Quantified active learning Kriging model for structural reliability analysis结构可靠性分析的量化主动学习 Kriging 模型
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Prentzas I, Fragiadakis M, 2024. Quantified active learning Kriging model for structural reliability analysis. Probabilistic Engineering Mechanics, 78: 103699.DOI: 10.1016/j.probengmech.2024.103699
提出了一类结构可靠性分析的量化主动学习 Kriging 方法 (quantified active learning Kriging-based, qAK)。所提方法主要基于极限状态面附近更新的概率密度函数 (probability density function, PDF)。采用改进学习函数 (称为最可能误分类函数) 权重构建该概率密度函数。因其对称量化了与模型精度相关的备选点不确定性，故而可作为更新 Kriging 模型的有效度量。所提方法精确近似了极限状态面上的点。此外，还提出了一类基于概率的终止准则。使用加权 K 均值算法和更新概率密度函数的样本来选取新的支撑点。因此，该方法不需要求解优化问题或采用抽样算法。所提量化主动学习 Kriging 方法相比于以前结构可靠性评估的 Kriging 方法，数值实现更为有效、鲁棒。所提方法是在标准可靠性方法 (即蒙特卡罗和子集模拟方法) 框架内提出的。通过六个算例研究验证了所提量化主动学习 Kriging 方法的有效性。关键词: 可靠性, 主动学习, Kriging 模型, 自适应精细化, 代理模型, 最可能误分类函数A quantified active learning Kriging-based (qAK) methodology for structural reliability analysis is presented. The proposed approach is based on an updated probability density function (PDF), which is dominant in the vicinity of the limit-state surface. This PDF is created using weights based on an improved learning function called the most probable misclassification function. This function is used as a metric for efficiently updating the Kriging model, as it symmetrically quantifies the uncertainty of candidate points in terms of the model’s accuracy. The proposed approach accurately approximates the points that lie on the limit-state surface. Moreover, a probabilistic-based stopping criterion is proposed. The new support points are selected using the weighted K-means algorithm and the sample from the updated PDF. Thus, the method does not require solving an optimization problem or using a sampling algorithm. The proposed qAK methods are more reliable and robust than previous implementations of the Kriging method for structural reliability assessment. The proposed approach is presented within the framework of standard reliability methods, i.e., the Monte Carlo and the Subset Simulation methods. The efficiency of the proposed qAK methods is demonstrated with the aid of six case studies.Keywords: Reliability; Active learning; Kriging; Adaptive refinement; Surrogate models; Most probable misclassification function.提出了一类可靠性分析的量化主动学习 Kriging 方法
基于更新概率密度函数进行主动学习
介绍了该方法的两类变体: 量化主动学习 Kriging 蒙特卡罗和量化主动学习 Kriging 子集模拟方法
- 针对 Kriging 训练阶段检验了基于概率的终止准则
量化主动学习 Kriging 子集模拟方法降低了主动学习 Kriging 方法对备选样本大小的敏感性
A quantified active learning Kriging method for reliability analysis is proposed.
Active learning is based on an updated probability density function.
- Two variants of the method are presented: the qAK-MCS and the qAK-SS method.
- A probabilistic-based stopping criterion is examined for the Kriging training phase.
The qAK-SS method reduces the sensitivity of AK methods to the candidate sample size.
图 1: 量化学习函数选取新训练点时的表现: (a) 原学习函数与量化学习函数的对比; (b) 采用原学习函数与量化学习函数为功能函数选取新训练点Fig. 1. Performance of the π_q learning function for selecting new training points: (a) Comparison between π, and π_q; (b) Selection of new training points for the G (x) = 5 - x_2 - 0.5 (x_1 - 0.1)^2 using π and π_q learning functions
图 2: 四分支极限状态函数: (a) 初始与更新概率密度函数样本; (b) 数据聚类与新支撑点选取Fig. 2. Section 6.1: Four-branch limit-state function: (a) Sample from initial and updated PDF; (b) Data clustering and selection of new support points
图 3: 量化主动学习 Kriging 方法流程图Fig. 3. Flowchart of the qAK method
Fig. 4. Monte Carlo sample X adopted for the first example
图 5: 迭代 1: (a) 极限状态函数云图上的新支撑点与初始支撑点; (b) 数据聚类与新支撑点选取Fig. 5. Iteration 1: (a) New and initial support points plotted against contours of the limit-state function G; (b) Data clustering and selection of new support points
图 6: 迭代 2: (a) 极限状态函数云图上的新支撑点与初始支撑点; (b) 数据聚类与新支撑点选取Fig. 6. Iteration 2: (a) New and initial support points plotted against contours of the limit-state function G; (b) Data clustering and selection of new support points
图 7: 迭代 3: (a) 极限状态函数云图上的新支撑点与初始支撑点; (b) 数据聚类与新支撑点选取Fig. 7. Iteration 3: (a) New and initial support points plotted against contours of the limit-state function G; (b) Data clustering and selection of new support points
图 8: 目标精度 1.5% 下四分支函数问题的量化主动学习 Kriging 蒙特卡罗模拟算法收敛过程Fig. 8. Convergence history of qAK-MCS algorithm for the problem of Eq. (32) and target accuracy δ_ε = 1.5%
图 9: 不同终止准则下量化学习函数的失效概率与调用函数箱线图Fig. 9. Boxplots of the results of (a) failure probability and (b) function calls for the proposed learning function π_q using different stopping criteria with ɛ = 0.5
图 10: 双组件串联系统功能函数云图及采用量化主动学习 Kriging 蒙特卡罗模拟的新训练点选取与系统可靠性问题的量化学习函数Fig. 10. Contour plot of Eq. (33) and the new training points selected using qAK-MCS and π_q learning function for the system reliability problem with two components
图 11: 迭代 4: (a) 变拓扑结构组件功能函数的三维图与等值线; (b) 变拓扑结构组件功能函数云图及采用量化主动学习 Kriging 蒙特卡罗模拟的新训练点选取与量化学习函数Fig. 11. Iteration 4: (a) 3D plot of Eq. (34) and the contour lines; (b) Contour plot of Eq. (34) and the new training points selected using qAK-MCS and π_q learning function
Fig. 12. Simplified building model adopted
图 13: 52 杆桁架: (a) 俯视图; (b) 侧视图Fig. 13. 52-bar truss: (a) Top view; (b) Side view
图 14: 三个不同学习函数下量化主动学习 Kriging 蒙特卡罗模拟方法的箱线图: (a) 失效概率; (b) 所需调用函数Fig. 14. Boxplots using the qAK-MCS method with three different learning functions: (a) Failure probability; (b) Required function calls
作者信息 | Authors
希腊雅典技术大学 (National Technical University of Athens) 土木工程学院
Michalis Fragiadakis, 通讯作者 (Corresp.)希腊雅典技术大学 (National Technical University of Athens) 土木工程学院Email: mfrag@mail.ntua.gr
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
