Development of an adaptive reliability analysis framework for reinforced concrete frame structures using uncertainty quantification 利用不确定性量化为钢筋混凝土框架结构开发自适应可靠性分析框架
Performing reliability analysis for reinforced concrete structures is a tedious and challenging task because it requires conducting a four-nested loop calculation procedure involving millions of data samples to account for the complex behaviours of the structures and multiple random variables. Therefore, the study proposes a novel, practical, and accurate reliability framework that is applicable for multi-component RC frame structures exhibiting different behaviours ranging from linear elastic and non-linear elastic to non-linear plastic. For this purpose, this study first employs a tree-based boosting ensemble model combined with quantile regression, dubbed as QR-LightGBM to calculate the structures’ limit state function and the associated uncertainty estimation at the same time. Next, an active learning process is implemented to improve the computed reliability results progressively. During each active learning step, relevant data samples with potentially high impacts on the model accuracy are determined based on their uncertainty, and then QR-LightGBM is retrained utilizing these samples. By doing so, the prediction performance of the surrogate model is enhanced with a minimized number of actual data samples, thus significantly reducing overall computational resources. The viability and effectiveness of the proposed framework are validated through three case studies involving a simple 1D reinforced concrete beam, a 2D three-story frame, and a 3D fivestory building structure. Furthermore, its performance is quantitatively demonstrated via comparison studies with competing methods such as Monte Carlo simulation, Kriging-based models, and an original LightGBM without active learning. 对钢筋混凝土结构进行可靠性分析是一项繁琐而具有挑战性的任务，因为它需要进行四层循环计算程序，涉及数百万个数据样本，以考虑结构的复杂行为和多个随机变量。因此，本研究提出了一种新颖、实用且精确的可靠性框架，适用于表现出从线性弹性、非线性弹性到非线性塑性等不同行为的多组件 RC 框架结构。为此，本研究首先采用了一种基于树的提升集合模型，并将其与量化回归相结合，称为 QR-LightGBM，以同时计算结构的极限状态函数和相关的不确定性估计。然后，实施主动学习过程，逐步改进计算出的可靠性结果。在每个主动学习步骤中，根据不确定性确定对模型准确性有潜在影响的相关数据样本，然后利用这些样本对 QR-LightGBM 进行再训练。通过这种方法，代用模型的预测性能在实际数据样本数量最少的情况下得到了提升，从而大大减少了总体计算资源。通过涉及简单一维钢筋混凝土梁、二维三层框架和三维四层建筑结构的三个案例研究，验证了所提框架的可行性和有效性。此外，通过与蒙特卡洛模拟、基于克里金的模型和无主动学习的原始 LightGBM 等竞争方法的比较研究，定量证明了该框架的性能。
Reinforced concrete (RC) structures are widely regarded as the most popular load-bearing structures in the real world, ranging from small houses to highrise buildings or largespan bridges. Therefore, ensuring their safety in the face of multiple highly fluctuating factors during their long-term service life is important. An effective way for this purpose is to conduct structural reliability analysis and estimate corresponding reliability indices and failure probabilities. However, structural reliability analysis for RC[1,2]\mathrm{RC}[1,2] is highly challenging because it typically involves numerous 钢筋混凝土（RC）结构被广泛认为是现实世界中最常见的承重结构，小到房屋，大到高层建筑或大跨度桥梁。因此，确保其在长期使用过程中面对多种剧烈波动因素时的安全性非常重要。为此，一种有效的方法就是进行结构可靠性分析，并估算相应的可靠性指数和失效概率。然而， RC[1,2]\mathrm{RC}[1,2] 的结构可靠性分析具有很高的挑战性，因为它通常涉及众多
random variables such as yield strength of steel, compressive strength of concrete, geometrical uncertainties, stochastic external loads [3], cross-section sizes [4], etc. On the other hand, it is commonly acknowledged that RC is an elastoplastic material, which requires nonlinear analysis to accurately modelling its complex behaviours [5]. The reliability analysis of RC could be further more complicated when accounting for time-varying factors such as corrosion, fire, earthquake, wind loads etc. [6, 7]. 随机变量，如钢材屈服强度、混凝土抗压强度、几何不确定性、随机外部荷载 [3]、截面尺寸 [4] 等。另一方面，人们普遍认为 RC 是一种弹塑性材料，需要进行非线性分析才能准确模拟其复杂行为 [5]。如果考虑到腐蚀、火灾、地震、风荷载等时变因素，RC 的可靠性分析可能会更加复杂[6, 7]。[6, 7].
In the authors’ opinion, the methods for addressing structural reliability analysis of RC structure can be categorized into three families: (semi-) analytical methods, simulation techniques, and surrogate models. The analytical approach first constructs a formula approximating the complex original limit state surfaces; after that, searching algorithms are utilized to find the most probable point and calculate the reliability index which is actually the distance between the latter and the origin point [8]. However, it is not always possible 作者认为，解决 RC 结构可靠性分析的方法可分为三类：（半）分析方法、模拟技术和代用模型。分析方法首先构建一个近似复杂原始极限状态曲面的公式，然后利用搜索算法找到最可能的点并计算可靠性指数，可靠性指数实际上就是后者与原点之间的距离[8]。然而，这并不总是可能的
to compute the gradient/Hessian of the limit state function, especially for non-linear problems such as RC structures’ behavior; hence, the applicability of the analytical approach is limited for these scenarios. On the other hand, the simulation technique is a more popular approach that is applicable to almost all structural problems including different structural configurations, various safety criteria, and multiple random variable inputs. Besides the conventional Monte Carlo simulation (MCS), additional sampling techniques such as Importance sampling, Subset simulation, etc. are helpful in reducing the required number of samples when working with problems featuring extremely small failure probabilities [9]. The third family of methods for reliability analysis which is relatively new compared to the other two, is to develop a surrogate model [10-12] that can predict the limit state function with considerably lower computation time compared to direct numerical simulation. 因此，分析方法在这些情况下的适用性有限。另一方面，模拟技术是一种更为流行的方法，几乎适用于所有结构问题，包括不同的结构配置、各种安全标准和多种随机变量输入。除了传统的蒙特卡罗模拟（MCS）外，重要度取样、子集模拟等其他取样技术也有助于在处理失效概率极小的问题时减少所需的取样数量[9]。与其他两种方法相比，可靠性分析的第三种方法相对较新，即开发代用模型 [10-12]，与直接数值模拟相比，它能以更短的计算时间预测极限状态函数。
In the context of structural reliability analysis, different surrogate models have been developed to replace the classical finite element method when assessing the structures’ responses and the associated limit state functions. MendozaLugo [13] proposed a non-parametric Bayesian Network when assessing the failure probability of reinforced concrete bridge pillars. The advantages of the Bayesian Network lie in its non-parametric nature and interpretable causal relationships between random variables and outputs of interests. Some authors incorporate machine/deep learning algorithms into reliability analysis frameworks to boost the calculation process. For example, Pan and Dias [14] adopted the support vector machine model, Hariri-Ardebili and Barak [15] utilized the Extreme gradient boosting method, Hung et al. [16, 17] employed deep learning algorithms such as Long Short term memory, and Transformer architecture, Zhang et al. [18] used Convolutional neural network, and so on. The most popular machine learning widely used in reliability analysis is Gaussian Process, a.k.a, Kriging model thanks to its unique ability to provide uncertainty estimation of prediction results. This capability opens the door for building active learning frameworks which can improve the surrogate model performance with a significantly reduced number of data samples. Some pioneering works leveraging the Kriging model are proposed by Echard et al. in [19-21]. However, the Kriging model needs to construct a large-size matrix for measuring the correlation between data samples. Moreover, the computational resources including the central processing unit (CPU) time and required memory are quadratically proportional to the data size. In fact, the performance of the Kriging model in handling the elastoplastic behavior of RC structure has not been thoroughly investigated in the reviewed works. The Bayesian Neural Network is another practical method able to provide uncertainty quantification, as demonstrated in the work of Li et al. [22] when performing uncertainty quantification in material property 在结构可靠性分析方面，人们开发了不同的替代模型，以取代传统的有限元方法来评估结构的响应和相关的极限状态函数。MendozaLugo [13] 在评估钢筋混凝土桥柱的失效概率时提出了一种非参数贝叶斯网络。贝叶斯网络的优势在于其非参数性质以及随机变量和相关输出之间可解释的因果关系。一些学者在可靠性分析框架中加入了机器/深度学习算法，以改进计算过程。例如，Pan 和 Dias[14]采用了支持向量机模型，Hariri-Ardebili 和 Barak[15]利用了极端梯度提升法，Hung 等人[16, 17]采用了长短期记忆和变压器架构等深度学习算法，Zhang 等人[18]使用了卷积神经网络，等等。在可靠性分析中广泛使用的最流行的机器学习是高斯过程（又称克里金模型），因为它具有对预测结果进行不确定性估计的独特能力。这种能力为构建主动学习框架打开了大门，该框架能在大幅减少数据样本数量的情况下提高代用模型的性能。Echard 等人在 [19-21] 中提出了一些利用克里金模型的开创性工作。然而，克里金模型需要构建一个大型矩阵来测量数据样本之间的相关性。此外，包括中央处理器（CPU）时间和所需内存在内的计算资源与数据大小成二次方。 事实上，Kriging 模型在处理 RC 结构弹塑性行为方面的性能还没有在综述作品中得到深入研究。贝叶斯神经网络是另一种能够提供不确定性量化的实用方法，Li 等人的研究证明了这一点。[22]在对材料特性进行不确定性量化时
prediction, the work of Hung et al. [23] when estimating the time-varying reliability of a bridge subjected to material degradation. Besides, the Dropout neural network is also a promising method for estimating uncertainty in the structures’ nonlinear responses [24]. It is noted that uncertainty quantification is necessary to conduct active learning frameworks, which can improve the surrogate model’s performance with a significantly reduced number of data samples. Various authors have employed active learning in the context of structural reliability analysis. For example, Zhao et al. [25] extended the adaptive Kriging Monte Carlo Simulation (AK-MCS) method, which only considers one new data sample at each iteration, to a parallel method, namely, P-AK-MCS, that can select multiple relevant data samples and append them to the current training dataset to improve the Kriging model. For this purpose, the authors proposed four parallel learning functions to determine various informative candidates without increasing the number of functional evaluations. When studying the safety of the construction equipment aerial building machine for high-rise buildings under extreme wind loads, Wang et al. [26] developed a time-saving active learning approach combining the classical Probability Density Evolution Method with deep learning and Query-by-Dropout-Committee techniques. The proposed method is proven to increase the efficiency by approximately 80%80 \% compared to the traditional PDEM while still maintaining reliability error only within 0.5%0.5 \%. Zhang et al. [27] stated that multiple sources of both time-variant and invariant uncertainties, including material performance degradation, external loads, and environmental conditions, should be taken into account when performing structural analysis. The authors utilized a gated recurrent unit neural network to predict the structures’ responses, Latin hypercube sampling to sample random variables, and an active learning process to improve the model accuracy and reduce the required amount of training data (Table 1). 例如，Hung 等人[23]在估算受材料退化影响的桥梁的时变可靠性时所采用的预测方法。此外，Dropout 神经网络也是估计结构非线性响应不确定性的一种有前途的方法[24]。有学者指出，不确定性量化是进行主动学习框架的必要条件，它可以在显著减少数据样本数量的情况下提高代用模型的性能。许多学者在结构可靠性分析中采用了主动学习方法。例如，Zhao 等人[25] 将每次迭代只考虑一个新数据样本的自适应克里金蒙特卡洛模拟（AK-MCS）方法扩展为一种并行方法，即 P-AK-MCS，它可以选择多个相关数据样本并将其附加到当前训练数据集，以改进克里金模型。为此，作者提出了四种并行学习函数，在不增加函数评估次数的情况下确定各种信息候选。在研究极端风载荷下高层建筑施工设备高空作业机的安全性时，Wang 等人[26]开发了一种省时的主动学习方法，该方法将经典的概率密度演化法与深度学习和逐滴查询技术相结合。事实证明，与传统的概率密度演化法相比，该方法的效率提高了约 80%80 \% ，同时可靠性误差仍保持在 0.5%0.5 \% 以内。Zhang 等人 [27]指出，在进行结构分析时，应考虑时变和不变不确定性的多种来源，包括材料性能退化、外部载荷和环境条件。作者利用门控递归单元神经网络来预测结构的响应，利用拉丁超立方采样法对随机变量进行采样，并利用主动学习过程来提高模型的准确性并减少所需的训练数据量（表 1）。
This study focuses on the reliability analysis of non-linear reinforced concrete structures with multiple random variables. The authors aim to develop a practical, high-performance and efficient framework for structural reliability analysis of reinforced concrete structures based on a machine learning-based surrogate model, active learning and uncertainty quantification. The framework is abbreviated as RCUQR (Uncertainty Quantification-based Reliability analysis framework for Reinforced Concrete structures). We first harness distributed parallel computing with a scripting approach to conduct extensive non-linear analysis. Next, we combine quantile regression with a high-performance machine learning algorithm for two purposes: predicting the structures’ responses and quantifying uncertainty. Third, an active learning process is leveraged to select the most relevant candidates, effectively improving the prediction model. Additionally, multiple utilities have been implemented, including 本研究重点关注具有多个随机变量的非线性钢筋混凝土结构的可靠性分析。作者旨在开发一种实用、高性能和高效的钢筋混凝土结构可靠性分析框架，该框架以基于机器学习的代用模型、主动学习和不确定性量化为基础。该框架简称为 RCUQR（基于不确定性量化的钢筋混凝土结构可靠性分析框架）。我们首先利用分布式并行计算和脚本方法来进行广泛的非线性分析。接下来，我们将量化回归与高性能机器学习算法相结合，以实现两个目的：预测结构的响应和量化不确定性。第三，利用主动学习过程来选择最相关的候选结构，从而有效改进预测模型。此外，还实现了多种实用功能，包括
Table 1 Existing works utilizing surrogate models and active learning in structural reliability analysis 表 1 在结构可靠性分析中利用代用模型和主动学习的现有研究成果
No 没有
Author 作者
Method 方法
Application 应用
Advantage 优势
Discussion 讨论
Hung et al. [23] Hung 等人[23]
Monte Carlo - Bayesian neural network 蒙特卡罗 - 贝叶斯神经网络
Bridge structure 桥梁结构
Realistic scenario 现实情景
Requirement of considerable amount of data samples for training 需要大量数据样本进行训练
Zhao et al. [25] 赵等人[25］
Parallel AK-MCS 并行 AK-MCS
Benchmark and engineering problems 基准和工程问题
Parallel computing 并行计算
Based on Kriging model 基于克里金模型
Diaz et al. [5] 迪亚兹等人[5］
Non-linear FEM and Response Surface method 非线性有限元和响应面法
Reinforced concrete beam 钢筋混凝土梁
High accuracy 高精度
No example with high-dimensional problems 无高维问题实例
4
Lieu et al. [12] Lieu 等人 [12]
Deep neural network 深度神经网络
Truss structure 桁架结构
Adaptive improvement 适应性改进
RC structures exhibits more complex behaviors than truss structures 与桁架结构相比，RC 结构的行为更为复杂
5
Hong et al. [28] Hong 等人 [28]
Surrogate model 代用模型
Benchmark and engineering problems 基准和工程问题
Beyond Kriging model 超越克里金模型
Emphasize the necessity of alternative surrogate models 强调替代模型的必要性
6
Nguyen et al. [16] Nguyen 等人 [16].
Surrogate model, subset simulation 代用模型，子集模拟
Engineering problems 工程问题
Rapid prediction 快速预测
Effort-intensive data preparation and training process 耗费大量精力的数据准备和培训过程
7
Zhang et al. [29] 张等人[29］
Physical-informed neural network 物理信息神经网络
Only benchmark problems 只有基准问题
Suitable for problems described by partial differential equations 适用于偏微分方程描述的问题
Highlight the importance of focusing to the vicinity of the limit state 强调将重点放在极限状态附近的重要性
8
Wang et al. [30] Wang 等人[30］
Convolution neural network 卷积神经网络
Engineering problems with geotechnical systems 岩土系统的工程问题
Reinforce concrete structures with different complexities 加固不同复杂程度的混凝土结构
High efficiency and straightforward to apply to various structures 效率高，可直接应用于各种结构
End-to-end framework with supplement utilities: model selection, visualization, performance and efficiency metrics 带有补充实用程序的端到端框架：模型选择、可视化、性能和效率指标
No Author Method Application Advantage Discussion
Hung et al. [23] Monte Carlo - Bayesian neural network Bridge structure Realistic scenario Requirement of considerable amount of data samples for training
Zhao et al. [25] Parallel AK-MCS Benchmark and engineering problems Parallel computing Based on Kriging model
Diaz et al. [5] Non-linear FEM and Response Surface method Reinforced concrete beam High accuracy No example with high-dimensional problems
4 Lieu et al. [12] Deep neural network Truss structure Adaptive improvement RC structures exhibits more complex behaviors than truss structures
5 Hong et al. [28] Surrogate model Benchmark and engineering problems Beyond Kriging model Emphasize the necessity of alternative surrogate models
6 Nguyen et al. [16] Surrogate model, subset simulation Engineering problems Rapid prediction Effort-intensive data preparation and training process
7 Zhang et al. [29] Physical-informed neural network Only benchmark problems Suitable for problems described by partial differential equations Highlight the importance of focusing to the vicinity of the limit state
8 Wang et al. [30] Convolution neural network Engineering problems with geotechnical systems Effectively handle soil properties' uncertainties Requirement of converting random variables to equivalent multi-channel "image"
9 Moustapha et al. [31] Kriging model + subset simulation Engineering problem with power transmission Multiple limit state functions Requirement of advanced knowledge on deep learning
10 Doan and Dinh [32] Surrogate models + limit state data Truss structures Accuracy of surrogate models is improved with limited amount of data Performance of Kriging model on complex structures is still questionable
11 Das and Tesfamariam [33] Probability density evolution method and stochastic spectral embedding Shear building frame subject to earthquake Computing failure probability with a small number of representative points It is argued that limit state data may not easily determined for high-dimensional nonlinear structures
12 Guo et al. [34] Probability density function-informed method Reinforced concrete beams under structural deterioration Account for various environmental actions Requirement of defining in advance orthonormal basis functions which may vary for different problems
Guardiani et al. [35] Support Vector Regression and Extreme Gradient Boosting Reliability of unsaturated slopes Assessing the mpact of different factors on the failure probability It is desirable to extend the method to more complex structures
This study LightGBM + active learning + quantile regression Reinforce concrete structures with different complexities High efficiency and straightforward to apply to various structures End-to-end framework with supplement utilities: model selection, visualization, performance and efficiency metrics| No | Author | Method | Application | Advantage | Discussion |
| :---: | :---: | :---: | :---: | :---: | :---: |
| | Hung et al. [23] | Monte Carlo - Bayesian neural network | Bridge structure | Realistic scenario | Requirement of considerable amount of data samples for training |
| | Zhao et al. [25] | Parallel AK-MCS | Benchmark and engineering problems | Parallel computing | Based on Kriging model |
| | Diaz et al. [5] | Non-linear FEM and Response Surface method | Reinforced concrete beam | High accuracy | No example with high-dimensional problems |
| 4 | Lieu et al. [12] | Deep neural network | Truss structure | Adaptive improvement | RC structures exhibits more complex behaviors than truss structures |
| 5 | Hong et al. [28] | Surrogate model | Benchmark and engineering problems | Beyond Kriging model | Emphasize the necessity of alternative surrogate models |
| 6 | Nguyen et al. [16] | Surrogate model, subset simulation | Engineering problems | Rapid prediction | Effort-intensive data preparation and training process |
| 7 | Zhang et al. [29] | Physical-informed neural network | Only benchmark problems | Suitable for problems described by partial differential equations | Highlight the importance of focusing to the vicinity of the limit state |
| 8 | Wang et al. [30] | Convolution neural network | Engineering problems with geotechnical systems | Effectively handle soil properties' uncertainties | Requirement of converting random variables to equivalent multi-channel "image" |
| 9 | Moustapha et al. [31] | Kriging model + subset simulation | Engineering problem with power transmission | Multiple limit state functions | Requirement of advanced knowledge on deep learning |
| 10 | Doan and Dinh [32] | Surrogate models + limit state data | Truss structures | Accuracy of surrogate models is improved with limited amount of data | Performance of Kriging model on complex structures is still questionable |
| 11 | Das and Tesfamariam [33] | Probability density evolution method and stochastic spectral embedding | Shear building frame subject to earthquake | Computing failure probability with a small number of representative points | It is argued that limit state data may not easily determined for high-dimensional nonlinear structures |
| 12 | Guo et al. [34] | Probability density function-informed method | Reinforced concrete beams under structural deterioration | Account for various environmental actions | Requirement of defining in advance orthonormal basis functions which may vary for different problems |
| | Guardiani et al. [35] | Support Vector Regression and Extreme Gradient Boosting | Reliability of unsaturated slopes | Assessing the mpact of different factors on the failure probability | It is desirable to extend the method to more complex structures |
| | This study | LightGBM + active learning + quantile regression | Reinforce concrete structures with different complexities | High efficiency and straightforward to apply to various structures | End-to-end framework with supplement utilities: model selection, visualization, performance and efficiency metrics |
a data generation function, scripts connecting Python with the OpenSees program, model selection, hyperparameter optimization, comparison studies, data visualization for failure cases, and convergence plots. Together, these utilities, along with the three main components mentioned above, form an end-to-end framework for the reliability analysis of non-linear RC structures. Compared with other existing reliability analysis tools such as UQPy [36], this solution offers additional advantages. The existing tools primarily focus on first-order reliability method (FORM) and second-order reliability method (SORM) for relatively simple problems with available analytical limit state functions and on Kriging models for numerical problems. Their current version do not support surrogate models based on high-performance machine learning (ML) or deep learning (DL) algorithms. Additionally, active learning is only available with Kriging models, which require predefined kernel functions and the computation of a correlation matrix. These requirements are not always applicable to other surrogate models, making them unsuitable for integrating active learning with modern high-performance ML/DL-based algorithms. The effectiveness and efficiency of RC-UQR are quantitatively demonstrated via three examples with increasing structural complexity. 数据生成功能、连接 Python 与 OpenSees 程序的脚本、模型选择、超参数优化、对比研究、失效案例数据可视化以及收敛图。这些实用工具与上述三个主要组件共同构成了非线性 RC 结构可靠性分析的端到端框架。与其他现有的可靠性分析工具（如 UQPy [36]）相比，该解决方案具有更多优势。现有的工具主要集中在一阶可靠性方法（FORM）和二阶可靠性方法（SORM）上，用于分析相对简单的问题和可用的极限状态函数，以及用于数值问题的克里金模型。其当前版本不支持基于高性能机器学习（ML）或深度学习（DL）算法的代用模型。此外，主动学习仅适用于 Kriging 模型，这需要预定义的核函数和相关矩阵的计算。这些要求并不总是适用于其他代用模型，因此不适合将主动学习与基于 ML/DL 的现代高性能算法集成。RC-UQR 的有效性和效率通过三个结构复杂度不断增加的示例得到了量化证明。
Contributions 捐款
In short, the main contributions of this study are summarized as follows: 总之，本研究的主要贡献概述如下：
From an application perspective, this paper presents an accessible framework for structural engineers, based on well-known knowledge and open source libraries such as quantile regression and the machine learning LightGBM model. It does not require any specialized software, commercial tools or computers with expensive GPU devices for its development and operation. 从应用的角度来看，本文基于众所周知的知识和开源库（如量化回归和机器学习 LightGBM 模型），为结构工程师提出了一个易于使用的框架。它的开发和运行不需要任何专业软件、商业工具或配备昂贵 GPU 设备的计算机。
Combining three techniques- uncertainty quantification, active learning, and machine learning- to perform reliability analysis is an appealing idea within the structural engineering community. However, according to the authors’ knowledge, this is the first time this approach has been applied to non-linear RC structures. This is likely due to the challenge of preparing data for non-linear RC structures, which requires specialized expertise in RC structure analysis and distributed parallel computing to conduct a large number of simulations on different CPUs simultaneously. In addition, complementary results have been presented to provide insights into different aspects of an end-to-end reliability analysis framework involving model selection, visualizing failure cases, the impact of the number of updated candidates, and providing graphical illustrations of the active learning effect. These com- 将不确定性量化、主动学习和机器学习这三种技术结合起来进行可靠性分析，在结构工程界是一个很有吸引力的想法。然而，据作者所知，这是首次将这种方法应用于非线性 RC 结构。这可能是由于为非线性 RC 结构准备数据所面临的挑战，这需要 RC 结构分析方面的专业知识和分布式并行计算，以便在不同的 CPU 上同时进行大量模拟。此外，还介绍了一些补充结果，以深入了解端到端可靠性分析框架的不同方面，包括模型选择、失效案例可视化、更新候选模型数量的影响，以及提供主动学习效果的图解。这些成果包括
prehensive results have not been previously presented or completed in existing works. 在现有的工作中，还没有提出或完成过全面的成果。
From a scientific perspective, the paper emphasizes that data selection is crucial for training a high-performance prediction model in structural reliability analysis. Consequently, directly using a state-of-the-art, well-tuned machine learning model such as LightGBM with randomly selected data samples cannot accurately approximate structures’ responses, leading to unreliable results. To select informative data samples, the active learning process utilizing uncertainty quantification is an effective way which can improve the accuracy of the surrogate model with a small number of additional FEM evaluations. The selected candidates are located either near the limit state surface or in regions where the surrogate model’s predictions are most uncertain. 本文从科学的角度强调，数据选择对于在结构可靠性分析中训练高性能预测模型至关重要。因此，直接使用最先进的、经过良好调整的机器学习模型（如 LightGBM）来随机选择数据样本，无法准确地近似结构的响应，从而导致不可靠的结果。为了选择有参考价值的数据样本，利用不确定性量化的主动学习过程是一种有效的方法，只需少量额外的有限元评估就能提高代用模型的准确性。所选候选样本要么位于极限状态面附近，要么位于代用模型预测最不确定的区域。
In the rest of the paper, Sect. 2 first presents the nonlinear analysis of reinforced concretes using finite element methods and fiber sections. After that, the overall working flow and main components of the RC-UQR framework are described in detail. In Sect. 2, the RC-UQR is applied to three examples including a 1D RC beam, a 2D three-story RC frame, and a 3D RC building structure. Finally, Sect. 2.1 concludes and suggests perspectives for the next steps of the study. 在本文的其余部分，第 2 节首先介绍了使用有限元方法和纤维截面对钢筋混凝土进行的非线性分析。随后，详细介绍了 RC-UQR 框架的整体工作流程和主要组成部分。在第 2 节中，RC-UQR 被应用于三个实例，包括一维 RC 梁、二维三层 RC 框架和三维 RC 建筑结构。最后，第 2.1 节进行了总结，并对下一步研究提出了展望。
2 Adaptive reliability analysis framework for reinforced concrete frame structures using uncertainty quantification 2 采用不确定性量化的钢筋混凝土框架结构自适应可靠性分析框架
2.1 Numerical simulation of reinforced concrete structures 2.1 钢筋混凝土结构的数值模拟
2.1.1 Fiber section modeling 2.1.1 纤维截面建模
Reinforced concrete structural elements encompass two materials that have completely different properties; thus, accurately simulating their behaviour under various load conditions requires an approach that should be theoretically supported by rigorous mathematical and physical foundations, but not too complicated to numerically implement. Among methods specifically dedicated to RC members, the forced-based distributed plasticity element, a.k.a, fiber elements can effectively handle both the plastic and elastic behaviors. Moreover, unlike the concentrated plasticity model which needs to assume in advance the positions of plastic hinges near the element ends, the distributed plasticity model can account for plasticity at multiple sections along the element length. This method first divides the element into a finite number of cross sections at some control points a priori selected by a numerical integration scheme 钢筋混凝土结构件包含两种性质完全不同的材料；因此，要准确模拟它们在各种荷载条件下的行为，需要一种既有严格的数学和物理基础作为理论支撑，又不至于过于复杂的数值计算方法。在专门针对 RC 构件的方法中，基于强制的分布塑性元素（又称纤维元素）可以有效地处理塑性和弹性行为。此外，集中塑性模型需要预先假定塑性铰链靠近构件两端的位置，而分布塑性模型则不同，它可以考虑构件长度上多个截面的塑性。这种方法首先通过数值积分方案在一些事先选定的控制点上将元素分成有限个截面
such as the Gauss-Lobatto quadrature rule. Subsequently, each of these sections is subdivided into N_(fib)N_{f i b} fibers (Fig. 1). The characteristics of a fiber ii involve its spatial location (y,z)(y, z), and the assigned area A_(i,fib)A_{i, f i b}. Once the stress-strain response of each individual longitudinal fiber is computed, the stress-strain state of the hosted section is calculated using the integration operator. The forced-based distributed plasticity element adopts the hypothesis of small displacements and deformations and the cross-section is assumed to be prismatic before, during and after deformation. This assumption implies the geometric linearity which simplifies the geometric relation between section deformations and fiber strains. It is recalled that the main source of nonlinearity in this study comes from material nonlinearity. 例如高斯-洛巴托正交规则。随后，每个截面被细分为 N_(fib)N_{f i b} 纤维（图 1）。纤维 ii 的特征涉及其空间位置 (y,z)(y, z) 和分配面积 A_(i,fib)A_{i, f i b} 。在计算出每根纵向纤维的应力应变响应后，就可以使用积分算子计算托管截面的应力应变状态。基于强制的分布式塑性元素采用了小位移和小变形假设，并假定横截面在变形前、变形中和变形后都是棱柱形的。这一假设意味着几何线性，简化了截面变形和纤维应变之间的几何关系。在本研究中，非线性的主要来源是材料非线性。
In this study, the numerical simulation of RC structures is realized with the help of the finite element program OpenSees [37] thanks to its widely acknowledged credibility and open-source nature that facilitates its integration into the proposed structural reliability framework. The typical steps of a numerical simulation are presented as below: 在本研究中，有限元程序 OpenSees [37]具有广受认可的可信度和开放源代码的特性，便于将其集成到建议的结构可靠性框架中，因此本研究借助 OpenSees 实现了对 RC 结构的数值模拟。数值模拟的典型步骤如下：
Step 1: Provide the structure’s geometrical configuration and dimensions. For this step, Computer-aid-design software such as AutoCAD is usually utilized to sketch the structures in 2D or 3D dimensional. Some FEM program has a built-in geometrical component for conducting this step, potentially equipping with programming support for automatically conducting parts of the geometric construction. 第 1 步：提供结构的几何构造和尺寸。在此步骤中，通常使用 AutoCAD 等计算机辅助设计软件绘制二维或三维结构草图。某些有限元程序内置了用于执行此步骤的几何组件，并可能配备程序支持，自动执行部分几何结构。
Step 2: Define node coordinates. This step involves specifically determining the Cartesian coordinates of the structures’ essential nodes, which are truss joints and beam-column joints. It is also necessary to number these 步骤 2：确定节点坐标。这一步涉及具体确定结构重要节点的笛卡尔坐标，即桁架连接点和梁柱连接点。还需要对这些节点进行编号
Fig. 1 Distributed plasticity element using fiber section for RC members 图 1 使用纤维截面的分布塑性元件用于 RC 构件
models since it is more practical to work with the nodes’ numbers rather than their coordinates. 模型，因为使用节点编号比使用节点坐标更实用。
Step 3: Define boundary conditions. In this step, nodes with supports such as rollers, pinned, or fixed are restrained in their corresponding degrees of freedom. 步骤 3：定义边界条件。在这一步中，带有滚柱、销轴或固定等支撑的节点会在其相应的自由度上受到约束。
Step 4: Define element connectivity. A structural element is defined by explicitly providing the node numbers of its ends. Furthermore, the structural elements need to be numbered. 步骤 4：定义元素连接。通过明确提供结构元素两端的节点编号来定义结构元素。此外，还需要对结构元素进行编号。
Step 5: Provide the materials’ properties, such as the elastic modulus, the compressive strength of concrete, the yield strength and hardening ratio of steel, the damping ratio, and so on. 步骤 5：提供材料属性，如弹性模量、混凝土抗压强度、钢材屈服强度和硬化比、阻尼比等。
Step 6: Section modelling using the fiber approach. This step is specifically applied to modelling reinforced concrete structural elements, which have been described in detail previously. 步骤 6：使用纤维法进行截面建模。这一步骤专门用于钢筋混凝土结构构件的建模，前面已经详细介绍过。
Step 7: Apply external loads. This step can be quickly done by specializing the load intensities and the number of elements/nodes at which external loads are applied. 步骤 7：施加外部载荷。这一步可以通过调整荷载强度和施加外部荷载的元素/节点数量来快速完成。
Step 8: Define desired outputs. Desired outputs can be nodal displacements, accelerations, and internal forces of elements. It is noted that for complex structures with numerous nodes/elements, it is more convenient to export outputs for a subset of critical nodes/ elements rather than every node/element. 步骤 8：定义期望输出。预期输出可以是节点位移、加速度和元素内力。需要注意的是，对于节点/元素众多的复杂结构，输出关键节点/元素子集的输出比输出每个节点/元素的输出更方便。
Step 9: Conduct analysis and store computed results. After these eight previous steps are in place, the analysis step is carried out by selecting appropriate algorithms, such as Newton Raphson iteration and the Newmark numerical integration scheme. 第 9 步：进行分析并存储计算结果。在上述八个步骤完成后，就可以通过选择合适的算法（如牛顿-拉斐尔森迭代和纽马克数值积分方案）进行分析。
Steps 1, 2, 3, and 4 are typical steps in constructing the geometrical configuration of structures in a finite element program. Meanwhile, the theoretical background of steps 5 and 6 is discussed in the previous subsection. Steps 7 and 8 vary depending on the specific problem. For step 9, the non-linear analysis of RC structures consists of a three-nested loop as below: 步骤 1、2、3 和 4 是在有限元程序中构建结构几何构型的典型步骤。同时，步骤 5 和 6 的理论背景已在上一小节中讨论过。第 7 和第 8 步因具体问题而异。对于第 9 步，RC 结构的非线性分析由以下三层循环组成：
The outer “For” loop: the applied load is discretized into a sequence of increment loads F_(0),F_(1)=F_(0)+DeltaF_(1),dots,F_(m)=F_(m-1)+DeltaF_(m)F_{0}, F_{1}=F_{0}+\Delta F_{1}, \ldots, F_{m}=F_{m-1}+\Delta F_{m}, where mm is the total number of load steps. 外部 "For "循环：外加载荷被离散化为一系列递增载荷 F_(0),F_(1)=F_(0)+DeltaF_(1),dots,F_(m)=F_(m-1)+DeltaF_(m)F_{0}, F_{1}=F_{0}+\Delta F_{1}, \ldots, F_{m}=F_{m-1}+\Delta F_{m} ，其中 mm 是载荷的总步数。
The intermediate “While” loop: At each load step, the Newton-Raphson iteration is conducted to numerically solve the structure’s equilibrium equation (Eq. 14). 中间的 "While "循环：在每个载荷步长上，进行牛顿-拉夫逊迭代，数值求解结构的平衡方程（公式 14）。
The inner “For” loop: This loop iterates through all structural elements, calculating their fiber stiffnesses, element stiffnesses, and then assembles them into the global structural stiffness matrix for the current load step. 内部 "For "循环：该循环遍历所有结构元素，计算它们的纤维刚度和元素刚度，然后将它们组合成当前载荷步长的全局结构刚度矩阵。
Moreover, in the context of structural reliability analysis, an additional fourth loop with a large number of samples is required to account for random variables. Collectively, this four-nested loop calculation procedure is highly computationally expensive, or even intractable for complex RC structures in the real world. 此外，在结构可靠性分析中，还需要额外的第四个循环，其中包含大量样本以考虑随机变量。总之，这种四嵌套循环计算程序的计算成本极高，对于现实世界中的复杂 RC 结构而言甚至难以实现。
2.2 Adaptive reliability analysis framework based on uncertainty quantification 2.2 基于不确定性量化的自适应可靠性分析框架
2.2.1 Quantile regression LightGBM model 2.2.1 定量回归 LightGBM 模型
Among a vast number of machine learning/deep learning models for data analysis, tree-based models are currently one of the most favourite ones thanks to their high performance and especially their expressivity, which is built from explicit rules based on the values of features. Another advantage of tree-based methods is that they have fewer hyperparameters to tune from problem to problem compared to the neural network-based counterparts. Furthermore, the performance of tree-based models can be improved by combining different tree models into an ensemble model which is more accurate, robust and applicable to problems featuring complex relationships between inputs and outputs. This is because according to the ensemble learning theory [38], ensemble models benefit from diversity, complementariness, and generalization. In other words, they do not solely depend on a single model configuration, or a specific set of 在大量用于数据分析的机器学习/深度学习模型中，基于树的模型是目前最受欢迎的模型之一，这要归功于它们的高性能，尤其是基于特征值的显式规则所带来的表现力。基于树的方法的另一个优势是，与基于神经网络的方法相比，它们在不同问题上需要调整的超参数较少。此外，还可以通过将不同的树模型组合成一个集合模型来提高树模型的性能，这种集合模型更准确、更稳健，而且适用于输入和输出之间关系复杂的问题。这是因为根据集合学习理论[38]，集合模型得益于多样性、互补性和泛化。换句话说，集合模型并不完全依赖于单一的模型配置，或一组特定的参数。
hyperparameters, and are able to leverage different strategies to discover the impacts of the features and their interrelationships on the outputs. 超参数，并能利用不同的策略来发现特征及其相互关系对输出的影响。
Individual trees in the ensemble model can grow in two fashions: leaf-wise or level-wise. Specifically, in level-wise fashion, at each calculation iteration, a new branch will be added to every leaf of the current level. Meanwhile, in leafwise fashion, only the leaf having the most impact on the loss function is deepened, i.e., prioritizing tree depth over width. This usually result in a lower loss function value with fewer leaves, leading to faster computational time. The popular leaf-wise ensemble model is LightGBM [39] whose mechanism is highlighted in Fig. 2 and mathematically described as follows. 集合模型中的单棵树可以以两种方式生长：按树叶生长或按层次生长。具体来说，在逐级增长时，每次计算迭代都会在当前级别的每一片叶子上添加一个新的分支。与此同时，在按树叶顺序计算时，只有对损失函数影响最大的树叶才会被加深，也就是说，树的深度优先于宽度。这通常会以较少的树叶获得较低的损失函数值，从而加快计算时间。常用的叶式集合模型是 LightGBM [39]，其机制如图 2 所示，数学描述如下。
Assuming that at learning iteration tt, the ensemble model consists of tt tree models: 假设在学习迭代 tt 时，集合模型由 tt 个树模型组成： widehat(f)(chi)=sum_(i=0)^(t-1) widehat(f)_(i)(chi)\widehat{f}(\chi)=\sum_{i=0}^{t-1} \widehat{f}_{i}(\chi),
Next, the residual error r(y_(i),f(x_(i)))r\left(y_{i}, f\left(x_{i}\right)\right) between the results predicted by the ensemble model and actual values is calculated. Subsequently, a new individual model widehat(f)_(t)(chi)=h(chi,theta)\widehat{f}_{t}(\chi)=h(\chi, \theta) developed from hat(f)_(t-1)(chi)\hat{f}_{t-1}(\chi) to estimate the residual is designed, whose parameters theta\theta are determined by: 接着，计算集合模型预测结果与实际值之间的残差 r(y_(i),f(x_(i)))r\left(y_{i}, f\left(x_{i}\right)\right) 。随后，设计一个由 hat(f)_(t-1)(chi)\hat{f}_{t-1}(\chi) 发展而来的新的单独模型 widehat(f)_(t)(chi)=h(chi,theta)\widehat{f}_{t}(\chi)=h(\chi, \theta) 来估计残差，其参数 theta\theta 由以下公式确定： theta_(t)=argmin(sum_(i=1)^(N_("sample "))(r_(i,t)-h(chi_(i),theta))^(2))\theta_{t}=\operatorname{argmin}\left(\sum_{i=1}^{N_{\text {sample }}}\left(r_{i, t}-h\left(\chi_{i}, \theta\right)\right)^{2}\right)
The ensemble model at learning iteration tt is then updated as follows: 然后，学习迭代 tt 中的集合模型更新如下：
Fig. 2 Illustration of the Light Gradient Boosting Machine (LightGBM) model 图 2 光梯度提升机（LightGBM）模型图示
Fig. 3 Schematic representation of the proposed RC-UQR framework for reliability analysis of non-linear reinforced concrete structures 图 3 用于非线性钢筋混凝土结构可靠性分析的拟议 RC-UQR 框架示意图 widehat(f)(chi)=sum_(i=0)^(t-1) widehat(f)_(i)(chi)+h(chi,theta_(t))\widehat{f}(\chi)=\sum_{i=0}^{t-1} \widehat{f}_{i}(\chi)+h\left(\chi, \theta_{t}\right)
In addition to the conventional LightGBM model which provides a single predicted value for a given input data, a variant version of LightGBM, namely quantile regressionLightGBM (QR-LightGBM) is derived to yield additional uncertainty estimation of prediction. Recall that a quantile value Q(tau)Q(\tau) is determined by: 除了为给定输入数据提供单一预测值的传统 LightGBM 模型外，我们还推导出了 LightGBM 的变体版本，即量化回归 LightGBM (QR-LightGBM)，以获得额外的预测不确定性估计。回想一下，量值 Q(tau)Q(\tau) 是通过以下方式确定的： Q(tau)={y inR:F(y) >= tau}Q(\tau)=\{y \in \mathbb{R}: F(y) \geq \tau\} with tau in(0,1)\tau \in(0,1)Q(tau)={y inR:F(y) >= tau}Q(\tau)=\{y \in \mathbb{R}: F(y) \geq \tau\} 与 tau in(0,1)\tau \in(0,1) 的关系
To yield the quantile value Q(tau)Q(\tau), the QR -LightGBM model will be trained by using the quantile regression loss function L_(tau)(y, hat(y))L_{\tau}(y, \hat{y}) instead of a normal regression loss function directly measuring the discrepancy between predicted and actual results such as MAE =|y- hat(y)|=|y-\hat{y}|. The quantile regression loss function is mathematically expressed as follows [40]: 为了得到量化值 Q(tau)Q(\tau) ，将使用量化回归损失函数 L_(tau)(y, hat(y))L_{\tau}(y, \hat{y}) 来训练 QR -LightGBM 模型，而不是使用直接测量预测结果与实际结果之间差异（如 MAE =|y- hat(y)|=|y-\hat{y}| ）的普通回归损失函数。量化回归损失函数的数学表达式如下 [40]： L_(tau)(y,( widehat(y_(tau))))={[tau(y-( widehat(y_(tau))))" if "y- hat(y) > 0],[(1-tau)(y-( widehat(y_(tau))))" otherwise "]:}L_{\tau}\left(y, \widehat{y_{\tau}}\right)=\left\{\begin{array}{c}\tau\left(y-\widehat{y_{\tau}}\right) \text { if } y-\hat{y}>0 \\ (1-\tau)\left(y-\widehat{y_{\tau}}\right) \text { otherwise }\end{array}\right.
Then, by calculating the upper and lower quantiles Q(tau_("high ")),Q(tau_("low "))Q\left(\tau_{\text {high }}\right), Q\left(\tau_{\text {low }}\right), we can estimate the prediction interval Gamma=[Q(tau_("low ")),Q(tau_("high "))]\Gamma=\left[Q\left(\tau_{\text {low }}\right), Q\left(\tau_{\text {high }}\right)\right]. For example, a prediction interval with a coverage of 90%90 \% is defined by Gamma=[Q(0.05),Q(0.95)]\Gamma=[Q(0.05), Q(0.95)]. Based on Gamma\Gamma, we can estimate the standard deviation of the predicted values, denoted by sigma_( widehat(g))(chi)\sigma_{\widehat{g}}(\chi). 然后，通过计算上量化值 Q(tau_("high ")),Q(tau_("low "))Q\left(\tau_{\text {high }}\right), Q\left(\tau_{\text {low }}\right) 和下量化值 Q(tau_("high ")),Q(tau_("low "))Q\left(\tau_{\text {high }}\right), Q\left(\tau_{\text {low }}\right) ，我们可以估算出预测区间 Gamma=[Q(tau_("low ")),Q(tau_("high "))]\Gamma=\left[Q\left(\tau_{\text {low }}\right), Q\left(\tau_{\text {high }}\right)\right] 。例如，覆盖范围为 90%90 \% 的预测区间由 Gamma=[Q(0.05),Q(0.95)]\Gamma=[Q(0.05), Q(0.95)] 定义。根据 Gamma\Gamma ，我们可以估算出预测值的标准偏差，用 sigma_( widehat(g))(chi)\sigma_{\widehat{g}}(\chi) 表示。
2.2.2 Adaptive reliability analysis framework for reinforced concrete structures using quantile regression LightGBM 2.2.2 使用量化回归 LightGBM 的钢筋混凝土结构自适应可靠性分析框架
