vector Machines. Structural Safety, 33(6): 343-353. doi: 10.1016/j.strusafe.2011.06.001
[19]	Bourinet J M. 2016. Rare-event probability estimation with adaptive support vector regression surrogates. Reliability Engineering & System Safety, 150: 210-221. doi: 10.1016/j.ress.2016.01.023
[20]	Brown M, Miller K. 1973. A Theory for Fatigue Failure under Multiaxial Stress-Strain Conditions. Proceedings of the Institution of Mechanical engineers, 187(1): 745-755. doi: 10.1243/PIME_PROC_1973_187_161_02
[21]	Cadini F, Santos F, Zio E. 2014. An improved adaptive kriging-based importance technique for sampling multiple failure regions of low probability. Reliability Engineering & System Safety, 131: 109-117. doi: 10.1016/j.ress.2014.06.023
[22]	Cao X, Zou L, Lu C. 2025. Augmentation method of fatigue data of welded structures based on physics-informed CTGAN. Fracture and Structural Integrity, 19(72): 162-178. doi: 10.3221/igf-esis.72.12
[23]	Cha Y-J, Ali R, Lewis J, et al. 2024. Deep learning-based structural health monitoring. Automation in Construction, 161: 105328. doi: 10.1016/j.autcon.2024.105328
[24]	Chaboche J L, Lesne P M. 1988. A NON-LINEAR CONTINUOUS FATIGUE DAMAGE MODEL. Fatigue & Fracture of Engineering Materials & Structures, 11(1): 1-17. doi: 10.1111/j.1460-2695.1988.tb01216.x
[25]	Chakraborty S. 2020. Simulation free reliability analysis: a physics-informed deep learning based approach. arXiv: 2005.01302.
[26]	Chen H, Yang F, Wu Z, et al. 2023. A nonlinear fatigue damage accumulation model under variable amplitude loading considering the loading sequence effect. International Journal of Fatigue, 177: 107945. doi: 10.1016/j.ijfatigue.2023.107945
[27]	Chen J, Liu S, Zhang W, et al. 2020. Uncertainty quantification of fatigue S-N curves with sparse data using hierarchical Bayesian data augmentation. International Journal of Fatigue, 134: 105511. doi: 10.1016/j.ijfatigue.2020.105511
[28]	Chen J, Liu Y. 2021a. Probabilistic physics-guided machine learning for fatigue data analysis. Expert Systems with Applications, 168: 114316. doi: 10.1016/j.eswa.2020.114316
[29]	Chen J, Liu Y. 2021b. Fatigue property prediction of additively manufactured Ti-6Al-4V using probabilistic physics-guided learning. Additive Manufacturing, 39: 101876. doi: 10.1016/j.addma.2021.101876
[30]	Chen W, Xu C, Shi Y, et al. 2019. A hybrid Kriging-based reliability method for small failure probabilities. Reliability Engineering & System Safety, 189: 31-41. doi: 10.1016/j.ress.2019.04.003
[31]	Chen X, Riaz A, Fassi S E. 2021. Application of artificial neural networks for efficient reliability-based design optimization. the 32nd Congress of the International Council of the Aeronautical Sciences, Shanghai, 1-17.
[32]	Cheng J, Li Q S. 2008. Reliability analysis of structures using artificial neural network based genetic algorithms. Computer Methods in Applied Mechanics and Engineering, 197(45-48): 3742-3750. doi: 10.1016/j.cma.2008.02.026
[33]	Cheng K, Lu Z. 2021. Adaptive Bayesian support vector regression model for structural reliability analysis. Reliability Engineering & System Safety, 206: 107286. doi: 10.1016/j.ress.2020.107286
[34]	Cheng K, Lu Z, Xiao S, et al. 2022. Estimation of small failure probability using generalized subset simulation. Mechanical Systems and Signal Processing, 163: 108114. doi: 10.1016/j.ymssp.2021.108114
[35]	Chocat R, Beaucaire P, Debeugny L, et al. 2019. Damage tolerance reliability analysis combining Kriging regression and support vector machine classification. Engineering Fracture Mechanics, 216: 106514. doi: 10.1016/j.engfracmech.2019.106514
[36]	Coffin L F, Jr. 2022. A Study of the Effects of Cyclic Thermal Stresses on a Ductile Metal. Transactions of the American Society of Mechanical Engineers, 76(6): 931-949. doi: 10.1115/1.4015020
[37]	Cornell C A. 1969. A probability-based structural code. Journal Proceedings. 66(12): 974-985
[38]	Dang C, Valdebenito M A, Wei P, et al. 2024. Bayesian active learning line sampling with log-normal process for rare-event probability estimation. Reliability Engineering & System Safety, 246: 110053. doi: 10.1016/j.ress.2024.110053
[39]	Deng X, Zhu S-P, Zhang S, et al. 2024. Physics-informed machine learning framework for creep-fatigue life prediction of a Ni-based superalloy using ensemble learning. Materials Today Communications, 41: 110260. doi: 10.1016/j.mtcomm.2024.110260
[40]	Deng X, Zhu S-P, Wang L, et al. 2025. Probabilistic framework for strain-based fatigue life prediction and uncertainty quantification using interpretable machine learning. International Journal of Fatigue, 190: 108647. doi: 10.1016/j.ijfatigue.2024.108647
[41]	Dong X-W, Li Z-A, Zhang H, et al. 2023. Neural network-based chaotic crossover method for structural reliability analysis considering time-dependent parameters. Structures, 53: 1186-1195. doi: 10.1016/j.istruc.2023.05.010
[42]	Dong X, Zhang H, Li Z, et al. 2025. Least Squares Support Vector Machines With Variable Selection and Hyperparameter Optimization for Complex Structures Reliability Assessment. Quality and Reliability Engineering International, 41(4): 1461-1470. doi: 10.1002/qre.3726
[43]	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. Reliability Engineering & System Safety, 111: 232-240. doi: 10.1016/j.ress.2012.10.008
[44]	Ellingwood B, Maes M, Michael Bartlett F, et al. 2025. Development of methods of structural reliability. Structural Safety, 113: 102474. doi: 10.1016/j.strusafe.2024.102474
[45]	Eshghi A T, Lee S. 2019. Adaptive improved response surface method for reliability-based design optimization. Engineering Optimization, 51(12): 2011-2029. doi: 10.1080/0305215X.2018.1561885
[46]	Fan J-L, Zhu G, Zhu M-L, et al. 2023. A data-physics integrated approach to life prediction in very high cycle fatigue regime. International Journal of Fatigue, 176: 107917. doi: 10.1016/j.ijfatigue.2023.107917
[47]	Feng F, Zhu T, Yang B, et al. 2025. Probabilistic fatigue life prediction in additive manufacturing materials with a physics-informed neural network framework. Expert Systems with Applications, 275: 127098. doi: 10.1016/j.eswa.2025.127098
[48]	Findley W N. 1957. Fatigue of Matals Under Combinations of Stresses. Transactions of the American Society of Mechanical Engineers, 79(6): 1337-1347. doi: 10.1115/1.4013320
[49]	Frost N E, Dugdale D S. 1958. The propagation of fatigue cracks in sheet specimens. Journal of the Mechanics and Physics of Solids, 6(2): 92-110. doi: 10.1016/0022-5096(58)90018-8
[50]	Gan L, Fan Z-M, Wu H, et al. 2025b. Prediction of multiaxial fatigue life with a data-driven knowledge transfer model. International Journal of Fatigue, 190: 108636. doi: 10.1016/j.ijfatigue.2024.108636
[51]	Gao J, Wang C, Xu Z, et al. 2022. Gaussian process regression based remaining fatigue life prediction for metallic materials under two-step loading. International Journal of Fatigue, 158: 106730. doi: 10.1016/j.ijfatigue.2022.106730
[52]	Gao J, Wang J, Xu Z, et al. 2023. Multiaxial fatigue prediction and uncertainty quantification based on back propagation neural network and Gaussian process regression. International Journal of Fatigue, 168: 107361. doi: 10.1016/j.ijfatigue.2022.107361
[53]	Gaspar B, Teixeira A P, Guedes Soares C. 2017. Adaptive surrogate model with active refinement combining Kriging and a trust region method. Reliability Engineering & System Safety, 165: 277-291. doi: 10.1016/j.ress.2017.03.035
[54]	Guo H Y, Luo C Q, Zhu S P, et al. 2025. Machine learning-based enhanced Monte Carlo simulation for low failure probability structural reliability analysis. Structures, 74: 108530. doi: 10.1016/j.istruc.2025.108530
[55]	Guo Q, Liu Y, Chen B, et al. 2020. An active learning Kriging model combined with directional importance sampling method for efficient reliability analysis. Probabilistic Engineering Mechanics, 60: 103054. doi: 10.1016/j.probengmech.2020.103054
[56]	Haddad M E, Topper T, Smith K. 1979. Prediction of non propagating cracks. Engineering Fracture Mechanics, 11(3): 573-584. doi: 10.1016/0013-7944(79)90081-X
[57]	Halamka J, Bartošák M, Španiel M. 2023. Using hybrid physics-informed neural networks to predict lifetime under multiaxial fatigue loading. Engineering Fracture Mechanics, 289: 109351. doi: 10.1016/j.engfracmech.2023.109351
[58]	Hao W Q, Tan L, Yang X G, et al. 2023. A physics-informed machine learning approach for notch fatigue evaluation of alloys used in aerospace. International Journal of Fatigue, 170: 107536. doi: 10.1016/j.ijfatigue.2023.107536
[59]	Hasofer A M, Lind N C. 1974. Exact and invariant second-moment code format. Journal of the Engineering Mechanics division, 100(1): 111-121. doi: 10.1061/jmcea3.0001848
[60]	Hawchar L, El Soueidy C-P, Schoefs F. 2018. Global kriging surrogate modeling for general time-variant reliability-based design optimization problems. Structural and Multidisciplinary Optimization, 58(3): 955-968. doi: 10.1007/s00158-018-1938-y
[61]	He G, Zhao Y, Yan C. 2024a. Uncertainty quantification in multiaxial fatigue life prediction using Bayesian neural networks. Engineering Fracture Mechanics, 298: 109961. doi: 10.1016/j.engfracmech.2024.109961
[62]	He J-C, Zhu S-P, Gao J-W, et al. 2024b. Microstructural size effect on the notch fatigue behavior of a Ni-based superalloy using crystal plasticity modelling approach. International Journal of Plasticity, 172: 103857. doi: 10.1016/j.ijplas.2023.103857
[63]	He J-C, Zhu S-P, Luo C, et al. 2024c. Probabilistic notch fatigue assessment under size effect using micromechanics-based critical distance theory. International Journal of Fatigue, 183: 108280. doi: 10.1016/j.ijfatigue.2024.108280
[64]	He J-C, Zarzoso G, Song X, et al. 2026. Microstructure sensitive fatigue life prediction and size effects in notched specimens: application to Ni-based superalloy GH4169. International Journal of Plasticity, 202: 104713. doi: 10.1016/j.ijplas.2026.104713
[65]	He L, Wang Z, Akebono H, et al. 2021. Machine learning-based predictions of fatigue life and fatigue limit for steels. Journal of Materials Science & Technology, 90: 9-19. doi: 10.1016/j.jmst.2021.02.021
[66]	Head A K. 1953. XCVIII. The growth of fatigue cracks. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 44(356): 925-938.
[67]	Hu Z, Mansour R, Olsson M, et al. 2021. Second-order reliability methods: a review and comparative study. Structural and Multidisciplinary Optimization, 64(6): 3233-3263. doi: 10.1007/s00158-021-03013-y
[68]	Huang S-Y, Zhang S-H, Liu L-L, et al. 2021. Efficient slope reliability analysis and risk assessment based on multiple Kriging metamodels. Computers and Geotechnics, 137: 104277. doi: 10.1016/j.compgeo.2021.104277
[69]	Huang X, Chen J, Zhu H. 2016. Assessing small failure probabilities by AK-SS: An active learning method combining Kriging and Subset Simulation. Structural Safety, 59: 86-95. doi: 10.1016/j.strusafe.2015.12.003
[70]	Huang X, Wang P, Xin F, et al. 2023. Line sampling based fuzzy simulation coupled with adaptive Kriging for estimating failure possibility of simplified turbine disk. Aerospace Science and Technology, 142: 108613. doi: 10.1016/j.ast.2023.108613
[71]	Hurtado J E. 2007. Filtered importance sampling with support vector margin: A powerful method for structural reliability analysis. Structural Safety, 29(1): 2-15. doi: 10.1016/j.strusafe.2005.12.002
[72]	J. Lanteigne P N-D. 1983. Energy balance approach to low-cycle fatigue. International Journal of Fracture, 23(4): R147-R149. doi: 10.1007/BF00020703
[73]	Jia G, Tabandeh A, Gardoni P. 2021. A density extrapolation approach to estimate failure probabilities. Structural Safety, 93: 102128. doi: 10.1016/j.strusafe.2021.102128
[74]	Jia Z, Ren L, Li H, et al. 2019. Pipeline leakage identification and localization based on the fiber Bragg grating hoop strain measurements and particle swarm optimization and support vector machine. Structural Control and Health Monitoring, 26(2): e2290. doi: 10.1002/stc.2290
[75]	Jiang X, Lu Z. 2022. Extended fuzzy first-order and second-moment method based on equivalent regularization for estimating failure credibility. Aerospace Science and Technology, 124: 107559. doi: 10.1016/j.ast.2022.107559
[76]	Kaymaz I. 2005. Application of kriging method to structural reliability problems. Structural Safety, 27(2): 133-151. doi: 10.1016/j.strusafe.2004.09.001
[77]	Keshtegar B, Chakraborty S. 2018. A hybrid self-adaptive conjugate first order reliability method for robust structural reliability analysis. Applied Mathematical Modelling, 53: 319-332. doi: 10.1016/j.apm.2017.09.017
[78]	Keshtegar B, Ben Seghier M E A, Zio E, et al. 2021. Novel efficient method for structural reliability analysis using hybrid nonlinear conjugate map-based support vector regression. Computer Methods in Applied Mechanics and Engineering, 381: 113818. doi: 10.1016/j.cma.2021.113818
[79]	Lelièvre N, Beaurepaire P, Mattrand C, et al. 2018. AK-MCSi: A Kriging-based method to deal with small failure probabilities and time-consuming models. Structural Safety, 73: 1-11.
[80]	Lemaitre J. 1985a. A Continuous Damage Mechanics Model for Ductile Fracture. Journal of Engineering Materials and Technology, 107(1): 83-89. doi: 10.1299/jsmecmd.2017.30.307
[81]	Lemaitre J. 1985b. Coupled elasto-plasticity and damage constitutive equations. Computer Methods in Applied Mechanics and Engineering, 51(1): 31-49. doi: 10.1016/0045-7825(85)90026-x
[82]	Li H, Sun G, Tian Z, et al. 2024. A physics‐informed neural network framework based on fatigue indicator parameters for very high cycle fatigue life prediction of an additively manufactured titanium alloy. Fatigue & Fracture of Engineering Materials & Structures, 47(9): 3171-3188. doi: 10.1111/ffe.14363
[83]	Li J, Ma Y, Li Z, et al. 2026. A Hybrid Framework for Multiaxial Fatigue Life Prediction Integrating Experiments, Simulations, and Physics-Informed Machine Learning. Acta Mechanica Solida Sinica.
[84]	Li W, Yang R, Qi Q, et al. 2021. A novel structural reliability method based on active Kriging and weighted sampling. Journal of Mechanical Science and Technology, 35(6): 2459-2469. doi: 10.1007/s12206-021-0517-0
[85]	Liang Q, Yang C, Lin Y, et al. 2025. Adaptive Kriging high-dimensional reliability assessment method based on multi-objective particle swarm optimization algorithm. Probabilistic Engineering Mechanics, 82: 103827. doi: 10.1016/j.probengmech.2025.103827
[86]	Liao D, Zhu S-P, Keshtegar B, et al. 2020. Probabilistic framework for fatigue life assessment of notched components under size effects. International Journal of Mechanical Sciences, 181: 105685. doi: 10.1016/j.ijmecsci.2020.105685
[87]	Lieu Q X, Nguyen K T, Dang K D, et al. 2022. An adaptive surrogate model to structural reliability analysis using deep neural network. Expert Systems with Applications, 189: 116104. doi: 10.1016/j.eswa.2021.116104
[88]	Ling C, Lu Z, Feng K, et al. 2019. A coupled subset simulation and active learning kriging reliability analysis method for rare failure events. Structural and Multidisciplinary Optimization, 60(6): 2325-2341. doi: 10.1007/s00158-019-02326-3
[89]	Liu X-X, Elishakoff I. 2020. A combined Importance Sampling and active learning Kriging reliability method for small failure probability with random and correlated interval variables. Structural Safety, 82: 101875. doi: 10.1016/j.strusafe.2019.101875
[90]	Liu Y-K, Fan J-L, Zhu G, et al. 2023. Data-driven approach to very high cycle fatigue life prediction. Engineering Fracture Mechanics, 292: 109630. doi: 10.1016/j.engfracmech.2023.109630
[91]	Lu C, Feng Y-W, Liem R P, et al. 2018. Improved Kriging with extremum response surface method for structural dynamic reliability and sensitivity analyses. Aerospace Science and Technology, 76: 164-175. doi: 10.1016/j.ast.2018.02.012
[92]	Lu Y, Lu Z, Feng K, et al. 2024. Meta model-based importance sampling combined with adaptive Kriging method for estimating failure probability function. Aerospace Science and Technology, 151: 109260. doi: 10.1016/j.ast.2024.109260
[93]	Lu Z, Liu Y. 2009. Crack growth-based multiaxial fatigue life prediction. 12th International Conference on Fracture, Ottawa, 2432-2441.
[94]	Luo C, Zhu S-P, Keshtegar B, et al. 2024. Active Kriging-based conjugate first-order reliability method for highly efficient structural reliability analysis using resample strategy. Computer Methods in Applied Mechanics and Engineering, 423: 116863. doi: 10.1016/j.cma.2024.116863
[95]	Luo C, Keshtegar B, Zhu S-P, et al. 2022a. EMCS-SVR: Hybrid efficient and accurate enhanced simulation approach coupled with adaptive SVR for structural reliability analysis. Computer Methods in Applied Mechanics and Engineering, 400: 115499. doi: 10.1016/j.cma.2022.115499
[96]	Luo C, Keshtegar B, Zhu S P, et al. 2022b. Hybrid enhanced Monte Carlo simulation coupled with advanced machine learning approach for accurate and efficient structural reliability analysis. Computer Methods in Applied Mechanics and Engineering, 388: 114218. doi: 10.1016/j.cma.2021.114218
[97]	Matin M, Azadi M. 2024. A novel machine learning-based model for predicting the transition fatigue lifetime in piston aluminum alloys. International Journal of Lightweight Materials and Manufacture, 7(5): 641-647. doi: 10.1016/j.ijlmm.2024.04.004
[98]	Meng D, Yang S, Lin T, et al. 2022. RBMDO Using Gaussian Mixture Model-Based Second-Order Mean-Value Saddlepoint Approximation. Computer Modeling in Engineering & Sciences, 132(2): 553-568. doi: 10.32604/cmes.2022.020756
[99]	Meng Z, Li G, Yang D, et al. 2017a. A new directional stability transformation method of chaos control for first order reliability analysis. Structural and Multidisciplinary Optimization, 55(2): 601-612. doi: 10.1007/s00158-016-1525-z
[100]	Meng Z, Pu Y, Zhou H. 2017b. Adaptive stability transformation method of chaos control for first order reliability method. Engineering with Computers, 34(4): 671-683. doi: 10.1007/s00366-017-0566-2
[101]	Meng Z, Qian Q, Xu M, et al. 2023. PINN-FORM: A new physics-informed neural network for reliability analysis with partial differential equation. Computer Methods in Applied Mechanics and Engineering, 414: 116172. doi: 10.1016/j.cma.2023.116172
[102]	Mohammad Amin Nabian, Rini Jasmine Gladstone, Meidani H. 2021. Efficient training of physics-informed Neural Networks via Importance Sampling. Computer-Aided Civil and Infrastructure Engineering, 36(8): 962-977. doi: 10.1111/mice.12685
[103]	Niu X, Zhu S-P, He J, et al. 2021. Fatigue reliability design and assessment of reactor pressure vessel structures: Concepts and validation. International Journal of Fatigue, 153: 106524. doi: 10.1016/j.ijfatigue.2021.106524
[104]	Niu X, Zhu S-P, He J-C, et al. 2022b. Defect tolerant fatigue assessment of AM materials: Size effect and probabilistic prospects. International Journal of Fatigue, 160: 106884. doi: 10.1016/j.ijfatigue.2022.106884
[105]	Niu X, Zhu S-P, He J-C, et al. 2023. Probabilistic and defect tolerant fatigue assessment of AM materials under size effect. Engineering Fracture Mechanics, 277: 109000. doi: 10.1016/j.engfracmech.2022.109000
[106]	Niu X, He C, Zhu S-P, et al. 2024. Defect sensitivity and fatigue design: Deterministic and probabilistic aspects in additively manufactured metallic materials. Progress in Materials Science, 144: 101290. doi: 10.1016/j.pmatsci.2024.101290
[107]	Nowell D, Nowell S C. 2019. A comparison of recent models for fatigue crack tip deformation. Theoretical and Applied Fracture Mechanics, 103: 102299. doi: 10.1016/j.tafmec.2019.102299
[108]	Pan Q, Dias D. 2017. An efficient reliability method combining adaptive Support Vector Machine and Monte Carlo Simulation. Structural Safety, 67: 85-95. doi: 10.1016/j.strusafe.2017.04.006
[109]	Papadopoulos H. 2013. Reliable probabilistic classification with neural networks. Neurocomputing, 107: 59-68. doi: 10.1016/j.neucom.2012.07.034
[110]	Papadopoulos V, Giovanis D G, Lagaros N D, et al. 2012. Accelerated subset simulation with neural networks for reliability analysis. Computer Methods in Applied Mechanics and Engineering, 223-224: 70-80.
[111]	Paris P, Erdogan F. 1963. A critical analysis of crack propagation laws. Journal of Basic Engineering, 85(4): 528-534. doi: 10.1115/1.3656900
[112]	Patel J, Choi S-K. 2011. Classification approach for reliability-based topology optimization using probabilistic neural networks. Structural and Multidisciplinary Optimization, 45(4): 529-543. doi: 10.1007/s00158-011-0711-2
[113]	Patel J, Choi S-K. 2012. An enhanced classification approach for reliability estimation of structural systems. Journal of Intelligent Manufacturing, 25(3): 505-519. doi: 10.1007/s10845-012-0702-1
[114]	Peng X, Wu S, Qian W, et al. 2022. The potency of defects on fatigue of additively manufactured metals. International Journal of Mechanical Sciences, 221: 107185. doi: 10.1016/j.ijmecsci.2022.107185
[115]	Rabotnov Y N, Leckie F A, Prager W. 1970. Creep Problems in Structural Members. Journal of Applied Mechanics, 37(1): 249-249. doi: 10.1115/1.3408479
[116]	Rackwitz R, Flessler B. 1978. Structural reliability under combined random load sequences. Computers & Structures, 9(5): 489-494. doi: 10.1016/0045-7949(78)90046-9
[117]	Razaaly N, Crommelin D, Congedo P M. 2020. Efficient estimation of extreme quantiles using adaptive kriging and importance sampling. International Journal for Numerical Methods in Engineering, 121(9): 2086-2105. doi: 10.1002/nme.6300
[118]	Richard B, Cremona C, Adelaide L. 2012. A response surface method based on support vector machines trained with an adaptive experimental design. Structural Safety, 39: 14-21. doi: 10.1016/j.strusafe.2012.05.001
[119]	Roudak M A, Shayanfar M A, Barkhordari M A, et al. 2017. A robust approximation method for nonlinear cases of structural reliability analysis. International Journal of Mechanical Sciences, 133: 11-20. doi: 10.1016/j.ijmecsci.2017.08.038
[120]	Roussouly N, Petitjean F, Salaun M. 2013. A new adaptive response surface method for reliability analysis. Probabilistic Engineering Mechanics, 32: 103-115. doi: 10.1016/j.probengmech.2012.10.001
[121]	Roy A, Chakraborty S. 2020. Support vector regression based metamodel by sequential adaptive sampling for reliability analysis of structures. Reliability Engineering & System Safety, 200: 106948. doi: 10.1016/j.ress.2020.106948
[122]	Roy A, Chakraborty S. 2022. Reliability analysis of structures by a three-stage sequential sampling based adaptive support vector regression model. Reliability Engineering & System Safety, 219: 108260. doi: 10.1016/j.ress.2021.108260
[123]	Roy A, Chatterjee T, Adhikari S. 2024. A physics-informed neural network enhanced importance sampling (PINN-IS) for data-free reliability analysis. Probabilistic Engineering Mechanics, 78: 103701. doi: 10.1016/j.probengmech.2024.103701
[124]	Sadananda K, Nani Babu M, Vasudevan A K. 2019. A review of fatigue crack growth resistance in the short crack growth regime. Materials Science and Engineering: A, 754: 674-701. doi: 10.1016/j.msea.2019.03.102
[125]	Salvati E, Tognan A, Laurenti L, et al. 2022. A defect-based physics-informed machine learning framework for fatigue finite life prediction in additive manufacturing. Materials & Design, 222: 111089. doi: 10.1016/j.matdes.2022.111089
[126]	Saraygord Afshari S, Enayatollahi F, Xu X, et al. 2022. Machine learning-based methods in structural reliability analysis: A review. Reliability Engineering & System Safety, 219: 108223. doi: 10.1016/j.ress.2021.108223
[127]	Shi T, Sun J, Li J, et al. 2023. Machine learning based very-high-cycle fatigue life prediction of AlSi10Mg alloy fabricated by selective laser melting. International Journal of Fatigue, 171: 107585. doi: 10.1016/j.ijfatigue.2023.107585
[128]	Shi Y, Beer M. 2024. Physics-informed neural network classification framework for reliability analysis. Expert Systems with Applications, 258: 125207. doi: 10.1016/j.eswa.2024.125207
[129]	Song H, Choi K K, Lee I, et al. 2012. Adaptive virtual support vector machine for reliability analysis of high-dimensional problems. Structural and Multidisciplinary Optimization, 47(4): 479-491. doi: 10.1115/detc2011-47538
[130]	Song J, Wei P, Valdebenito M, et al. 2020. Adaptive reliability analysis for rare events evaluation with global imprecise line sampling. Computer Methods in Applied Mechanics and Engineering, 372: 113344. doi: 10.1016/j.cma.2020.113344
[131]	Song J, Wei P, Valdebenito M, et al. 2021. Active learning line sampling for rare event analysis. Mechanical Systems and Signal Processing, 147: 107113. doi: 10.1016/j.ymssp.2020.107113
[132]	Sun Z, Wang J, Li R, et al. 2017. LIF: A new Kriging based learning function and its application to structural reliability analysis. Reliability Engineering & System Safety, 157: 152-165. doi: 10.1016/j.ress.2016.09.003
[133]	Tanaka K, Akiniwa Y. 2024. Short Fatigue-Crack Growth from Crack-like Defects under Completely Reversed Loading Predicted Based on Cyclic R-Curve. Materials, 17(18): 4484. doi: 10.3390/ma17184484
[134]	Tognan A, Patanè A, Laurenti L, et al. 2024. A Bayesian defect-based physics-guided neural network model for probabilistic fatigue endurance limit evaluation. Computer Methods in Applied Mechanics and Engineering, 418: 116521. doi: 10.1016/j.cma.2023.116521
[135]	Wagner F, Latz J, Papaioannou I, et al. 2020. Multilevel sequential importance sampling for rare event estimation. SIAM Journal on Scientific Computing, 42(4): A2062-A2087. doi: 10.1137/19M1289601
[136]	Wang D, Qiu H, Gao L, et al. 2021b. A single-loop Kriging coupled with subset simulation for time-dependent reliability analysis. Reliability Engineering & System Safety, 216: 107931. doi: 10.1016/j.ress.2021.107931
[137]	Wang H, Li B, Gong J, et al. 2023a. Machine learning-based fatigue life prediction of metal materials: Perspectives of physics-informed and data-driven hybrid methods. Engineering Fracture Mechanics, 284: 109242. doi: 10.1016/j.engfracmech.2023.109242
[138]	Wang J, Lu Z, Wang L. 2022. An efficient method for estimating failure probability bounds under random-interval mixed uncertainties by combining line sampling with adaptive Kriging. International Journal for Numerical Methods in Engineering, 124(2): 308-333. doi: 10.1002/nme.7122
[139]	Wang L, Zhu S-P, Luo C, et al. 2023b. Defect driven physics-informed neural network framework for fatigue life prediction of additively manufactured materials. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 381(2260): 20220386.
[140]	Wang L, Zhu S-P, Luo C, et al. 2023c. Physics-guided machine learning frameworks for fatigue life prediction of AM materials. International Journal of Fatigue, 172: 107658. doi: 10.1016/j.ijfatigue.2023.107658
[141]	Wang L, Zhu S-P, Wu B, et al. 2025. Multi-fidelity physics-informed machine learning framework for fatigue life prediction of additive manufactured materials. Computer Methods in Applied Mechanics and Engineering, 439: 117924. doi: 10.1016/j.cma.2025.117924
[142]	Wang Y, Qiu Y, Li J, et al. 2024. Two Fatigue Life Prediction Models Based on the Critical Plane Theory and Artificial Neural Networks [M], Metals: 938.
[143]	Wang Z, Shafieezadeh A. 2021. Metamodel-based subset simulation adaptable to target computational capacities: the case for high-dimensional and rare event reliability analysis. Structural and Multidisciplinary Optimization, 64(2): 649-675. doi: 10.1007/s00158-021-02864-9
[144]	Weng H, Bamer F, Luo C, et al. 2025. Physics-informed neural network for constitutive modeling of cyclic crystal plasticity considering deformation mechanism. International Journal of Mechanical Sciences, 302: 110491. doi: 10.1016/j.ijmecsci.2025.110491
[145]	Xia W, Liao Z. 2025. Enhanced generalized subset simulation with multiple importance sampling for reliability estimation. Computers & Structures, 313: 107741. doi: 10.1016/j.compstruc.2025.107741
[146]	Xiang Z, Chen J, Bao Y, et al. 2020. An active learning method combining deep neural network and weighted sampling for structural reliability analysis. Mechanical Systems and Signal Processing, 140: 106684. doi: 10.1016/j.ymssp.2020.106684
[147]	Xiao L, Wang G, Long W, et al. 2024. Fatigue life prediction of the FCC-based multi-principal element alloys via domain knowledge-based machine learning. Engineering Fracture Mechanics, 296: 109860. doi: 10.1016/j.engfracmech.2024.109860
[148]	Xiao M, Zhang J, Gao L, et al. 2019. An efficient Kriging-based subset simulation method for hybrid reliability analysis under random and interval variables with small failure probability. Structural and Multidisciplinary Optimization, 59(6): 2077-2092. doi: 10.1007/s00158-018-2176-z
[149]	Xie Y-H, Liu Q, Zhu S-P, et al. 2023. Burst speed prediction and reliability assessment of turbine disks: Experiments and probabilistic aspects. Engineering Failure Analysis, 145: 107053. doi: 10.1016/j.engfailanal.2023.107053
[150]	Xiong B, Tan H. 2017. New structural reliability method with focus on important region and based on adaptive support vector machines. Advances in Mechanical Engineering, 9(6): 168781401771058.
[151]	Xu C, Chen W, Ma J, et al. 2020. AK-MSS: An adaptation of the AK-MCS method for small failure probabilities. Structural Safety, 86: 101971. doi: 10.1016/j.strusafe.2020.101971
[152]	Yan Y, Lu Z. 2025. Adaptive Physics-Informed Neural Network Based Directional Sampling Method for Efficient Reliability Analysis. AIAA Journal, 63(6): 2532-2544. doi: 10.2514/1.J064926
[153]	Yang D, Liu Y, Li S, et al. 2017. Fatigue crack growth prediction of 7075 aluminum alloy based on the GMSVR model optimized by the artificial bee colony algorithm. Engineering Computations, 34(4): 1034-1053. doi: 10.1108/EC-11-2015-0362
[154]	Yang J, Kang G, Kan Q. 2022. Rate-dependent multiaxial life prediction for polyamide-6 considering ratchetting: Semi-empirical and physics-informed machine learning models. International Journal of Fatigue, 163: 107086. doi: 10.1016/j.ijfatigue.2022.107086
[155]	Yang S, Meng D, Yang H, et al. 2025. Enhanced soft Monte Carlo simulation coupled with support vector regression for structural reliability analysis. Proceedings of the Institution of Civil Engineers-Transport. Emerald Publishing Limited, 178(7): 459-474 doi: 10.1680/jtran.24.00128
[156]	Yang X, Cheng X. 2020. Active learning method combining Kriging model and multimodal-optimization-based importance sampling for the estimation of small failure probability. International Journal for Numerical Methods in Engineering, 121(21): 4843-4864. doi: 10.1002/nme.6495
[157]	Yang X, Liu Y, Fang X, et al. 2018. Estimation of low failure probability based on active learning Kriging model with a concentric ring approaching strategy. Structural and Multidisciplinary Optimization, 58(3): 1175-1186. doi: 10.1007/s00158-018-1960-0
[158]	Yang Z, Yin C, Li X, et al. 2024. Efficient slope reliability and sensitivity analysis using quantile-based first-order second-moment method. Journal of Rock Mechanics and Geotechnical Engineering, 16(10): 4192-4203. doi: 10.1016/j.jrmge.2024.04.007
[159]	Yi P, Wei K, Kong X, et al. 2015. Cumulative PSO-Kriging model for slope reliability analysis. Probabilistic Engineering Mechanics, 39: 39-45. doi: 10.1016/j.probengmech.2014.12.001
[160]	Yu C, Yang Q, Hu X. 2025. A novel framework of neural network for notch fatigue life prediction by integrating self-attention mechanism and implicit physical constraints. Engineering Fracture Mechanics, 319: 110994. doi: 10.1016/j.engfracmech.2025.110994
[161]	Yu X L, Yan Q S. 2011. Reliability Analysis of Self-Anchored Suspension Bridge by Improved Response Surface Method. Applied Mechanics and Materials, 90-93: 869-873.
[162]	Yuan X, Zheng W, Zhao C, et al. 2024a. Line sampling for time-variant failure probability estimation using an adaptive combination approach. Reliability Engineering & System Safety, 243: 109885. doi: 10.1016/j.ress.2023.109885
[163]	Yuan Y-l, Hu C-m, Li L, et al. 2024b. Efficient slope reliability analysis using a surrogate-assisted normal search particle swarm optimization algorithm. Journal of Computational Design and Engineering, 11(1): 173-194.
[164]	Yunoh M F M, Abdullah S, Saad M H M, et al. 2015. FATIGUE FEATURE EXTRACTION ANALYSIS BASED ON A K-MEANS CLUSTERING APPROACH. Journal of Mechanical Engineering and Sciences, 8: 1275-1282. doi: 10.15282/jmes.8.2015.2.0124
[165]	Zhan L, Liu J, Zhang M, et al. 2020. One-Class Support Vector Machine Based Schemes for Structural Reliability Assessment Under Imbalanced Sample Conditions. IEEE Access, 8: 184350-184359. doi: 10.1109/ACCESS.2020.3027815
[166]	Zhang C, Shafieezadeh A. 2022. Simulation-free reliability analysis with active learning and Physics-Informed Neural Network. Reliability Engineering & System Safety, 226: 108716. doi: 10.1016/j.ress.2022.108716
[167]	Zhang D, Han X, Jiang C, et al. 2017. Time-dependent reliability analysis through response surface method. Journal of Mechanical Design, 139(4): 041404. doi: 10.1115/1.4035860
[168]	Zhang S, Wang L, Zhu S-P, et al. 2024. Physics-informed neural network for creep-fatigue life prediction of Inconel 617 and interpretation of influencing factors. Materials & Design, 245: 113267. doi: 10.1016/j.matdes.2024.113267
[169]	Zhang W, Guan Y, Wang Z, et al. 2025. A novel active learning Kriging based on improved Metropolis-Hastings and importance sampling for small failure probabilities. Computer Methods in Applied Mechanics and Engineering, 435: 117658. doi: 10.1016/j.cma.2024.117658
[170]	Zhang X-C, Gong J-G, Xuan F-Z. 2021a. A physics-informed neural network for creep-fatigue life prediction of components at elevated temperatures. Engineering Fracture Mechanics, 258: 108130. doi: 10.1016/j.engfracmech.2021.108130
[171]	Zhang X, Zhu S-P, He J-C, et al. 2026. Probabilistic crystal plasticity modelling framework for notch fatigue assessment under material variability. Acta Mechanica Sinica, 42(3): 425068. doi: 10.1007/s10409-025-25068-x
[172]	Zhang X, Lu Z, Cheng K. 2021b. AK-DS: An adaptive Kriging-based directional sampling method for reliability analysis. Mechanical Systems and Signal Processing, 156: 107610. doi: 10.1016/j.ymssp.2021.107610
[173]	Zhang X, Wang L, Sørensen J D. 2019. REIF: A novel active-learning function toward adaptive Kriging surrogate models for structural reliability analysis. Reliability Engineering & System Safety, 185: 440-454. doi: 10.1016/j.ress.2019.01.014
[174]	Zhao B, Song J, Xie L, et al. 2023. Multiaxial fatigue life prediction method based on the back-propagation neural network. International Journal of Fatigue, 166: 107274. doi: 10.1016/j.ijfatigue.2022.107274
[175]	Zhao Y, Zhang D, Yang M, et al. 2024. On efficient time-dependent reliability analysis method through most probable point-oriented Kriging model combined with importance sampling. Structural and Multidisciplinary Optimization, 67(1): 6. doi: 10.1007/s00158-023-03721-7
[176]	Zheng W, Yuan X, Bao X, et al. 2025. Adaptive support vector machine for time-variant failure probability function estimation. Reliability Engineering & System Safety, 253: 110510. doi: 10.1016/j.ress.2024.110510
[177]	Zhou T, Sun X, Chen X. 2023a. A multiaxial low-cycle fatigue prediction method under irregular loading by ANN model with knowledge-based features. International Journal of Fatigue, 176: 107868. doi: 10.1016/j.ijfatigue.2023.107868
[178]	Zhou T, Sun X, Chen X. 2023b. A physics-guided modelling method of artificial neural network for multiaxial fatigue life prediction under irregular loading. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 381(2260): 20220392.
[179]	Zhou T, Jiang S, Han T, et al. 2023c. A physically consistent framework for fatigue life prediction using probabilistic physics-informed neural network. International Journal of Fatigue, 166: 107234. doi: 10.1016/j.ijfatigue.2022.107234
[180]	Zhu S-P, Huang H-Z, Peng W, et al. 2016. Probabilistic Physics of Failure-based framework for fatigue life prediction of aircraft gas turbine discs under uncertainty. Reliability Engineering & System Safety, 146: 1-12. doi: 10.1016/j.ress.2015.10.002
[181]	Zhu S-P, Hao Y-Z, Liao D. 2020. Probabilistic modeling and simulation of multiple surface crack propagation and coalescence. Applied Mathematical Modelling, 78: 383-398. doi: 10.1016/j.apm.2019.09.045
[182]	Zhu S-P, Wang L, Luo C, et al. 2023. Physics-informed machine learning and its structural integrity applications: state of the art. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 381(2260): 20220406.
[183]	Zou Q, Wen J. 2023. Bayesian model averaging for probabilistic S-N curves with probability distribution model form uncertainty. International Journal of Fatigue, 177: 107955. doi: 10.1016/j.ijfatigue.2023.107955