A digital twin framework for civil engineering structures
土木工程结构的数字孪生框架
A R T I C L E I N F O
A R T I C L E I N F O R M A T I O N
Keywords: 关键词：
Digital twins 数字双胞胎
Predictive maintenance 预测性维护
Bayesian networks 贝叶斯网络
Deep learning 深度学习
Structural health monitoring
结构健康监测
结构健康监测
Model order reduction 减少模型顺序
Abstract 摘要
A B S T R A C T The digital twin concept represents an appealing opportunity to advance condition-based and predictive maintenance paradigms for civil engineering systems, thus allowing reduced lifecycle costs, increased system safety, and increased system availability. This work proposes a predictive digital twin approach to the health monitoring, maintenance, and management planning of civil engineering structures. The asset-twin coupled dynamical system is encoded employing a probabilistic graphical model, which allows all relevant sources of uncertainty to be taken into account. In particular, the time-repeating observations-to-decisions flow is modeled using a dynamic Bayesian network. Real-time structural health diagnostics are provided by assimilating sensed data with deep learning models. The digital twin state is continually updated in a sequential Bayesian inference fashion. This is then exploited to inform the optimal planning of maintenance and management actions within a dynamic decision-making framework. A preliminary offline phase involves the population of training datasets through a reduced-order numerical model and the computation of a health-dependent control policy. The strategy is assessed on two synthetic case studies, involving a cantilever beam and a railway bridge, demonstrating the dynamic decision-making capabilities of health-aware digital twins
A B S T R A C T 数字孪生概念为推进土木工程系统基于状态的预测性维护范例提供了一个极具吸引力的机会，从而可以降低生命周期成本、提高系统安全性和系统可用性。本研究提出了一种预测性数字孪生方法，用于土木工程结构的健康监测、维护和管理规划。资产-孪生耦合动态系统采用概率图形模型进行编码，可将所有相关的不确定性来源考虑在内。特别是，使用动态贝叶斯网络对从观测到决策的时间重复流进行建模。通过深度学习模型同化感知数据，提供实时结构健康诊断。数字孪生状态以连续贝叶斯推理的方式不断更新。然后在动态决策框架内，利用这些数据为维护和管理行动的优化规划提供信息。初步离线阶段包括通过简化的数字模型生成训练数据集，并计算与健康状况相关的控制策略。该策略在涉及悬臂梁和铁路桥梁的两个合成案例研究中进行了评估，展示了健康感知数字孪生的动态决策能力。
1. Introduction 1.导言
The optimal management of deteriorating structural systems is an important challenge in modern engineering. In particular, the failure or non-optimized maintenance planning of civil structures may entail high safety, economic, and social costs. Within this context, enabling a digital twin (DT) perspective for structural systems that are critical for either safety or operative reasons, is crucial to allow for condition-based or predictive maintenance practices, in place of customarily employed time-based ones. Indeed, having an up-to-date digital replica of the physical asset of interest can yield several benefits spanning its entire lifecycle, including performance and health monitoring, as well as maintenance, inspection, and management planning [1].
对老化的结构系统进行优化管理是现代工程学面临的一项重要挑战。特别是，民用结构的故障或未优化的维护规划可能会带来高昂的安全、经济和社会成本。在这种情况下，对那些因安全或运行原因而至关重要的结构系统采用数字孪生（DT）视角，对于实现基于状态或预测的维护实践，以取代通常采用的基于时间的维护实践至关重要。事实上，拥有相关物理资产的最新数字副本可以在其整个生命周期内产生多种益处，包括性能和健康监测，以及维护、检查和管理规划[1]。
对老化的结构系统进行优化管理是现代工程学面临的一项重要挑战。特别是，民用结构的故障或未优化的维护规划可能会带来高昂的安全、经济和社会成本。在这种情况下，对那些因安全或运行原因而至关重要的结构系统采用数字孪生（DT）视角，对于实现基于状态或预测的维护实践，以取代通常采用的基于时间的维护实践至关重要。事实上，拥有相关物理资产的最新数字副本可以在其整个生命周期内产生多种益处，包括性能和健康监测，以及维护、检查和管理规划[1]。
The DT concept [2-6] has been recently applied to several fields for operational monitoring, control, and decision support, including structural health monitoring (SHM) and predictive maintenance [7,8], additive manufacturing [9], smart cities [10], urban sustainability [11], and railway systems management [12]. It allows for a personalized characterization of a physical asset, in the form of computational models and parameters of interest, that evolves over time and is kept synchronized with its physical counterpart by means of data-collecting devices. Within a civil SHM framework, such a twinning perspective can be enabled by
DT 概念[2-6]最近被应用于多个领域的运行监测、控制和决策支持，包括结构健康监测（SHM）和预测性维护[7,8]、增材制造[9]、智能城市[10]、城市可持续性[11]和铁路系统管理[12]。它允许以计算模型和相关参数的形式对物理资产进行个性化描述，并随着时间的推移而不断变化，通过数据采集设备与其物理对应物保持同步。在民用 SHM 框架内，可以通过以下方式实现这种孪生观点
DT 概念[2-6]最近被应用于多个领域的运行监测、控制和决策支持，包括结构健康监测（SHM）和预测性维护[7,8]、增材制造[9]、智能城市[10]、城市可持续性[11]和铁路系统管理[12]。它允许以计算模型和相关参数的形式对物理资产进行个性化描述，并随着时间的推移而不断变化，通过数据采集设备与其物理对应物保持同步。在民用 SHM 框架内，可以通过以下方式实现这种孪生观点
Fig. 1. Predictive digital twin framework for civil engineering structures: graphical abstraction of the end-to-end information flow enabled by the probabilistic graphical model.
图 1.土木工程结构的预测性数字孪生框架：概率图形模型实现的端到端信息流的图形抽象。
图 1.土木工程结构的预测性数字孪生框架：概率图形模型实现的端到端信息流的图形抽象。
the assimilation of data through data-driven structural health diagnostics (from physical to digital), possibly accommodating the quantification and propagation of relevant uncertainties related to, e.g., measurement noise, modeling assumptions, environmental and operational variabilities [13-17]. The resulting updated digital state should then enable prediction of the physical system evolution, as well as inform optimal planning of maintenance and management actions (from digital to physical).
通过数据驱动的结构健康诊断（从物理到数字）对数据进行同化，并可能考虑到与测量噪声、建模假设、环境和运行变异等相关的不确定性的量化和传播[13-17]。由此更新的数字状态应能预测物理系统的演变，并为维护和管理行动的优化规划提供信息（从数字到物理）。
通过数据驱动的结构健康诊断（从物理到数字）对数据进行同化，并可能考虑到与测量噪声、建模假设、环境和运行变异等相关的不确定性的量化和传播[13-17]。由此更新的数字状态应能预测物理系统的演变，并为维护和管理行动的优化规划提供信息（从数字到物理）。
In this work, we propose a DT framework for civil engineering structures. The overall computational strategy is based upon a probabilistic graphical model (PGM) inspired by the foundational model proposed in [18], which provides a general framework to carry out data assimilation, state estimation, prediction, planning, and learning. Formally, such a PGM is a dynamic Bayesian network with the addition of decision nodes, i.e., a dynamic decision network
在这项工作中，我们提出了土木工程结构的 DT 框架。整体计算策略基于概率图形模型（PGM），该模型受文献[18]中提出的基础模型的启发，为数据同化、状态估计、预测、规划和学习提供了一个通用框架。从形式上看，这种 PGM 是一个增加了决策节点的动态贝叶斯网络，即动态决策网络
在这项工作中，我们提出了土木工程结构的 DT 框架。整体计算策略基于概率图形模型（PGM），该模型受文献[18]中提出的基础模型的启发，为数据同化、状态估计、预测、规划和学习提供了一个通用框架。从形式上看，这种 PGM 是一个增加了决策节点的动态贝叶斯网络，即动态决策网络
- From physical to digital. Structural response data are gathered from the physical system and assimilated with deep learning (DL) models, see e.g., [21,22], to estimate the current structural health in terms of presence, location, and severity of structural damage. To solve this inverse problem, we refer to vibration-based SHM techniques, see e.g., [23-26], which exploit the aforementioned collected data, such as displacement or acceleration time histories. This first estimate of the digital state is then employed to estimate an updated digital state, according to control-dependent transition dynamics models describing how the structural health is expected to evolve.
从物理到数字。从物理系统中收集结构响应数据，并与深度学习（DL）模型同化（参见文献[21,22]），以估计当前结构的健康状况，包括结构损坏的存在、位置和严重程度。为解决这一逆向问题，我们参考了基于振动的 SHM 技术，如 [23-26]，该技术利用了上述收集的数据，如位移或加速度时间历程。然后，根据描述结构健康状况预期演变方式的控制相关过渡动力学模型，利用对数字状态的首次估计来估计更新的数字状态。 - From digital to physical. The updated digital state is exploited to predict the future evolution of the physical system and the associated uncertainty, thereby enabling predictive decision-making about maintenance and management actions feeding back to the physical system.
从数字到物理。利用更新后的数字状态来预测物理系统的未来演变和相关的不确定性，从而对反馈到物理系统的维护和管理行动做出预测性决策。 - Offline learning phase. The DT setup considered in this work takes advantage of a preliminary offline learning phase. This phase involves the training of the DL models underlying the structural health identification, and learning the control policy to be applied at each time step of the online phase. The DL models are trained in a supervised fashion, with labeled data pertaining to specific damage conditions generated by exploiting physics-based numerical models. To efficiently assemble a training dataset representative of potential damage and operational conditions the structure might undergo during its lifetime, we exploit a reduced-order modeling strategy for parametrized systems relying on the reduced basis method [27]. The health-dependent
离线学习阶段。本工作中考虑的 DT 设置利用了初步离线学习阶段。该阶段包括对结构健康识别所依据的 DL 模型进行训练，以及学习在线阶段每个时间步骤所要应用的控制策略。DL 模型采用监督方式进行训练，利用基于物理的数值模型生成的与特定损坏条件相关的标记数据。为了有效地收集代表结构在其寿命期内可能发生的潜在损坏和运行条件的训练数据集，我们采用了一种参数化系统的降阶建模策略，该策略依赖于降阶基方法[27]。与健康相关的
Fig. 2. Dynamic decision network encoding the asset-twin coupled dynamical system. Circle nodes denote random variables, square nodes denote actions, and diamond nodes denote the objective function. Bold outlines denote observed quantities, while thin outlines denote estimated quantities. Directed solid edges represent the variables' dependencies encoded via conditional probability distributions, while directed dashed edges represent the variables' dependencies encoded via deterministic functions.
图 2.编码资产-双胞胎耦合动态系统的动态决策网络。圆形节点表示随机变量，方形节点表示行动，菱形节点表示目标函数。粗体轮廓表示观测量，细体轮廓表示估计量。有向实线表示通过条件概率分布编码的变量依赖关系，有向虚线表示通过确定性函数编码的变量依赖关系。
图 2.编码资产-双胞胎耦合动态系统的动态决策网络。圆形节点表示随机变量，方形节点表示行动，菱形节点表示目标函数。粗体轮廓表示观测量，细体轮廓表示估计量。有向实线表示通过条件概率分布编码的变量依赖关系，有向虚线表示通过确定性函数编码的变量依赖关系。
control policy is also computed offline, by maximizing the expected future rewards for the planning problem induced by the PGM.
控制策略也是离线计算的，方法是使 PGM 诱导的规划问题的预期未来回报最大化。
控制策略也是离线计算的，方法是使 PGM 诱导的规划问题的预期未来回报最大化。
The elements of novelty that characterize this work are the following: (i) the adaptation of the PGM digital twinning framework to the health monitoring, maintenance, and management planning of civil engineering structures; (ii) the assimilation of vibration response data is carried out by exploiting DL models, which allow automated selection and extraction of optimized damagesensitive features and real-time assessment of the structural state. This work shows how to incorporate in the DT framework high-dimensional multivariate time series describing the sensor measurements, while tracking the associated uncertainties. The proposed computational framework is made available in the public repository digital-twin-SHM [28]. The code implements the PGM framework as a dynamic decision network. This enables us to easily specify the graph topology from a few time slices, and then unroll it for any number of time steps in the future.
这项工作的新颖之处在于以下几点：(i) PGM 数字孪生框架适用于土木工程结构的健康监测、维护和管理规划；(ii) 利用 DL 模型对振动响应数据进行同化，从而自动选择和提取优化的损伤敏感特征，并对结构状态进行实时评估。这项工作展示了如何将描述传感器测量的高维多变量时间序列纳入 DT 框架，同时跟踪相关的不确定性。提出的计算框架可在公共存储库 digital-twin-SHM [28] 中获取。代码以动态决策网络的形式实现了 PGM 框架。这使我们能够轻松地从几个时间片中指定图拓扑结构，然后在未来任意数量的时间步骤中展开它。
这项工作的新颖之处在于以下几点：(i) PGM 数字孪生框架适用于土木工程结构的健康监测、维护和管理规划；(ii) 利用 DL 模型对振动响应数据进行同化，从而自动选择和提取优化的损伤敏感特征，并对结构状态进行实时评估。这项工作展示了如何将描述传感器测量的高维多变量时间序列纳入 DT 框架，同时跟踪相关的不确定性。提出的计算框架可在公共存储库 digital-twin-SHM [28] 中获取。代码以动态决策网络的形式实现了 PGM 框架。这使我们能够轻松地从几个时间片中指定图拓扑结构，然后在未来任意数量的时间步骤中展开它。
The remainder of this paper is organized as follows. In Section 2, we describe the proposed DT framework. In Section 3, the computational procedure is assessed on two test cases, respectively related to a cantilever beam and a railway bridge. Conclusions and future developments are drawn in Section 4.
本文的其余部分安排如下。在第 2 节中，我们介绍了所提出的 DT 框架。第 3 节中，我们在两个测试案例中对计算程序进行了评估，这两个案例分别与悬臂梁和铁路桥梁有关。第 4 节是结论和未来发展。
本文的其余部分安排如下。在第 2 节中，我们介绍了所提出的 DT 框架。第 3 节中，我们在两个测试案例中对计算程序进行了评估，这两个案例分别与悬臂梁和铁路桥梁有关。第 4 节是结论和未来发展。
2. Predictive digital twins using physics-based models and machine learning
2.利用物理模型和机器学习预测数字双胞胎
In this section, we describe the methodology characterizing our DT framework in terms of the PGM encoding the assettwin coupled dynamical system in Section 2.1; the population of training datasets exploiting physics-based numerical models in Section 2.2; and the DL models underlying the structural health identification in Section 2.3.
在本节中，我们将在第 2.1 节中介绍 DT 框架的方法论特征，包括编码资产双耦合动力系统的 PGM；第 2.2 节中介绍利用基于物理的数值模型的训练数据集群；以及第 2.3 节中介绍结构健康识别所依据的 DL 模型。
在本节中，我们将在第 2.1 节中介绍 DT 框架的方法论特征，包括编码资产双耦合动力系统的 PGM；第 2.2 节中介绍利用基于物理的数值模型的训练数据集群；以及第 2.3 节中介绍结构健康识别所依据的 DL 模型。
2.1. Probabilistic graphical model for predictive digital twins
2.1.预测数字双胞胎的概率图模型
The digital twin assimilates vibration recordings shaped as multivariate time series
数字孪生系统将振动记录同化为多变量时间序列
数字孪生系统将振动记录同化为多变量时间序列
The PGM that defines the elements comprising the asset-twin coupled dynamical system, and mathematically describes the relevant interactions via observed data and control inputs, is the dynamic decision network sketched in Fig. 2. Circle nodes in the graph denote random variables at discrete times, square nodes denote actions, and diamond nodes denote the objective function. Bold outlines denote observed quantities, while thin outlines denote estimated quantities. The directed acyclic structure of the PGM encodes the assumed conditional dependencies. Edges in the graph represent dependencies between random variables. Solid edges represent the variables' dependencies encoded via conditional probability distributions, while dashed edges represent the variables' dependencies encoded via deterministic functions.
图 2 所示的动态决策网络定义了资产-孪生兄弟耦合动态系统的组成要素，并通过观测数据和控制输入对相关的相互作用进行了数学描述。图中的圆形节点表示离散时间的随机变量，方形节点表示行动，菱形节点表示目标函数。粗体轮廓表示观察到的数量，细体轮廓表示估计的数量。PGM 的有向无循环结构编码了假定的条件依赖关系。图中的边代表随机变量之间的依赖关系。实线表示通过条件概率分布编码的变量依赖关系，虚线表示通过确定性函数编码的变量依赖关系。
图 2 所示的动态决策网络定义了资产-孪生兄弟耦合动态系统的组成要素，并通过观测数据和控制输入对相关的相互作用进行了数学描述。图中的圆形节点表示离散时间的随机变量，方形节点表示行动，菱形节点表示目标函数。粗体轮廓表示观察到的数量，细体轮廓表示估计的数量。PGM 的有向无循环结构编码了假定的条件依赖关系。图中的边代表随机变量之间的依赖关系。实线表示通过条件概率分布编码的变量依赖关系，虚线表示通过确定性函数编码的变量依赖关系。
We consider a non-dimensional time discretization, and denote discrete time steps by
我们考虑非一维时间离散化，用
我们考虑非一维时间离散化，用
The physical state
物理状态
物理状态
The key assumptions behind our PGM are that the physical state is only observable indirectly via the sensed structural response, and the physical and digital states evolve according to a Markovian process. This implies that the conditional probabilities associated with the random variables at one time step depend only on the random variables at the previous time step, and are independent of all past states. The resulting graph topology encodes a conditional independence structure that allows us to conveniently cast the asset tracking within a sequential Bayesian inference framework. Indeed, by exploiting conditional independence and Bayes rule, the joint distributions over variables can be factorized up to the current time step
我们的 PGM 背后的关键假设是，物理状态只能通过感应到的结构响应间接观测到，物理状态和数字状态按照马尔可夫过程演化。这意味着与某一时间步的随机变量相关的条件概率仅取决于前一时间步的随机变量，而与过去的所有状态无关。由此产生的图拓扑结构编码了一种条件独立性结构，使我们能够方便地将资产跟踪置于顺序贝叶斯推理框架内。事实上，利用条件独立性和贝叶斯规则，变量的联合分布可以因式分解到当前时间步
我们的 PGM 背后的关键假设是，物理状态只能通过感应到的结构响应间接观测到，物理状态和数字状态按照马尔可夫过程演化。这意味着与某一时间步的随机变量相关的条件概率仅取决于前一时间步的随机变量，而与过去的所有状态无关。由此产生的图拓扑结构编码了一种条件独立性结构，使我们能够方便地将资产跟踪置于顺序贝叶斯推理框架内。事实上，利用条件独立性和贝叶斯规则，变量的联合分布可以因式分解到当前时间步
with factors: 因素：
The control policy
控制策略
控制策略
Fig. 3. Dynamic decision network employed to predict the future evolution of the digital state and the associated uncertainty. Circle nodes denote random variables, and diamond nodes denote the objective function. Directed solid edges represent the variables' dependencies encoded via conditional probability distributions, while directed dashed edges represent the dependencies encoded via deterministic functions
图 3.用于预测未来数字状态演变和相关不确定性的动态决策网络。圆圈节点表示随机变量，菱形节点表示目标函数。有向实线表示通过条件概率分布编码的变量依赖关系，有向虚线表示通过确定性函数编码的变量依赖关系。
图 3.用于预测未来数字状态演变和相关不确定性的动态决策网络。圆圈节点表示随机变量，菱形节点表示目标函数。有向实线表示通过条件概率分布编码的变量依赖关系，有向虚线表示通过确定性函数编码的变量依赖关系。
where
其中
其中
Herein,
在这里，
在这里，
Starting from the updated digital state
从当前时间步骤
从当前时间步骤
The algorithmic description of the online phase of the proposed digital twinning framework is reported in Algorithm 1. The operations repeat each time new observational data are provided. Note that the considered PGM digital twinning framework is general, and can easily be adapted to deal with physical assets other than civil engineering structures by reorganizing the topology of the graph, if necessary.
拟议数字孪生框架在线阶段的算法说明见算法 1。每次提供新的观测数据时，都会重复这些操作。需要注意的是，所考虑的 PGM 数字孪生框架是通用的，如有必要，可以通过重新组织图的拓扑结构，很容易地适应于处理土木工程结构以外的其他有形资产。
拟议数字孪生框架在线阶段的算法说明见算法 1。每次提供新的观测数据时，都会重复这些操作。需要注意的是，所考虑的 PGM 数字孪生框架是通用的，如有必要，可以通过重新组织图的拓扑结构，很容易地适应于处理土木工程结构以外的其他有形资产。
Algorithm 1 Online phase - algorithmic description Input: observational data \(O_{t}=o_{t}\) assimilate \(o_{t}\) with the DL models to provide \(D_{t}^{\mathrm{NN}}=d_{t}^{\mathrm{NN}}\). \(\quad \triangleright\left(O_{t}\right) \rightarrow\left(D_{t}^{\mathrm{NN}}\right)\) infer \(D_{t}\) and \(U_{t}\) by updating \(d_{t-1}\), given \(u_{t-1}^{A}, d_{t}^{\mathrm{NN}}\), and the CPTs encoding \(\phi_{t}^{\text {history }}, \phi_{t}^{\mathrm{NN}}\) and \(\phi_{t}^{\text {control }} \triangleright\left(D_{t-1}, D_{t}^{\mathrm{NN}}, U_{t-1}^{A},\right) \rightarrow\left(D_{t}, U_{t}\right)\) infer the future evolution of \(D_{t}\) and \(U_{t}\), given the updated \(d_{t}\), and the CPTs encoding \(\phi_{t}^{\text {history }}\) and \(\phi_{t}^{\text {control }} . \quad \triangleright\left(D_{t_{c}}\right) \rightarrow\left(D_{t_{p}}, U_{t_{p}}\right)\) select \(U_{t}^{A}=u_{t}^{A}\) as the best point estimate of \(U_{t}=u_{t} . \quad \triangleright\left(U_{t}\right) \rightarrow\left(U_{t}^{A}\right)\) return control input to be enacted \(u_{t}^{A}\), expected evolution of \(D_{t}\) and \(U_{t}\).
2.2. Numerical models for simulation-based damage identification
2.2.基于模拟的损伤识别数值模型
As anticipated in the previous section, the assimilation of structural response data to identify the structural state is carried out through DL models. A simulation-based strategy is exploited to train the DL models on the basis of vibration responses. The training data are numerically generated by simulating physics-based models so that the effect of damage on the structural response can be systematically reproduced [32]. In particular, the structure to be monitored is modeled as a linear-elastic continuum, discretized in space through finite elements. Its dynamic response to the applied loadings, under the assumption of linearized kinematics, is described by the following semi-discretized form of the elasto-dynamic problem:
如上一节所述，通过 DL 模型对结构响应数据进行同化以识别结构状态。在振动响应的基础上，采用基于模拟的策略来训练 DL 模型。训练数据是通过模拟物理模型数值生成的，因此可以系统地再现损伤对结构响应的影响[32]。具体而言，需要监测的结构被模拟为线性弹性连续体，并通过有限元在空间中离散化。在线性化运动学假设下，其对外加载荷的动态响应由以下弹性动力问题的半离散形式描述：
如上一节所述，通过 DL 模型对结构响应数据进行同化以识别结构状态。在振动响应的基础上，采用基于模拟的策略来训练 DL 模型。训练数据是通过模拟物理模型数值生成的，因此可以系统地再现损伤对结构响应的影响[32]。具体而言，需要监测的结构被模拟为线性弹性连续体，并通过有限元在空间中离散化。在线性化运动学假设下，其对外加载荷的动态响应由以下弹性动力问题的半离散形式描述：
which is referred to as the full-order model (FOM). Here
称为全阶模型（FOM）。
称为全阶模型（FOM）。
With reference to civil structures, we focus on the early detection of damage patterns characterized by a small evolution rate, whose prompt identification can reduce lifecycle costs and increase the safety and availability of the structure. In this context, a localized reduction of the material stiffness stands as the simplest damage mechanism resulting from a time scale separation between damage growth and damage assessment, see e.g., [34-36]. Here, local stiffness reduction is obtained by parametrizing the stiffness matrix via two variables
在民用结构方面，我们的重点是早期检测以小演化率为特征的损伤模式，对其进行及时识别可以降低生命周期成本，提高结构的安全性和可用性。在这种情况下，材料刚度的局部降低是最简单的损坏机制，其产生于损坏增长与损坏评估之间的时间尺度分离，参见 [34-36]。在这里，局部刚度减小是通过参数向量
在民用结构方面，我们的重点是早期检测以小演化率为特征的损伤模式，对其进行及时识别可以降低生命周期成本，提高结构的安全性和可用性。在这种情况下，材料刚度的局部降低是最简单的损坏机制，其产生于损坏增长与损坏评估之间的时间尺度分离，参见 [34-36]。在这里，局部刚度减小是通过参数向量
As
随着
随着
The ROM approximation to the solution of Problem (11) is obtained by linearly combining
通过线性组合
通过线性组合
The solution of this reduced-order system is advanced in time using the same strategy employed for the FOM model, and then projected onto the original FOM space as
采用与 FOM 模型相同的策略，在时间上推进这一简化系统的求解，然后投影到原 FOM 空间，即
采用与 FOM 模型相同的策略，在时间上推进这一简化系统的求解，然后投影到原 FOM 空间，即
The basis matrix
基矩阵
基矩阵
To populate the training dataset
为了填充训练数据集
为了填充训练数据集
where the vibration recordings
其中，与
其中，与
where
其中，
其中，
2.3. Data assimilation via artificial neural networks
2.3.通过人工神经网络进行数据同化
The
在我们的 PGM 中，
在我们的 PGM 中，
Every time new observational data
每次获取新的观测数据
每次获取新的观测数据
The aforementioned classification and regression tasks are addressed by means of DL models. The use of DL models for SHM purposes has the advantage of automating the feature engineering stage characterizing the pattern recognition paradigm for SHM [35,51]. Indeed, a DL model is trained to select and extract optimized damage-sensitive features from raw sensor recordings through an end-to-end learning process. Moreover, since the DL model is learned offline, the structural state can be next assessed in real-time regardless of considering continuous or discrete variables, which would be difficult to achieve with other optimization techniques, such as nonlinear programming, stochastic optimization, and metaheuristic methods.
上述分类和回归任务可通过 DL 模型来完成。将 DL 模型用于 SHM 目的具有将特征工程阶段自动化的优势，这也是 SHM 模式识别范例的特点 [35,51]。事实上，通过端到端的学习过程，DL 模型可以从原始传感器记录中选择和提取优化的损伤敏感特征。此外，由于 DL 模型是离线学习的，因此无论考虑连续变量还是离散变量，都可以对结构状态进行下一步实时评估，而这是其他优化技术（如非线性编程、随机优化和元搜索方法）难以实现的。
上述分类和回归任务可通过 DL 模型来完成。将 DL 模型用于 SHM 目的具有将特征工程阶段自动化的优势，这也是 SHM 模式识别范例的特点 [35,51]。事实上，通过端到端的学习过程，DL 模型可以从原始传感器记录中选择和提取优化的损伤敏感特征。此外，由于 DL 模型是离线学习的，因此无论考虑连续变量还是离散变量，都可以对结构状态进行下一步实时评估，而这是其他优化技术（如非线性编程、随机优化和元搜索方法）难以实现的。
The model
该模型
该模型
The estimated counterpart of
When
当利用
当利用
The model
Since the space of digital states in the PGM is discrete, the outcomes of
由于 PGM 中的数字状态空间是离散的，
由于 PGM 中的数字状态空间是离散的，
The models
模型
模型
The dataset used to learn
用于学习
用于学习
where
其中
其中
The set of weights and biases parametrizing
对
对
which provides a measure of the distance between the discrete probability distribution describing
它提供了描述
它提供了描述
The set of weights and biases
Fig. 4. L-shaped cantilever beam: details of synthetic recordings related to displacements
图 4.L 形悬臂梁：与位移
图 4.L 形悬臂梁：与位移
which provides a measure of the distance between the target magnitude of the stiffness reduction
它提供了刚度降低的目标幅度
它提供了刚度降低的目标幅度
The algorithmic description of the procedures and computations characterizing the preliminary offline phase of the proposed digital twinning framework is reported in Algorithm 2. The implementation details of the deep learning models are reported in the Appendix.
算法 2 报告了拟议数字孪生框架初步离线阶段的程序和计算的算法描述。深度学习模型的实现细节见附录。
算法 2 报告了拟议数字孪生框架初步离线阶段的程序和计算的算法描述。深度学习模型的实现细节见附录。
Algorithm 2 Preliminary offline phase - algorithmic description Input: parametrization of the operational and damage conditions PGM implementing the prediction graph set up the physics-based numerical model of the structure to be monitored. assemble the snapshot matrix of the structural response via FOM analyses. compute the POD basis functions via singular value decomposition of the snapshots matrix. use the ROM to populate the training dataset \(\mathcal{D}\) with vibration recordings at sensor location. use the recordings and labels in \(\mathcal{D}\) to derive \(\mathcal{D}_{\mathrm{CL}}\) and \(\mathcal{D}_{\mathrm{RG}}\). train the classification model \(\mathrm{NN}_{\mathrm{CL}}\) on \(\mathcal{D}_{\mathrm{CL}}\) and the regression models \(\mathrm{NN}_{\mathrm{RG}}\) on \(\mathcal{D}_{\mathrm{RG}}\). test the generalization capabilities of \(\mathrm{NN}_{\mathrm{CL}}\) and \(\mathrm{NN}_{\mathrm{RG}}\) on noisy FOM data. compute the confusion matrix encoding the \(\phi_{t}^{\mathrm{NN}}\) factor. compute the control policy \(\pi\left(D_{t}\right)\) by solving the planning problem induced by the PGM. return trained DL models, \(\phi_{t}^{\mathrm{NN}}\) factor, control policy \(\pi\left(D_{t}\right)\).
3. Numerical experiments
3.数值实验
This section demonstrates the proposed methodology for two test cases: an L-shaped cantilever beam and a railway bridge.
本节针对两个测试案例（L 型悬臂梁和铁路桥）演示了所提出的方法。
本节针对两个测试案例（L 型悬臂梁和铁路桥）演示了所提出的方法。
The FOM and ROM in Problem (11) and Problem (12) are implemented in the Matlab environment, using the redbKIT library [53]. The PGM framework for predictive digital twins is implemented in Python, using the pgmpy library [54]. All computations have been carried out on a PC featuring an AMD Ryzen
问题（11）和问题（12）中的 FOM 和 ROM 是在 Matlab 环境中使用 redbKIT 库[53]实现的。预测数字双胞胎的 PGM 框架在 Python 中使用 pgmpy 库[54]实现。所有计算均在一台配备 AMD Ryzen
问题（11）和问题（12）中的 FOM 和 ROM 是在 Matlab 环境中使用 redbKIT 库[53]实现的。预测数字双胞胎的 PGM 框架在 Python 中使用 pgmpy 库[54]实现。所有计算均在一台配备 AMD Ryzen
3.1. L-shaped cantilever beam
3.1.L 型悬臂梁
The first test case deals with the L-shaped cantilever beam depicted in Fig. 4. The structure is made of two arms, each one having a length of
第一个测试案例是图 4 所示的 L 形悬臂梁。该结构由两个臂组成，每个臂的长度为
第一个测试案例是图 4 所示的 L 形悬臂梁。该结构由两个臂组成，每个臂的长度为
Fig. 5. L-shaped cantilever beam - Confusion matrix measuring the offline performance of the DL models in correctly categorizing the digital state. Results are reported in terms of classification accuracy, measuring how observational data are classified with respect to the ground truth digital state. Digital states are ordered first for damage location and then for damage level.
图 5.L 形悬臂梁 - 衡量 DL 模型在正确分类数字状态方面离线性能的混淆矩阵。结果以分类准确率的形式进行报告，衡量观测数据如何根据基本真实数字状态进行分类。数字状态首先根据损坏位置排序，然后根据损坏程度排序。
图 5.L 形悬臂梁 - 衡量 DL 模型在正确分类数字状态方面离线性能的混淆矩阵。结果以分类准确率的形式进行报告，衡量观测数据如何根据基本真实数字状态进行分类。数字状态首先根据损坏位置排序，然后根据损坏程度排序。
3.1.1. Dataset assembly 3.1.1.数据集组装
Synthetic displacement time histories
合成位移时间历程
合成位移时间历程
In addition to the damage-free baseline condition, damage is simulated by considering
如图 4 所示，除了无损伤基线条件外，还通过考虑
如图 4 所示，除了无损伤基线条件外，还通过考虑
The FOM is obtained with a finite element discretization using linear tetrahedral elements and resulting in
FOM 是使用线性四面体元素进行有限元离散化得到的，结果是
FOM 是使用线性四面体元素进行有限元离散化得到的，结果是
The dataset
数据集
数据集
3.1.2. Digital twin framework
3.1.2.数字孪生框架
The two structural health parameters within the digital state are
数字状态下的两个结构健康参数是
数字状态下的两个结构健康参数是
The confusion matrix reported in Fig. 5 measures the offline performance of
图 5 中的混淆矩阵衡量了
图 5 中的混淆矩阵衡量了
For the present case, we consider four possible control inputs, each provided with a CPT modeling the transition probability
在本例中，我们考虑了四种可能的控制输入，每种输入都有一个 CPT，模拟了在采取
在本例中，我们考虑了四种可能的控制输入，每种输入都有一个 CPT，模拟了在采取
- Do nothing (DN) action. There is no maintenance action planned in this case and the physical state will evolve according to a stochastic deterioration process.
不采取任何行动（DN）。在这种情况下，没有计划进行维护，物理状态将按照随机劣化过程发展。 - Minor imperfect maintenance (MI) action. A maintenance action is performed and the asset may be restored from its current condition to a healthier damage state. This can be traced back to, e.g., patching and sealing cracked surfaces, rectifying and replacing expansion joints, or tightening/replacing loose/missing knot bolts for steel members.
轻微不完善维护 (MI) 行动。执行维护行动后，资产可能会从当前状态恢复到较健康的损坏状态。这可以追溯到诸如修补和密封开裂的表面、纠正和更换伸缩缝或拧紧/更换钢构件松动/缺失的连接螺栓等。 - Major imperfect maintenance (MA) action. A maintenance action is performed and the asset may be restored from its current condition to a healthier damage state, with a higher probability of improvements than in the previous case. This can be traced back to, e.g., repairing heavily damaged slabs, piers, and steel members, and retrofitting compromised structural elements.
主要不完善维护 (MA) 行动。执行维护行动后，资产可能会从目前的状态恢复到较健康的损坏状态，改善的概率高于前一种情况。这可以追溯到修复严重损坏的楼板、桥墩和钢构件，以及改造受损的结构部件等。 - Perfect maintenance (PM) action. A maintenance action is performed and the asset is restored from its current condition to the damage-free state. This can be traced back to the replacement of excessively compromised structural elements.
完美维护 (PM) 行动。执行维护行动后，资产将从当前状态恢复到无损坏状态。这可以追溯到更换过度损坏的结构元件。
3.1.3. Results: two available actions
3.1.3.结果：两个可用行动
We first illustrate the DT capabilities to assimilate observational data and track the structural health evolution, by restricting the available actions to DN and PM. The (unknown) ground truth evolution of structural health varies depending on the most recently applied control input, which can be either DN or PM. In the absence of maintenance, the physical state evolves following a deterioration process. We prescribe a (simulated) stochastic degradation process that monotonically deteriorates the structural health. The degradation process features a probability of damage inception
我们首先通过将可用操作限制为 DN 和 PM 来说明 DT 吸收观测数据和跟踪结构健康演变的能力。结构健康状况的（未知）基本真实演变取决于最近应用的控制输入，控制输入可以是 DN 或 PM。在没有维护的情况下，物理状态会随着恶化过程而变化。我们预设了一个（模拟的）随机退化过程，使结构健康状况单调恶化。退化过程的特点是损坏开始的概率
我们首先通过将可用操作限制为 DN 和 PM 来说明 DT 吸收观测数据和跟踪结构健康演变的能力。结构健康状况的（未知）基本真实演变取决于最近应用的控制输入，控制输入可以是 DN 或 PM。在没有维护的情况下，物理状态会随着恶化过程而变化。我们预设了一个（模拟的）随机退化过程，使结构健康状况单调恶化。退化过程的特点是损坏开始的概率
The state transition model encoding
编码
编码
At each time step, the DT selects a control input
在每个时间步长内，DT 都会选择对资产实施控制输入
在每个时间步长内，DT 都会选择对资产实施控制输入
where
其中
其中
During the offline phase, we solve the planning problem induced by the PGM to compute the control policy
在离线阶段，我们通过解决 PGM 诱导的规划问题来计算控制策略
在离线阶段，我们通过解决 PGM 诱导的规划问题来计算控制策略
Fig. 6 depicts a simulated online phase of the DT up to time step
图 6 描述了直至时间步长
图 6 描述了直至时间步长
Fig. 6. L-shaped cantilever beam - Online phase of the digital twin framework with two possible actions: DN (do nothing), and PM (perfect maintenance). Probabilistic and best point estimates of: (top) digital state evolution against the ground truth digital state; (bottom) control inputs informed by the digital twin, against the optimal control input under ground truth. In the top panels the background color corresponds to
图 6.L 型悬臂梁 - 数字孪生框架的在线阶段，有两种可能的操作：DN（什么都不做）和 PM（完美维护）。概率和最佳点估算：(上图）数字状态演变与地面数字状态的对比；（下图）数字孪生提供的控制输入与地面数字状态下的最佳控制输入的对比。在上图中，背景颜色对应于
图 6.L 型悬臂梁 - 数字孪生框架的在线阶段，有两种可能的操作：DN（什么都不做）和 PM（完美维护）。概率和最佳点估算：(上图）数字状态演变与地面数字状态的对比；（下图）数字孪生提供的控制输入与地面数字状态下的最佳控制输入的对比。在上图中，背景颜色对应于
Fig. 7. L-shaped cantilever beam - Digital twin future predictions with two possible actions: DN (do nothing), and PM (perfect maintenance). The starting time is
图 7.L 型悬臂梁--数字孪生未来预测，有两种可能的操作：DN（什么都不做）和 PM（完美维护）。起始时间为
图 7.L 型悬臂梁--数字孪生未来预测，有两种可能的操作：DN（什么都不做）和 PM（完美维护）。起始时间为
Fig. 7 depicts the predicted evolution of the digital state and of the corresponding informed control inputs, starting from
图 7 描述了从
图 7 描述了从
3.1.4. Results: four available actions
3.1.4.结果：四项可用行动
We now consider all four possible control inputs. We prescribe a stochastic degradation process with a probability of damage inception
现在我们考虑所有四种可能的控制输入。我们预设了一个随机降解过程，损坏开始的概率
现在我们考虑所有四种可能的控制输入。我们预设了一个随机降解过程，损坏开始的概率
Fig. 8. L-shaped cantilever beam - Online phase of the digital twin framework with four possible actions: DN (do nothing), PM (perfect maintenance), MI (minor imperfect maintenance), and MA (major imperfect maintenance). Probabilistic and best point estimates of: (top) digital state evolution against the ground truth digital state; (bottom) control inputs informed by the digital twin, against the optimal control input under ground truth. In the top panels the background color corresponds to
图 8.L 型悬臂梁--数字孪生框架的在线阶段，有四种可能的操作：DN（什么都不做）、PM（完美维护）、MI（轻微不完美维护）和 MA（重大不完美维护）。概率和最佳点估算：(上图）数字状态演变与基本真实数字状态的对比；（下图）数字孪生提供的控制输入与基本真实下的最优控制输入的对比。在上图中，背景颜色对应于
图 8.L 型悬臂梁--数字孪生框架的在线阶段，有四种可能的操作：DN（什么都不做）、PM（完美维护）、MI（轻微不完美维护）和 MA（重大不完美维护）。概率和最佳点估算：(上图）数字状态演变与基本真实数字状态的对比；（下图）数字孪生提供的控制输入与基本真实下的最优控制输入的对比。在上图中，背景颜色对应于
while the effect of a MA action is modeled with
而 MA 作用的影响是通过
而 MA 作用的影响是通过
The transition model
与 MI 操作相关的过渡模型
与 MI 操作相关的过渡模型
The two reward functions in Eq. (9) are chosen as:
公式 (9) 中的两个奖励函数选为
公式 (9) 中的两个奖励函数选为
We assume a discount factor
我们假设贴现率为
我们假设贴现率为
Fig. 9. L-shaped cantilever beam - Digital twin future predictions with four possible actions: DN (do nothing), PM (perfect maintenance), MI (minor imperfect maintenance), and MA (major imperfect maintenance). The starting time is
图 9.L 型悬臂梁--数字孪生未来预测，四种可能的操作：DN（什么都不做）、PM（完美维护）、MI（轻微不完美维护）和 MA（重大不完美维护）。起始时间为
图 9.L 型悬臂梁--数字孪生未来预测，四种可能的操作：DN（什么都不做）、PM（完美维护）、MI（轻微不完美维护）和 MA（重大不完美维护）。起始时间为
Fig. 8 depicts a simulated online phase of the DT up to
图 8 描述了 DT 的模拟在线阶段，最高可达
图 8 描述了 DT 的模拟在线阶段，最高可达
Fig. 9 depicts the predicted evolution of the digital state and control inputs, from
图 9 描述了从
图 9 描述了从
3.2. Railway bridge 3.2.铁路桥
The second case study concerns the railway bridge depicted in Fig. 10. It is an integral concrete portal frame bridge located along the Bothnia line in the Swedish suburbs of Hörnefors. It features a span of
第二个案例研究涉及图 10 所示的铁路桥。这是一座整体式混凝土门式框架桥，位于瑞典赫内福斯郊区的 Bothnia 线沿线。它的跨度为
第二个案例研究涉及图 10 所示的铁路桥。这是一座整体式混凝土门式框架桥，位于瑞典赫内福斯郊区的 Bothnia 线沿线。它的跨度为
The bridge is subjected to the transit of Gröna Tåget trains type, at a speed
该桥承受 Gröna Tåget 型列车的通过，速度为
该桥承受 Gröna Tåget 型列车的通过，速度为
3.2.1. Dataset assembly 3.2.1.数据集组装
Synthetic displacement time histories
合成位移时间历程
合成位移时间历程
In addition to the undamaged condition, the presence of damage in the structure is accounted for using a localized stiffness reduction that can take place within
如图 11 所示，除了未损坏状态外，结构中存在的损坏可通过在
如图 11 所示，除了未损坏状态外，结构中存在的损坏可通过在
Fig. 10. Hörnefors railway bridge.
图 10.赫内福斯铁路桥。
图 10.赫内福斯铁路桥。
Fig. 11. Railway bridge: details of synthetic recordings related to displacements
图 11.铁路桥：与位移相关的合成记录详情
图 11.铁路桥：与位移相关的合成记录详情
The FOM features
FOM 的特点是
FOM 的特点是
The ROM is obtained from a snapshot matrix S, assembled through 400 evaluations of the FOM for different values of parameters
ROM 由一个快照矩阵 S 得出，该快照矩阵是通过对不同参数值的 FOM 进行 400 次评估得出的
ROM 由一个快照矩阵 S 得出，该快照矩阵是通过对不同参数值的 FOM 进行 400 次评估得出的
The training dataset
训练数据集
训练数据集
3.2.2. Digital twin framework
3.2.2.数字孪生框架
As in the previous case, the two structural health parameters within the digital state are
与前一种情况一样，数字状态下的两个结构健康参数是
与前一种情况一样，数字状态下的两个结构健康参数是
The confusion matrix measuring the offline performance of
衡量
衡量
For the present case, we consider the following three possible control inputs:
在本例中，我们考虑了以下三种可能的控制输入：
在本例中，我们考虑了以下三种可能的控制输入：
- Do nothing (DN) action. There is no maintenance action planned in this case and the physical state will evolve according to a stochastic deterioration process.
不采取任何行动（DN）。在这种情况下，没有计划进行维护，物理状态将按照随机劣化过程发展。
Fig. 12. Railway bridge - Confusion matrix measuring the offline performance of the DL models in correctly categorizing the digital state. Results are reported in terms of classification accuracy, measuring how observational data are classified with respect to the ground truth digital state. Digital states are ordered first for damage location and then for damage level.
图 12.铁路桥梁 - 衡量 DL 模型在正确分类数字状态方面离线性能的混淆矩阵。结果以分类准确率的形式报告，衡量了观测数据相对于基本真实数字状态的分类情况。数字状态首先根据损坏位置排序，然后根据损坏程度排序。
图 12.铁路桥梁 - 衡量 DL 模型在正确分类数字状态方面离线性能的混淆矩阵。结果以分类准确率的形式报告，衡量了观测数据相对于基本真实数字状态的分类情况。数字状态首先根据损坏位置排序，然后根据损坏程度排序。
- Perfect maintenance (PM) action. A maintenance action is performed and the asset is restored from its current condition to the damage-free state.
完美维护 (PM) 行动。执行维护操作后，资产将从当前状态恢复到无损坏状态。 - Restrict operational conditions (RE) action. The operational conditions of the bridge are restricted by allowing only lightweight trains, carrying less than 18 ton per axle, to travel across the bridge. Such a restriction results in a lower deterioration rate, but also yields a lower revenue generated by the infrastructure.
限制运行条件（RE）行动。限制大桥的运营条件，只允许每轴载重量低于 18 吨的轻型列车通过大桥。这种限制降低了老化率，但也降低了基础设施的收益。
In the cases where the most recently issued control input is either DN or RE, the physical state undergoes a degradation process that monotonically deteriorates the structural health. When operational conditions are not restricted, we prescribe a stochastic degradation process featuring a probability of damage inception
在最近发布的控制输入为 DN 或 RE 的情况下，物理状态会经历一个退化过程，使结构健康状况单调恶化。在运行条件不受限制的情况下，我们规定了一个随机退化过程，其特点是损坏开始的概率
在最近发布的控制输入为 DN 或 RE 的情况下，物理状态会经历一个退化过程，使结构健康状况单调恶化。在运行条件不受限制的情况下，我们规定了一个随机退化过程，其特点是损坏开始的概率
The transition model
与 DN 操作相关的过渡模型
与 DN 操作相关的过渡模型
In this case, the two reward functions in Eq. (9) are chosen as:
在这种情况下，公式 (9) 中的两个奖励函数选为
在这种情况下，公式 (9) 中的两个奖励函数选为
where the last contribution in
其中
其中
3.2.3. Results 3.2.3.结果
During the offline phase, we solve the planning problem in Eq. (8) by assuming a discount factor
在离线阶段，我们通过假设贴现因子
在离线阶段，我们通过假设贴现因子
Fig. 13. Railway bridge - Online phase of the digital twin framework with three possible actions: DN (do nothing), PM (perfect maintenance), and RE (restrict operational conditions). Probabilistic and best point estimates of: (top) digital state evolution against the ground truth digital state; (bottom) control inputs informed by the digital twin, against the optimal control input under ground truth. In the top panels the background color corresponds to
图 13.铁路桥梁 - 数字孪生框架的在线阶段，有三种可能的操作：DN（什么都不做）、PM（完美维护）和 RE（限制运行条件）。概率和最佳点估算：(上图）数字状态演变与数字状态基本事实的对比；（下图）数字孪生提供的控制输入与基本事实下的最佳控制输入的对比。在上图中，背景颜色对应于
图 13.铁路桥梁 - 数字孪生框架的在线阶段，有三种可能的操作：DN（什么都不做）、PM（完美维护）和 RE（限制运行条件）。概率和最佳点估算：(上图）数字状态演变与数字状态基本事实的对比；（下图）数字孪生提供的控制输入与基本事实下的最佳控制输入的对比。在上图中，背景颜色对应于
Fig. 14. Railway bridge - Digital twin future predictions with three possible actions: DN (do nothing), PM (perfect maintenance), and RE (restrict operational conditions). The starting time is
图 14.铁路桥--数字孪生未来预测与三种可能的行动：DN（什么都不做）、PM（完美维护）和 RE（限制运行条件）。起始时间为
图 14.铁路桥--数字孪生未来预测与三种可能的行动：DN（什么都不做）、PM（完美维护）和 RE（限制运行条件）。起始时间为
Fig. 13 reports a sample simulation of the DT online phase up to time step
图 13 报告了 DT 在线阶段到时间步长
图 13 报告了 DT 在线阶段到时间步长
Fig. 14 reports the predicted evolution of the digital state and control inputs, from
图 14 报告了数字状态和控制输入的预测变化，从
truth value, the prediction is also too optimistic in terms of deterioration rate, which suggests the use of a more refined transition model.
在真值方面，预测的恶化率也过于乐观，这就需要使用一个更加完善的过渡模型。
图 14 报告了数字状态和控制输入的预测变化，从
truth value, the prediction is also too optimistic in terms of deterioration rate, which suggests the use of a more refined transition model.
在真值方面，预测的恶化率也过于乐观，这就需要使用一个更加完善的过渡模型。
4. Conclusions 4.结论
In this work we have proposed a predictive digital twin approach to the health monitoring, maintenance, and management planning of civil structures, to advance condition-based and predictive maintenance practices. The presented strategy relies upon a probabilistic graphical model inspired by [18]. This framework is used to encode the asset-twin coupled dynamical system, the relevant end-to-end information flow via observational data (physical to digital) and control inputs (digital to physical), and its evolution over time, all with quantified uncertainty. The assimilation of observational data is carried out with deep learning models, leveraging the capabilities of convolutional layers to automatically select and extract damage-sensitive features from raw vibration recordings. The structural health parameters comprising the digital state are used to capture the variability of the physical asset. They are continually updated in a sequential Bayesian inference fashion, according to control-dependent transition dynamics models describing how the structural health is expected to evolve. The updated digital state is eventually exploited to predict the future evolution of the physical system and the associated uncertainty. This enables predictive decision-making about maintenance and management actions.
在这项工作中，我们提出了一种预测性数字孪生方法，用于土建结构的健康监测、维护和管理规划，以推进基于状态的预测性维护实践。所提出的策略依赖于受文献 [18] 启发而开发的概率图形模型。该框架用于编码资产-双耦合动态系统、通过观测数据（物理到数字）和控制输入（数字到物理）进行的相关端到端信息流及其随时间的演变，所有这些都具有量化的不确定性。观测数据的同化采用深度学习模型，利用卷积层的功能，从原始振动记录中自动选择和提取损伤敏感特征。构成数字状态的结构健康参数用于捕捉物理资产的可变性。这些参数根据描述结构健康状况预期演变方式的控制相关过渡动力学模型，以贝叶斯推理的方式进行连续更新。更新后的数字状态最终被用来预测物理系统的未来演变和相关的不确定性。这样就能对维护和管理行动做出预测性决策。
在这项工作中，我们提出了一种预测性数字孪生方法，用于土建结构的健康监测、维护和管理规划，以推进基于状态的预测性维护实践。所提出的策略依赖于受文献 [18] 启发而开发的概率图形模型。该框架用于编码资产-双耦合动态系统、通过观测数据（物理到数字）和控制输入（数字到物理）进行的相关端到端信息流及其随时间的演变，所有这些都具有量化的不确定性。观测数据的同化采用深度学习模型，利用卷积层的功能，从原始振动记录中自动选择和提取损伤敏感特征。构成数字状态的结构健康参数用于捕捉物理资产的可变性。这些参数根据描述结构健康状况预期演变方式的控制相关过渡动力学模型，以贝叶斯推理的方式进行连续更新。更新后的数字状态最终被用来预测物理系统的未来演变和相关的不确定性。这样就能对维护和管理行动做出预测性决策。
The computational procedure takes advantage of a preliminary offline phase which involves: (i) using physics-based numerical models and reduced order modeling, to overcome the lack of experimental data for civil applications under varying damage and operational conditions while populating the datasets for training the deep learning models; (ii) learning the health-dependent control policy to be applied at each time step of the online phase, to map the belief over the digital state onto actions feeding back to the physical asset.
计算程序利用了初步离线阶段，包括：(i) 使用基于物理的数值模型和降序建模，以克服在不同损坏和运行条件下民用应用实验数据的缺乏，同时填充用于训练深度学习模型的数据集；(ii) 学习在线阶段每个时间步应用的健康控制策略，将对数字状态的信念映射到反馈到物理资产的行动上。
计算程序利用了初步离线阶段，包括：(i) 使用基于物理的数值模型和降序建模，以克服在不同损坏和运行条件下民用应用实验数据的缺乏，同时填充用于训练深度学习模型的数据集；(ii) 学习在线阶段每个时间步应用的健康控制策略，将对数字状态的信念映射到反馈到物理资产的行动上。
The proposed strategy has been assessed against the simulated monitoring of an L-shaped cantilever beam and a railway bridge. In the absence of experimental data, the tests have been carried out considering high-fidelity simulation data, corrupted with an additive Gaussian noise. The obtained results have proved the digital twin capabilities of accurately tracking the digital state evolution under varying operational conditions, with relatively low uncertainty. The framework is also able to promptly suggest the appropriate control input, within at most two time steps of when the (unknown) ground truth structural health demands it.
针对 L 型悬臂梁和铁路桥梁的模拟监测，对所提出的策略进行了评估。在缺乏实验数据的情况下，测试采用了高保真模拟数据，并加入了高斯噪声。获得的结果证明，数字孪生具有在不同运行条件下准确跟踪数字状态演变的能力，而且不确定性相对较低。当（未知）地面真实结构健康状况需要时，该框架还能在最多两个时间步骤内及时建议适当的控制输入。
针对 L 型悬臂梁和铁路桥梁的模拟监测，对所提出的策略进行了评估。在缺乏实验数据的情况下，测试采用了高保真模拟数据，并加入了高斯噪声。获得的结果证明，数字孪生具有在不同运行条件下准确跟踪数字状态演变的能力，而且不确定性相对较低。当（未知）地面真实结构健康状况需要时，该框架还能在最多两个时间步骤内及时建议适当的控制输入。
Although the capabilities of health-aware digital twins are showcased in the specific context of monitoring the structural integrity of civil structures to advance predictive maintenance practices, the applicability of the presented framework is general. Indeed, the proposed framework can be adapted for various types of structures and engineering systems by adjusting the components within the dynamic Bayesian network to align with the specific characteristics of the problem at hand. The solution to the inverse problem (if any) can be estimated by assimilating available observational data using methods other than deep neural networks, for instance through Markov chain Monte Carlo sampling algorithms. Similarly, the state transition models are closely tied to the employed parametrization of the digital state and the availability of historical data. The same applies to the available control inputs, which are likely to vary for different structures, such as those in mechanical or aerospace systems, and the method chosen for solving the associated planning problem. Additionally, the graph topology can be easily reorganized to adapt to situations where observational data are not acquired after issuing a control input, or when control inputs are issued with a different frequency than that governing the digital twin update.
虽然健康感知数字孪生的功能是在监测土木工程结构完整性以推进预测性维护实践的特定背景下展示的，但所提出的框架具有普遍适用性。事实上，通过调整动态贝叶斯网络中的组件，使其符合当前问题的具体特征，所提出的框架可适用于各种类型的结构和工程系统。逆问题的解决方案（如果有的话）可以通过使用深度神经网络以外的方法（例如马尔科夫链蒙特卡罗采样算法）吸收可用的观测数据来估算。同样，状态转换模型也与所采用的数字状态参数化和历史数据的可用性密切相关。这同样适用于可用的控制输入，不同的结构（如机械或航空航天系统中的结构）可能会有不同的控制输入，以及解决相关规划问题所选择的方法。此外，图形拓扑结构可以轻松重组，以适应在发出控制输入后未获取观测数据的情况，或发出控制输入的频率与数字孪生更新频率不同的情况。
虽然健康感知数字孪生的功能是在监测土木工程结构完整性以推进预测性维护实践的特定背景下展示的，但所提出的框架具有普遍适用性。事实上，通过调整动态贝叶斯网络中的组件，使其符合当前问题的具体特征，所提出的框架可适用于各种类型的结构和工程系统。逆问题的解决方案（如果有的话）可以通过使用深度神经网络以外的方法（例如马尔科夫链蒙特卡罗采样算法）吸收可用的观测数据来估算。同样，状态转换模型也与所采用的数字状态参数化和历史数据的可用性密切相关。这同样适用于可用的控制输入，不同的结构（如机械或航空航天系统中的结构）可能会有不同的控制输入，以及解决相关规划问题所选择的方法。此外，图形拓扑结构可以轻松重组，以适应在发出控制输入后未获取观测数据的情况，或发出控制输入的频率与数字孪生更新频率不同的情况。
Future research lines will investigate the ability of the digital twin to update the transition dynamics models by learning from previous data. As suggested by the railway bridge case study, this will allow for a more accurate prediction of the expected evolution of the digital state, thus enabling predictive decision-making better tailored to the monitored asset. Another aspect of interest concerns solving the planning problem induced by the probabilistic model using reinforcement learning algorithms, capable of taking into account a finite planning horizon representing the design lifetime of the asset.
未来的研究方向将研究数字孪生系统通过学习以前的数据来更新过渡动态模型的能力。正如铁路桥梁案例研究中提出的那样，这将允许对数字状态的预期演变进行更准确的预测，从而使预测性决策更适合受监控的资产。另一个值得关注的方面是利用强化学习算法解决概率模型引起的规划问题，该算法能够考虑到代表资产设计寿命的有限规划范围。
未来的研究方向将研究数字孪生系统通过学习以前的数据来更新过渡动态模型的能力。正如铁路桥梁案例研究中提出的那样，这将允许对数字状态的预期演变进行更准确的预测，从而使预测性决策更适合受监控的资产。另一个值得关注的方面是利用强化学习算法解决概率模型引起的规划问题，该算法能够考虑到代表资产设计寿命的有限规划范围。
Code availability 代码可用性
The implementation code used for the experiments presented in Section 3 is available in the public repository digitaltwin-SHM [28]. The code implements the proposed digital twin framework and can be used to generate the graphs of digital state estimation and prediction reported in this paper. The DL models trained according to the implementation details reported in the Appendix are also made available in the same repository.
第 3 节中介绍的实验所使用的实现代码可从公共存储库 digitaltwin-SHM [28] 中获取。该代码实现了所提出的数字孪生框架，可用于生成本文所报告的数字状态估计和预测图。根据附录中报告的实现细节训练的数字孪生模型也可在同一资源库中获取。
第 3 节中介绍的实验所使用的实现代码可从公共存储库 digitaltwin-SHM [28] 中获取。该代码实现了所提出的数字孪生框架，可用于生成本文所报告的数字状态估计和预测图。根据附录中报告的实现细节训练的数字孪生模型也可在同一资源库中获取。
Declaration of competing interest
利益冲突声明
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
作者声明，他们没有任何可能会影响本文所报告工作的已知经济利益或个人关系。
作者声明，他们没有任何可能会影响本文所报告工作的已知经济利益或个人关系。
Data availability 数据可用性
The observational data used to run the experiments presented in Section 3 are available in the public repository digitaltwin-SHM [28]. The Matlab library for finite element simulation and reduced-order modeling of partial differential equations employed to generate these data is available in the public repository Redbkit [53].
用于运行第 3 节中介绍的实验的观测数据可在公共资源库 digitaltwin-SHM [28] 中获取。用于生成这些数据的有限元模拟和偏微分方程降阶建模的 Matlab 库可从公共资源库 Redbkit [53] 中获取。
用于运行第 3 节中介绍的实验的观测数据可在公共资源库 digitaltwin-SHM [28] 中获取。用于生成这些数据的有限元模拟和偏微分方程降阶建模的 Matlab 库可从公共资源库 Redbkit [53] 中获取。
Acknowledgments 致谢
Matteo Torzoni acknowledges the financial support from the Politecnico di Milano, Italy through the interdisciplinary Ph.D. Grant "Physics-informed deep learning for structural health monitoring". Marco Tezzele and Karen E. Willcox acknowledge support from the "NASA University Leadership Initiative" under Cooperative Agreement 80NSSC21M0071. Andrea Manzoni acknowledges the project "Dipartimento di Eccellenza 2023-2027", funded by MUR, Italy, and the project "FAIR (Future Artificial Intelligence Research)", funded by the NextGenerationEU program within the PNRR-PE-AI scheme (M4C2, Investment 1.3, Line on Artificial Intelligence).
Matteo Torzoni 感谢意大利米兰理工大学通过跨学科博士论文 "结构健康监测的物理信息深度学习 "提供的资金支持。Marco Tezzele 和 Karen E. Willcox 感谢 "美国国家航空航天局大学领导力计划 "根据合作协议 80NSSC21M0071 提供的支持。Andrea Manzoni 感谢由意大利 MUR 资助的 "Dipartimento di Eccellenza 2023-2027 "项目，以及由 PNRR-PE-AI 计划内的 NextGenerationEU 项目（M4C2，投资 1.3，人工智能项目）资助的 "FAIR（未来人工智能研究）"项目。
Matteo Torzoni 感谢意大利米兰理工大学通过跨学科博士论文 "结构健康监测的物理信息深度学习 "提供的资金支持。Marco Tezzele 和 Karen E. Willcox 感谢 "美国国家航空航天局大学领导力计划 "根据合作协议 80NSSC21M0071 提供的支持。Andrea Manzoni 感谢由意大利 MUR 资助的 "Dipartimento di Eccellenza 2023-2027 "项目，以及由 PNRR-PE-AI 计划内的 NextGenerationEU 项目（M4C2，投资 1.3，人工智能项目）资助的 "FAIR（未来人工智能研究）"项目。
Appendix. Implementation details
附录。实施细节
In this Appendix, we discuss the implementation details of the DL models described in Section 2.3. The architectures, as well as the relevant hyperparameters and training options, have been chosen through a preliminary study, aimed at minimizing
在本附录中，我们将讨论第 2.3 节中描述的 DL 模型的实现细节。这些架构以及相关的超参数和训练选项都是通过初步研究选择的，目的是尽量减少
在本附录中，我们将讨论第 2.3 节中描述的 DL 模型的实现细节。这些架构以及相关的超参数和训练选项都是通过初步研究选择的，目的是尽量减少
In the present work,
在本研究中，
在本研究中，
Table A. 1 表 A. 1
| (a) | (b) | ||||
| Layer 层数 | Output shape 输出形状 | Activation 激活 | Input 输入 | ||
| 0 - Input 0 - 输入 |
|
None 无 | - | Convolution kernel size: 卷积核大小 |
|
| 1 - Conv1D |
|
Tanh | 0 | Dropout rate: 辍学率： |
|
| 2 - MaxPooling1D |
|
None 无 | 1 | Weight initializer: 重量初始化器： | Xavier 泽维尔 |
| 3 - Dropout 3 - 中途退学 |
|
None 无 | 2 |
|
|
| 4 - Conv1D |
|
Tanh | 3 | Optimizer: 优化器： | Adam 亚当 |
| 5 - MaxPooling1D |
|
None 无 | 4 | Batch size: 批量大小： |
|
| 6 - Dropout 6 - 中途退学 |
|
None 无 | 5 | Initial learning rate: 初始学习率： |
|
| 7 - Conv1D |
|
Tanh | 6 | Allowed epochs: 允许的年代： | 250 |
| 8 - MaxPooling1D |
|
None 无 | 7 | Learning schedule: 学习时间表： |
|
| 9 - Dropout 9 - 中途退学 |
|
None 无 | 8 | Weight decay: 重量衰减： | 0.05 |
| 10 - Flatten 10 - 压平 |
|
None 无 | 9 | Early stop patience: 早期停止的耐心： | 15 epochs 15 个纪元 |
| 11 - Dense 11 - 密集 |
|
Tanh | 10 | Train-val split: 火车与汽车分离： |
|
| 12 - Dense 12 - 密集 |
|
Tanh | 11 | ||
| 13 - Dense 13 - 密集 |
|
Softmax 软磁 | 12 | ||
Table A. 2 表 A. 2
| (a) | (b) | |||||
| Layer 层数 | Output shape 输出形状 | Activation 激活 | Input 输入 | |||
| 0 - Input 0 - 输入 |
|
None 无 | - | Convolution kernel size: 卷积核大小 |
|
|
| 1 - Conv1D |
|
Tanh | 0 | Dropout rate: 辍学率： |
|
|
| 2 - MaxPooling1D |
|
None 无 |
|
Weight initializer: 重量初始化器： | Xavier 泽维尔 | |
| 3 - Dropout 3 - 中途退学 |
|
None 无 | 2 |
|
|
|
| 4 - Conv1D |
|
Tanh | 3 | Optimizer: 优化器： | Adam 亚当 | |
| 5 - MaxPooling1D |
|
None 无 | 4 | Batch size: 批量大小： |
|
|
| 6 - Dropout 6 - 中途退学 |
|
None 无 | 5 | Initial learning rate: 初始学习率： |
|
|
| 7 - Conv1D |
|
Tanh | 6 | Allowed epochs: 允许的年代： | 250 | |
| 8 - MaxPooling1D |
|
None 无 | 7 | Learning schedule: 学习时间表： |
|
|
| 9 - Dropout 9 - 中途退学 |
|
None 无 | 8 | Weight decay: 重量衰减： | 0.05 | |
| 10 - Flatten 10 - 压平 |
|
None 无 | 9 | Early stop patience: 提前停止的耐心： | 15 epochs 15 个纪元 | |
| 11 - Dense 11 - 密集 |
|
Tanh | 10 | Train-val split: 火车与汽车分离： |
|
|
| 12 - Dense 12 - 密集 |
|
Tanh | 11 | |||
| 13 - Dense 13 - 密集 |
|
None 无 | 12 | |||
Using the Xavier's weight initialization [61],
利用 Xavier 权重初始化方法 [61]，
利用 Xavier 权重初始化方法 [61]，
where
其中
其中
References 参考资料
[1] A. Thelen, X. Zhang, O. Fink, Y. Lu, S. Ghosh, B.D. Youn, M.D. Todd, S. Mahadevan, C. Hu, Z. Hu, A comprehensive review of digital twin - part 1: modeling and twinning enabling technologies, Struct. Multidiscip. Optim. 65 (2022) 354, http://dx.doi.org/10.1007/s00158-022-03425-4.
[1] A. Thelen, X. Zhang, O. Fink, Y. Lu, S. Ghosh, B.D. Youn, M.D. Todd, S. Mahadevan, C. Hu, Z. Hu, A comprehensive review of digital twin - part 1: modeling and twinning enabling technologies, Struct.Multidiscip.Optim.65 (2022) 354, http://dx.doi.org/10.1007/s00158-022-03425-4.
[1] A. Thelen, X. Zhang, O. Fink, Y. Lu, S. Ghosh, B.D. Youn, M.D. Todd, S. Mahadevan, C. Hu, Z. Hu, A comprehensive review of digital twin - part 1: modeling and twinning enabling technologies, Struct.Multidiscip.Optim.65 (2022) 354, http://dx.doi.org/10.1007/s00158-022-03425-4.
[2] E.J. Tuegel, A.R. Ingraffea, T.G. Eason, S.M. Spottswood, Reengineering aircraft structural life prediction using a digital twin, Int. J. Aerosp. Eng. 2011 (2011) 154798, http://dx.doi.org/10.1155/2011/154798.
[2] E.J. Tuegel, A.R. Ingraffea, T.G. Eason, S.M. Spottswood, Reengineering aircraft structural life prediction using a digital twin, Int. J. Aerosp.J. Aerosp.2011 (2011) 154798, http://dx.doi.org/10.1155/2011/154798.
[2] E.J. Tuegel, A.R. Ingraffea, T.G. Eason, S.M. Spottswood, Reengineering aircraft structural life prediction using a digital twin, Int. J. Aerosp.J. Aerosp.2011 (2011) 154798, http://dx.doi.org/10.1155/2011/154798.
[3] E. Tuegel, The airframe digital twin: Some challenges to realization, in: 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2012, http://dx.doi.org/10.2514/6.2012-1812.
[3] E. Tuegel, The airframe digital twin: Some challenges to realization, in：53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2012, http://dx.doi.org/10.2514/6.2012-1812.
[3] E. Tuegel, The airframe digital twin: Some challenges to realization, in：53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2012, http://dx.doi.org/10.2514/6.2012-1812.
[4] M. Grieves, J. Vickers, Digital twin: Mitigating unpredictable, undesirable emergent behavior in complex systems, in: Transdisciplinary Perspectives on Complex Systems: New Findings and Approaches, Springer, 2017, pp. 85-113, http://dx.doi.org/10.1007/978-3-319-38756-7_4.
[4] M. Grieves, J. Vickers, Digital twin: Mitigating unpredictable, undesirable emergent behavior in complex systems, in：Transdisciplinary Perspectives on Complex Systems：新发现与新方法》，施普林格出版社，2017 年，第 85-113 页，http://dx.doi.org/10.1007/978-3-319-38756-7_4。
[4] M. Grieves, J. Vickers, Digital twin: Mitigating unpredictable, undesirable emergent behavior in complex systems, in：Transdisciplinary Perspectives on Complex Systems：新发现与新方法》，施普林格出版社，2017 年，第 85-113 页，http://dx.doi.org/10.1007/978-3-319-38756-7_4。
[5] A. Rasheed, O. San, T. Kvamsdal, Digital twin: Values, challenges and enablers from a modeling perspective, IEEE Access 8 (2020) 21980-22012, http://dx.doi.org/10.1109/ACCESS.2020.2970143.
[6] S.A. Niederer, M.S. Sacks, M. Girolami, K. Willcox, Scaling digital twins from the artisanal to the industrial, Nat. Comput. Sci. 1 (5) (2021) 313-320, http://dx.doi.org/10.1038/s43588-021-00072-5.
[6] S.A. Niederer, M.S. Sacks, M. Girolami, K. Willcox, Scaling digital twins from the artisanal to the industrial, Nat.Comput.1 (5) (2021) 313-320, http://dx.doi.org/10.1038/s43588-021-00072-5.
[6] S.A. Niederer, M.S. Sacks, M. Girolami, K. Willcox, Scaling digital twins from the artisanal to the industrial, Nat.Comput.1 (5) (2021) 313-320, http://dx.doi.org/10.1038/s43588-021-00072-5.
[7] Y. Ye, Q. Yang, F. Yang, Y. Huo, S. Meng, Digital twin for the structural health management of reusable spacecraft: a case study, Eng. Fract. Mech. 234 (2020) 107076, http://dx.doi.org/10.1016/j.engfracmech.2020.107076.
[7] Y. Ye, Q. Yang, F. Yang, Y. Huo, S. Meng, Digital twin for the structural health management of reusable spacecraft: a case study, Eng.Fract.Mech.234 (2020) 107076, http://dx.doi.org/10.1016/j.engfracmech.2020.107076.
[7] Y. Ye, Q. Yang, F. Yang, Y. Huo, S. Meng, Digital twin for the structural health management of reusable spacecraft: a case study, Eng.Fract.Mech.234 (2020) 107076, http://dx.doi.org/10.1016/j.engfracmech.2020.107076.
[8] M. Podskarbi, D.J. Knezevic, Digital twin for operations - present applications and future digital thread, in: OTC Offshore Technology Conference, Houston, Texas, 2020, http://dx.doi.org/10.4043/30553-MS.
[8] M. Podskarbi, D.J. Knezevic, Digital twin for operations - Present applications and future digital thread, in：OTC Offshore Technology Conference, Houston, Texas, 2020, http://dx.doi.org/10.4043/30553-MS.
[8] M. Podskarbi, D.J. Knezevic, Digital twin for operations - Present applications and future digital thread, in：OTC Offshore Technology Conference, Houston, Texas, 2020, http://dx.doi.org/10.4043/30553-MS.
[9] A. Phua, C. Davies, G. Delaney, A digital twin hierarchy for metal additive manufacturing, Comput. Ind. 140 (2022) 103667, http://dx.doi.org/10.1016/ j.compind.2022.103667.
[9] A. Phua, C. Davies, G. Delaney, A digital twin hierarchy for metal additive manufacturing, Comput.Ind.140 (2022) 103667, http://dx.doi.org/10.1016/ j.compind.2022.103667.
[9] A. Phua, C. Davies, G. Delaney, A digital twin hierarchy for metal additive manufacturing, Comput.Ind.140 (2022) 103667, http://dx.doi.org/10.1016/ j.compind.2022.103667.
[10] M. Jans-Singh, K. Leeming, R. Choudhary, M. Girolami, Digital twin of an urban-integrated hydroponic farm, Data-Cent. Eng. 1 (2020) 20, http:
[10] M. Jans-Singh、K. Leeming、R. Choudhary、M. Girolami，城市综合水培农场的数字孪生，Data-Cent.Eng.1 (2020) 20, http:
[10] M. Jans-Singh、K. Leeming、R. Choudhary、M. Girolami，城市综合水培农场的数字孪生，Data-Cent.Eng.1 (2020) 20, http:
[11] A. Tzachor, S. Sabri, C.E. Richards, A. Rajabifard, M. Acuto, Potential and limitations of digital twins to achieve the sustainable development goals, Nat. Sustain. 5 (10) (2022) 822-829, http://dx.doi.org/10.1038/s41893-022-00923-7.
[11] A. Tzachor, S. Sabri, C.E. Richards, A. Rajabifard, M. Acuto, Potential and limitations of digital twins to achieve the sustainable development goals, Nat.Sustain.5 (10) (2022) 822-829, http://dx.doi.org/10.1038/s41893-022-00923-7.
[11] A. Tzachor, S. Sabri, C.E. Richards, A. Rajabifard, M. Acuto, Potential and limitations of digital twins to achieve the sustainable development goals, Nat.Sustain.5 (10) (2022) 822-829, http://dx.doi.org/10.1038/s41893-022-00923-7.
[12] G. Arcieri, C. Hoelzl, O. Schwery, D. Straub, K.G. Papakonstantinou, E. Chatzi, Bridging POMDPs and Bayesian decision making for robust maintenance planning under model uncertainty: An application to railway systems, Reliab. Eng. Syst. Saf. (2023) 109496, http://dx.doi.org/10.1016/j.ress.2023.109496.
[12] G. Arcieri, C. Hoelzl, O. Schwery, D. Straub, K.G. Papakonstantinou, E. Chatzi, Bridging POMDPs and Bayesian decision making for robust maintenance planning under model uncertainty：An application to railway systems, Reliab.Eng.Syst.Saf.(2023) 109496, http://dx.doi.org/10.1016/j.ress.2023.109496.
[12] G. Arcieri, C. Hoelzl, O. Schwery, D. Straub, K.G. Papakonstantinou, E. Chatzi, Bridging POMDPs and Bayesian decision making for robust maintenance planning under model uncertainty：An application to railway systems, Reliab.Eng.Syst.Saf.(2023) 109496, http://dx.doi.org/10.1016/j.ress.2023.109496.
[13] M.G. Kapteyn, D.J. Knezevic, D. Huynh, M. Tran, K.E. Willcox, Data-driven physics-based digital twins via a library of component-based reduced-order models, Internat. J. Numer. Methods Engrg. 123 (13) (2020) 2986-3003, http://dx.doi.org/10.1002/nme.6423.
[13] M.G. Kapteyn, D.J. Knezevic, D. Huynh, M. Tran, K.E. Willcox, Data-driven physics-based digital twins via a library of component-based reduced-order models, Internat.J. Numer.Methods Engrg.123 (13) (2020) 2986-3003, http://dx.doi.org/10.1002/nme.6423.
[13] M.G. Kapteyn, D.J. Knezevic, D. Huynh, M. Tran, K.E. Willcox, Data-driven physics-based digital twins via a library of component-based reduced-order models, Internat.J. Numer.Methods Engrg.123 (13) (2020) 2986-3003, http://dx.doi.org/10.1002/nme.6423.
[14] B. Moya, A. Badías, I. Alfaro, F. Chinesta, E. Cueto, Digital twins that learn and correct themselves, Internat. J. Numer. Methods Engrg. 123 (13) (2022) 3034-3044, http://dx.doi.org/10.1002/nme.6535.
[14] B. Moya, A. Badías, I. Alfaro, F. Chinesta, E. Cueto, Digital twins that learn and correct themselves, Internat.J. Numer.Methods Engrg.123 (13) (2022) 3034-3044, http://dx.doi.org/10.1002/nme.6535.
[14] B. Moya, A. Badías, I. Alfaro, F. Chinesta, E. Cueto, Digital twins that learn and correct themselves, Internat.J. Numer.Methods Engrg.123 (13) (2022) 3034-3044, http://dx.doi.org/10.1002/nme.6535.
[15] W. Sisson, P. Karve, S. Mahadevan, Digital twin approach for component health-informed rotorcraft flight parameter optimization, AIAA J. 60 (3) (2022) 1923-1936, http://dx.doi.org/10.2514/1.J060770.
[16] B. Moya, I. Alfaro, D. Gonzalez, F. Chinesta, E. Cueto, Physically sound, self-learning digital twins for sloshing fluids, PLoS One 15 (6) (2020) e0234569, http://dx.doi.org/10.1371/journal.pone.0234569.
[17] K. Cherifi, V. Mehrmann, P. Schulze, Simulations in a digital twin of an electrical machine, 2022, http://dx.doi.org/10.48550/arXiv.2207.02171, arXiv preprint arXiv:2207.02171.
[18] M.G. Kapteyn, J.V. Pretorius, K.E. Willcox, A probabilistic graphical model foundation for enabling predictive digital twins at scale, Nat. Comput. Sci. 1 (5) (2021) 337-347, http://dx.doi.org/10.1038/s43588-021-00069-0.
[18] M.G. Kapteyn, J.V. Pretorius, K.E. Willcox, A probabilistic graphical model foundation for enabling predictive digital twins at scale, Nat.Comput.1 (5) (2021) 337-347, http://dx.doi.org/10.1038/s43588-021-00069-0.
[18] M.G. Kapteyn, J.V. Pretorius, K.E. Willcox, A probabilistic graphical model foundation for enabling predictive digital twins at scale, Nat.Comput.1 (5) (2021) 337-347, http://dx.doi.org/10.1038/s43588-021-00069-0.
[19] D. Koller, N. Friedman, Probabilistic Graphical Models: Principles and Techniques, MIT Press, Cambridge, MA, 2009.
[19] D. Koller, N. Friedman, Probabilistic Graphical Models：Principles and Techniques, MIT Press, Cambridge, MA, 2009.
[19] D. Koller, N. Friedman, Probabilistic Graphical Models：Principles and Techniques, MIT Press, Cambridge, MA, 2009.
[20] S.J. Russell, Artificial Intelligence: A Modern Approach, Pearson Education, London, UK, 2020.
[20] S.J. Russell，Artificial Intelligence：A Modern Approach, Pearson Education, London, UK, 2020.
[20] S.J. Russell，Artificial Intelligence：A Modern Approach, Pearson Education, London, UK, 2020.
[21] M. Torzoni, A. Manzoni, S. Mariani, Structural health monitoring of civil structures: a diagnostic framework powered by deep metric learning, Comput. Struct. 271 (2022) 106858, http://dx.doi.org/10.1016/j.compstruc.2022.106858.
[21] M. Torzoni, A. Manzoni, S. Mariani, Structural health monitoring of civil structures: a diagnostic framework powered by deep metric learning, Comput.Struct.271 (2022) 106858, http://dx.doi.org/10.1016/j.compstruc.2022.106858.
[21] M. Torzoni, A. Manzoni, S. Mariani, Structural health monitoring of civil structures: a diagnostic framework powered by deep metric learning, Comput.Struct.271 (2022) 106858, http://dx.doi.org/10.1016/j.compstruc.2022.106858.
[22] V. Giglioni, I. Venanzi, V. Poggioni, A. Milani, F. Ubertini, Autoencoders for unsupervised real-time bridge health assessment, Comput.-Aided Civ. Infrastruct. Eng. 38 (2023) 959-974, http://dx.doi.org/10.1111/mice.12943.
[22] V. Giglioni、I. Venanzi、V. Poggioni、A. Milani、F. Ubertini，Autoencoders for unsupervised real-time bridge health assessment，Comput.-Aided Civ. Infrastruct.Infrastruct.Eng.38 (2023) 959-974, http://dx.doi.org/10.1111/mice.12943.
[22] V. Giglioni、I. Venanzi、V. Poggioni、A. Milani、F. Ubertini，Autoencoders for unsupervised real-time bridge health assessment，Comput.-Aided Civ. Infrastruct.Infrastruct.Eng.38 (2023) 959-974, http://dx.doi.org/10.1111/mice.12943.
[23] M. Torzoni, A. Manzoni, S. Mariani, A multi-fidelity surrogate model for structural health monitoring exploiting model order reduction and artificial neural networks, Mech. Syst. Signal Process. 197 (2023) 110376, http://dx.doi.org/10.1016/j.ymssp.2023.110376.
[23] M. Torzoni, A. Manzoni, S. Mariani, A multi-fidelity surrogate model for structural health monitoring exploiting model order reduction and artificial neural networks, Mech.Syst.Signal Process.197 (2023) 110376, http://dx.doi.org/10.1016/j.ymssp.2023.110376.
[23] M. Torzoni, A. Manzoni, S. Mariani, A multi-fidelity surrogate model for structural health monitoring exploiting model order reduction and artificial neural networks, Mech.Syst.Signal Process.197 (2023) 110376, http://dx.doi.org/10.1016/j.ymssp.2023.110376.
[24] M. Torzoni, A. Manzoni, S. Mariani, A deep neural network, multi-fidelity surrogate model approach for Bayesian model updating in SHM, in: European Workshop on Structural Health Monitoring, Springer International Publishing, 2023, pp. 1076-1086, http://dx.doi.org/10.1007/978-3-031-07258-1_108.
[24] M. Torzoni, A. Manzoni, S. Mariani, A deep neural network, multi-fidelity surrogate model approach for Bayesian model updating in SHM, in：European Workshop on Structural Health Monitoring, Springer International Publishing, 2023, pp.
[24] M. Torzoni, A. Manzoni, S. Mariani, A deep neural network, multi-fidelity surrogate model approach for Bayesian model updating in SHM, in：European Workshop on Structural Health Monitoring, Springer International Publishing, 2023, pp.
[25] L. Rosafalco, M. Torzoni, A. Manzoni, S. Mariani, A. Corigliano, A self-adaptive hybrid model/data-driven approach to SHM based on model order reduction and deep learning, in: Structural Health Monitoring Based on Data Science Techniques, Springer International Publishing, 2022, pp. 165-184, http://dx.doi.org/10.1007/978-3-030-81716-9.
[25] L. Rosafalco, M. Torzoni, A. Manzoni, S. Mariani, A. Corigliano, A self-adaptive hybrid model/data-driven approach to SHM based on model order reduction and deep learning, in：基于数据科学技术的结构健康监测》，施普林格国际出版公司，2022 年，第 165-184 页，http://dx.doi.org/10.1007/978-3-030-81716-9。
[25] L. Rosafalco, M. Torzoni, A. Manzoni, S. Mariani, A. Corigliano, A self-adaptive hybrid model/data-driven approach to SHM based on model order reduction and deep learning, in：基于数据科学技术的结构健康监测》，施普林格国际出版公司，2022 年，第 165-184 页，http://dx.doi.org/10.1007/978-3-030-81716-9。
[26] S. Eftekhar Azam, S. Mariani, Online damage detection in structural systems via dynamic inverse analysis: A recursive Bayesian approach, Eng. Struct. 159 (2018) 28-45, http://dx.doi.org/10.1016/j.engstruct.2017.12.031.
[26] S. Eftekhar Azam、S. Mariani，《通过动态逆分析在线检测结构系统中的损坏》：A recursive Bayesian approach, Eng.Struct.159 (2018) 28-45, http://dx.doi.org/10.1016/j.engstruct.2017.12.031.
[26] S. Eftekhar Azam、S. Mariani，《通过动态逆分析在线检测结构系统中的损坏》：A recursive Bayesian approach, Eng.Struct.159 (2018) 28-45, http://dx.doi.org/10.1016/j.engstruct.2017.12.031.
[27] A. Quarteroni, A. Manzoni, F. Negri, Reduced Basis Methods for Partial Differential Equations: An Introduction, Vol. 92, Springer, 2015, http://dx.doi, org
[27] A. Quarteroni, A. Manzoni, F. Negri, Reduced Basis Methods for Partial Differential Equations：An Introduction, Vol. 92, Springer, 2015, http://dx.doi, org
[27] A. Quarteroni, A. Manzoni, F. Negri, Reduced Basis Methods for Partial Differential Equations：An Introduction, Vol. 92, Springer, 2015, http://dx.doi, org
[28] M. Torzoni, M. Tezzele, Digital-twin-SHM, 2023, https://github.com/mtezzele/digital-twin-SHM.
[29] G. Morcous, Performance prediction of bridge deck systems using Markov chains, J. Perform. Constr. Facil. 20 (2) (2006) 146-155, http://dx.doi.org/10. 1061/(ASCE)0887-3828(2006)20:2(146).
[29] G. Morcous, Performance prediction of bridge deck systems using Markov chains, J. Perform.Constr.Facil.20 (2) (2006) 146-155, http://dx.doi.org/10.1061/(ASCE)0887-3828(2006)20:2(146).
[29] G. Morcous, Performance prediction of bridge deck systems using Markov chains, J. Perform.Constr.Facil.20 (2) (2006) 146-155, http://dx.doi.org/10.1061/(ASCE)0887-3828(2006)20:2(146).
[30] P.D. Thompson, M.B. Johnson, Markovian bridge deterioration: developing models from historical data, Struct. Infrastr. Eng. 1 (1) (2005) 85-91, http://dx.doi.org/10.1080/15732470412331289332.
[30] P.D. Thompson, M.B. Johnson, Markovian bridge deterioration: developing models from historical data, Struct.Infrastr.Eng.1 (1) (2005) 85-91, http://dx.doi.org/10.1080/15732470412331289332.
[30] P.D. Thompson, M.B. Johnson, Markovian bridge deterioration: developing models from historical data, Struct.Infrastr.Eng.1 (1) (2005) 85-91, http://dx.doi.org/10.1080/15732470412331289332.
[31] R.S. Sutton, A.G. Barto, Reinforcement Learning: An Introduction, MIT Press, Cambridge, MA, 2018.
[31] R.S.萨顿、A.G.巴托：《强化学习》（Reinforcement Learning：导论》，麻省理工学院出版社，马萨诸塞州剑桥市，2018 年。
[31] R.S.萨顿、A.G.巴托：《强化学习》（Reinforcement Learning：导论》，麻省理工学院出版社，马萨诸塞州剑桥市，2018 年。
[32] T. Taddei, J.D. Penn, M. Yano, A.T. Patera, Simulation-based classification; a model-order-reduction approach for structural health monitoring, Arch. Comput. Methods Eng. 25 (2018) 23-45, http://dx.doi.org/10.1007/s11831-016-9185-0.
[32] T. Taddei, J.D. Penn, M. Yano, A.T. Patera, Simulation-based classification; a model-order-reduction approach for structural health monitoring, Arch.Comput.Methods Eng. 25 (2018) 23-45, http://dx.doi.org/10.1007/s11831-016-9185-0.
[32] T. Taddei, J.D. Penn, M. Yano, A.T. Patera, Simulation-based classification; a model-order-reduction approach for structural health monitoring, Arch.Comput.Methods Eng. 25 (2018) 23-45, http://dx.doi.org/10.1007/s11831-016-9185-0.
[33] N.M. Newmark, A method of computation for structural dynamics, J. Eng. Mech. 85 (3) (1959) 67-94, http://dx.doi.org/10.1061/JMCEA3.0000098.
[33] N.M. Newmark, A method of computation for structural dynamics, J. Eng. Mech.Mech.85 (3) (1959) 67-94, http://dx.doi.org/10.1061/JMCEA3.0000098.
[33] N.M. Newmark, A method of computation for structural dynamics, J. Eng. Mech.Mech.85 (3) (1959) 67-94, http://dx.doi.org/10.1061/JMCEA3.0000098.
[34] A. Teughels, J. Maeck, G. De Roeck, Damage assessment by FE model updating using damage functions, Comput. Struct. 80 (25) (2002) 1869-1879, http://dx.doi.org/10.1016/S0045-7949(02)00217-1.
[34] A. Teughels, J. Maeck, G. De Roeck, Damage assessment by FE model updating using damage functions, Comput.80 (25) (2002) 1869-1879, .80 (25) (2002) 1869-1879, http://dx.doi.org/10.1016/S0045-7949(02)00217-1.
[34] A. Teughels, J. Maeck, G. De Roeck, Damage assessment by FE model updating using damage functions, Comput.80 (25) (2002) 1869-1879, .80 (25) (2002) 1869-1879, http://dx.doi.org/10.1016/S0045-7949(02)00217-1.
[35] C. Farrar, K. Worden, Structural Health Monitoring: A Machine Learning Perspective, John Wiley & Sons, 2013, http://dx.doi.org/10.1002/9781118443118.
[35] C. Farrar, K. Worden, Structural Health Monitoring：A Machine Learning Perspective, John Wiley & Sons, 2013, http://dx.doi.org/10.1002/9781118443118.
[35] C. Farrar, K. Worden, Structural Health Monitoring：A Machine Learning Perspective, John Wiley & Sons, 2013, http://dx.doi.org/10.1002/9781118443118.
[36] A. Pandey, M. Biswas, Damage detection in structures using changes in flexibility, J. Sound Vib. 169 (1) (1994) 3-17, http://dx.doi.org/10.1006/jsvi. 1994.1002 .
[36] A. Pandey, M. Biswas, Damage detection in structures using changes in flexibility, J. Sound Vib.169 (1) (1994) 3-17, http://dx.doi.org/10.1006/jsvi.1994.1002 .
[36] A. Pandey, M. Biswas, Damage detection in structures using changes in flexibility, J. Sound Vib.169 (1) (1994) 3-17, http://dx.doi.org/10.1006/jsvi.1994.1002 .
[37] L. Rosafalco, A. Manzoni, S. Mariani, A. Corigliano, Fully convolutional networks for structural health monitoring through multivariate time series classification, Adv. Model. Simul. Eng. Sci. 7 (2020) 38, http://dx.doi.org/10.1186/s40323-020-00174-1.
[37] L. Rosafalco, A. Manzoni, S. Mariani, A. Corigliano, Fully convolutional networks for structural health monitoring through multivariate time series classification, Adv. Model.Simul.Simul.7 (2020) 38, http://dx.doi.org/10.1186/s40323-020-00174-1.
[37] L. Rosafalco, A. Manzoni, S. Mariani, A. Corigliano, Fully convolutional networks for structural health monitoring through multivariate time series classification, Adv. Model.Simul.Simul.7 (2020) 38, http://dx.doi.org/10.1186/s40323-020-00174-1.
[38] L. Rosafalco, M. Torzoni, A. Manzoni, S. Mariani, A. Corigliano, Online structural health monitoring by model order reduction and deep learning algorithms, Comput. Struct. 255 (2021) 106604, http://dx.doi.org/10.1016/j.compstruc.2021.106604.
[38] L. Rosafalco、M. Torzoni、A. Manzoni、S. Mariani、A. Corigliano，《通过模型阶次减少和深度学习算法进行在线结构健康监测》，Comput.Struct.255 (2021) 106604, http://dx.doi.org/10.1016/j.compstruc.2021.106604.
[38] L. Rosafalco、M. Torzoni、A. Manzoni、S. Mariani、A. Corigliano，《通过模型阶次减少和深度学习算法进行在线结构健康监测》，Comput.Struct.255 (2021) 106604, http://dx.doi.org/10.1016/j.compstruc.2021.106604.
[39] M. Torzoni, L. Rosafalco, A. Manzoni, S. Mariani, A. Corigliano, SHM under varying environmental conditions: an approach based on model order reduction and deep learning, Comput. Struct. 266 (2022) 106790, http://dx.doi.org/10.1016/j.compstruc.2022.106790.
[39] M. Torzoni, L. Rosafalco, A. Manzoni, S. Mariani, A. Corigliano, SHM under varying environmental conditions: an approach based on model order reduction and deep learning, Comput.Struct.266 (2022) 106790, http://dx.doi.org/10.1016/j.compstruc.2022.106790.
[39] M. Torzoni, L. Rosafalco, A. Manzoni, S. Mariani, A. Corigliano, SHM under varying environmental conditions: an approach based on model order reduction and deep learning, Comput.Struct.266 (2022) 106790, http://dx.doi.org/10.1016/j.compstruc.2022.106790.
[40] F. Chinesta, A. Huerta, G. Rozza, K. Willcox, Model reduction methods, in: E. Stein, R. de Borst, T.J.R. Hughes (Eds.), Encyclopedia of Computational Mechanics, second ed., John Wiley & Sons, 2017, pp. 1-36.
[40] F. Chinesta, A. Huerta, G. Rozza, K. Willcox, Model reduction methods, in：E. Stein, R. de Borst, T.J.R. Hughes (Eds.), Encyclopedia of Computational Mechanics, second ed., John Wiley & Sons, 2017, pp.
[40] F. Chinesta, A. Huerta, G. Rozza, K. Willcox, Model reduction methods, in：E. Stein, R. de Borst, T.J.R. Hughes (Eds.), Encyclopedia of Computational Mechanics, second ed., John Wiley & Sons, 2017, pp.
[41] P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Silveira (Eds.), Volume 1: System- and Data-Driven Methods and Algorithms, De Gruyter, Berlin, Boston, 2021, http://dx.doi.org/10.1515/9783110498967.
[42] G. Rozza, G. Stabile, F. Ballarin, in: G. Rozza, G. Stabile, F. Ballarin (Eds.), Advanced Reduced Order Methods and Applications in Computational Fluid Dynamics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2022, http://dx.doi.org/10.1137/1.9781611977257.
[42] G. Rozza, G. Stabile, F. Ballarin, in：G. Rozza, G. Stabile, F. Ballarin (Eds.), Advanced Reduced Order Methods and Applications in Computational Fluid Dynamics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2022, http://dx.doi.org/10.1137/1.9781611977257.
[42] G. Rozza, G. Stabile, F. Ballarin, in：G. Rozza, G. Stabile, F. Ballarin (Eds.), Advanced Reduced Order Methods and Applications in Computational Fluid Dynamics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2022, http://dx.doi.org/10.1137/1.9781611977257.
[43] G. Kerschen, J.C. Golinval, Physical interpretation of the proper orthogonal modes using the singular value decomposition, J. Sound Vib. 249 (5) (2002) 849-865, http://dx.doi.org/10.1006/jsvi.2001.3930.
[43] G. Kerschen, J.C. Golinval, Physical interpretation of the proper orthogonal modes using the singular value decomposition, J. Sound Vib.249 (5) (2002) 849-865, http://dx.doi.org/10.1006/jsvi.2001.3930.
[43] G. Kerschen, J.C. Golinval, Physical interpretation of the proper orthogonal modes using the singular value decomposition, J. Sound Vib.249 (5) (2002) 849-865, http://dx.doi.org/10.1006/jsvi.2001.3930.
[44] G. Kerschen, J.C. Golinval, A. Vakakis, L.A. Bergman, The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An overview, Nonlinear Dynam. 41 (1) (2005) 147-169, http://dx.doi.org/10.1007/s11071-005-2803-2.
[44] G. Kerschen, J.C. Golinval, A. Vakakis, L.A. Bergman, The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems. An overview, Nonlinear Dynam：An overview, Nonlinear Dynam.41 (1) (2005) 147-169, http://dx.doi.org/10.1007/s11071-005-2803-2.
[44] G. Kerschen, J.C. Golinval, A. Vakakis, L.A. Bergman, The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems. An overview, Nonlinear Dynam：An overview, Nonlinear Dynam.41 (1) (2005) 147-169, http://dx.doi.org/10.1007/s11071-005-2803-2.
[45] M. Guo, J.S. Hesthaven, Reduced order modeling for nonlinear structural analysis using Gaussian process regression, Comput. Methods Appl. Mech. Engrg. 341 (2018) 807-826, http://dx.doi.org/10.1016/j.cma.2018.07.017.
[45] M. Guo, J.S. Hesthaven, Reduced order modeling for nonlinear structural analysis using Gaussian process regression, Comput.Methods Appl.Engrg.341 (2018) 807-826, http://dx.doi.org/10.1016/j.cma.2018.07.017.
[45] M. Guo, J.S. Hesthaven, Reduced order modeling for nonlinear structural analysis using Gaussian process regression, Comput.Methods Appl.Engrg.341 (2018) 807-826, http://dx.doi.org/10.1016/j.cma.2018.07.017.
[46] M. Tezzele, L. Fabris, M. Sidari, M. Sicchiero, G. Rozza, A multi-fidelity approach coupling parameter space reduction and non-intrusive POD with application to structural optimization of passenger ship hulls, Internat. J. Numer. Methods Engrg. 124 (5) (2023) 1193-1210, http://dx.doi.org/10.1002/ nme. 7159 .
[46] M. Tezzele, L. Fabris, M. Sidari, M. Sicchiero, G. Rozza, A multi-fidelity approach coupling parameter space reduction and non-intrusive POD with application to structural optimization of passenger ship hulls, Internat.J. Numer.Methods Engrg.124 (5) (2023) 1193-1210, http://dx.doi.org/10.1002/ nme.7159 .
[46] M. Tezzele, L. Fabris, M. Sidari, M. Sicchiero, G. Rozza, A multi-fidelity approach coupling parameter space reduction and non-intrusive POD with application to structural optimization of passenger ship hulls, Internat.J. Numer.Methods Engrg.124 (5) (2023) 1193-1210, http://dx.doi.org/10.1002/ nme.7159 .
[47] M. Sieber, C.O. Paschereit, K. Oberleithner, Spectral proper orthogonal decomposition, J. Fluid Mech. 792 (2016) 798-828, http://dx.doi.org/10.1017/ jfm.2016.103.
[47] M. Sieber, C.O. Paschereit, K. Oberleithner, Spectral proper orthogonal decomposition, J. Fluid Mech.792 (2016) 798-828, http://dx.doi.org/10.1017/ jfm.2016.103.
[47] M. Sieber, C.O. Paschereit, K. Oberleithner, Spectral proper orthogonal decomposition, J. Fluid Mech.792 (2016) 798-828, http://dx.doi.org/10.1017/ jfm.2016.103.
[48] A. Towne, O.T. Schmidt, T. Colonius, Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis, J. Fluid Mech. 847 (2018) 821-867, http://dx.doi.org/10.1017/jfm.2018.283.
[48] A. Towne, O.T. Schmidt, T. Colonius, Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis, J. Fluid Mech.847 (2018) 821-867, http://dx.doi.org/10.1017/jfm.2018.283.
[48] A. Towne, O.T. Schmidt, T. Colonius, Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis, J. Fluid Mech.847 (2018) 821-867, http://dx.doi.org/10.1017/jfm.2018.283.
[49] A. Lario, R. Maulik, O.T. Schmidt, G. Rozza, G. Mengaldo, Neural-network learning of SPOD latent dynamics, J. Comput. Phys. 468 (2022) 111475, http://dx.doi.org/10.1016/j.jcp.2022.111475.
[49] A. Lario、R. Maulik、O.T. Schmidt、G. Rozza、G. Mengaldo，SPOD 潜在动力学的神经网络学习，J. Comput.468 (2022) 111475, http://dx.doi.org/10.1016/j.jcp.2022.111475.
[49] A. Lario、R. Maulik、O.T. Schmidt、G. Rozza、G. Mengaldo，SPOD 潜在动力学的神经网络学习，J. Comput.468 (2022) 111475, http://dx.doi.org/10.1016/j.jcp.2022.111475.
[50] K.R. dos Santos, D.G. Giovanis, K. Kontolati, D. Loukrezis, M.D. Shields, Grassmannian diffusion maps based surrogate modeling via geometric harmonics, Internat. J. Numer. Methods Engrg. 123 (15) (2022) 3507-3529, http://dx.doi.org/10.1002/nme.6977.
[50] K.R. dos Santos, D.G. Giovanis, K. Kontolati, D. Loukrezis, M.D. Shields, Grassmannian diffusion maps based surrogate modeling via geometric harmonics, Internat.J. Numer.Methods Engrg.123 (15) (2022) 3507-3529, http://dx.doi.org/10.1002/nme.6977.
[50] K.R. dos Santos, D.G. Giovanis, K. Kontolati, D. Loukrezis, M.D. Shields, Grassmannian diffusion maps based surrogate modeling via geometric harmonics, Internat.J. Numer.Methods Engrg.123 (15) (2022) 3507-3529, http://dx.doi.org/10.1002/nme.6977.
[51] C.M. Bishop, Pattern Recognition and Machine Learning, in: Information Science and Statistics, Springer, New York, NY, 2006.
[51] C.M. Bishop, Pattern Recognition and Machine Learning, in：信息科学与统计学》，施普林格出版社，纽约州纽约市，2006 年。
[51] C.M. Bishop, Pattern Recognition and Machine Learning, in：信息科学与统计学》，施普林格出版社，纽约州纽约市，2006 年。
[52] I. Goodfellow, Y. Bengio, A. Courville, Deep Learning, MIT Press, Cambridge, MA, 2016.
[53] F. Negri, redbKIT, version 2.2, 2016, http://redbkit.github.io/redbKIT.
[54] A. Ankan, A. Panda, pgmpy: Probabilistic graphical models using python, in: Proceedings of the 14th Python in Science Conference (SCIPY 2015), Citeseer, 2015.
[54] A. Ankan, A. Panda, pgmpy：Probabilistic graphical models using python, in：Proceedings of the 14th Python in Science Conference (SCIPY 2015), Citeseer, 2015.
[54] A. Ankan, A. Panda, pgmpy：Probabilistic graphical models using python, in：Proceedings of the 14th Python in Science Conference (SCIPY 2015), Citeseer, 2015.
[55] F. Chollet, et al., Keras, 2015, https://keras.io.
[56] K. Papakonstantinou, M. Shinozuka, Planning structural inspection and maintenance policies via dynamic programming and Markov processes. Part II: POMDP implementation, Reliab. Eng. Syst. Saf. 130 (2014) 214-224, http://dx.doi.org/10.1016/j.ress.2014.04.006.
[56] K. Papakonstantinou, M. Shinozuka, Planning structural inspection and maintenance policies via dynamic programming and Markov processes.Part II: POMDP implementation, Reliab.Eng.Syst.Saf.130 (2014) 214-224, http://dx.doi.org/10.1016/j.ress.2014.04.006.
[56] K. Papakonstantinou, M. Shinozuka, Planning structural inspection and maintenance policies via dynamic programming and Markov processes.Part II: POMDP implementation, Reliab.Eng.Syst.Saf.130 (2014) 214-224, http://dx.doi.org/10.1016/j.ress.2014.04.006.
[57] M. Ülker-Kaustell, Some Aspects of the Dynamic Soil-Structure Interaction of a Portal Frame Railway Bridge (Ph.D. thesis), KTH Royal Institute of Technology, 2009.
[57] M. Ülker-Kaustell，Some Aspects of the Dynamic Soil-Structure Interaction of a Portal Frame Railway Bridge（博士论文），KTH Royal Institute of Technology，2009.
[57] M. Ülker-Kaustell，Some Aspects of the Dynamic Soil-Structure Interaction of a Portal Frame Railway Bridge（博士论文），KTH Royal Institute of Technology，2009.
[58] T. Arvidsson, J. Li, Dynamic Analysis of a Portal Frame Railway Bridge Using Frequency Dependent Soil Structure Interaction (Ph.D. thesis), KTH Royal Institute of Technology, 2011.
[58] T. Arvidsson，J. Li，《利用频率相关土体结构相互作用对门式刚架铁路桥进行动态分析》（博士论文），KTH 皇家理工学院，2011 年。
[58] T. Arvidsson，J. Li，《利用频率相关土体结构相互作用对门式刚架铁路桥进行动态分析》（博士论文），KTH 皇家理工学院，2011 年。
[59] European Committee for Standardization, Part 2: Traffic loads on bridges, in: EN 1991-2 Eurocode 1: Actions on Structures, 2003, pp. 66-74.
[59] 欧洲标准化委员会，《第 2 部分：桥梁上的交通荷载》，EN 1991-2 Eurocode 1： Actions on Structures，2003 年，第 66-74 页：EN 1991-2 Eurocode 1: Actions on Structures, 2003, pp.
[59] 欧洲标准化委员会，《第 2 部分：桥梁上的交通荷载》，EN 1991-2 Eurocode 1： Actions on Structures，2003 年，第 66-74 页：EN 1991-2 Eurocode 1: Actions on Structures, 2003, pp.
[60] P. Battaglia, J.B.C. Hamrick, V. Bapst, A. Sanchez, V. Zambaldi, M. Malinowski, A. Tacchetti, D. Raposo, A. Santoro, R. Faulkner, C. Gulcehre, F. Song, A. Ballard, J. Gilmer, G.E. Dahl, A. Vaswani, K. Allen, C. Nash, V.J. Langston, C. Dyer, N. Heess, D. Wierstra, P. Kohli, M. Botvinick, O. Vinyals, Y. Li, R. Pascanu, Relational inductive biases, deep learning, and graph networks, 2018, http://dx.doi.org/10.48550/arXiv.1806.01261, arXiv preprint arXiv:1806.01261.
[60] P. Battaglia, J.B.C. Hamrick, V. Bapst, A. Sanchez, V. Zambaldi, M. Malinowski, A. Tacchetti, D. Raposo, A. Santoro, R. Faulkner, C. Gulcehre, F. Song, A. Ballard, J. Gilmer, G.E. Dahl, A. Vaswani, K. Allen, C. Nash, V.J. Langston, C. Dyer, N. Heess, D. Wierstra, P. Kohli, M. Botvinick, O. Vinyals, Y. Li, R. Wierstra.Nash, V.J. Langston, C. Dyer, N. Heess, D. Wierstra, P. Kohli, M. Botvinick, O. Vinyals, Y. Li, R. Pascanu, Relational inductive biases, deep learning, and graph networks, 2018, http://dx.doi.org/10.48550/arXiv.1806.01261, arXiv preprint arXiv:1806.01261.
[60] P. Battaglia, J.B.C. Hamrick, V. Bapst, A. Sanchez, V. Zambaldi, M. Malinowski, A. Tacchetti, D. Raposo, A. Santoro, R. Faulkner, C. Gulcehre, F. Song, A. Ballard, J. Gilmer, G.E. Dahl, A. Vaswani, K. Allen, C. Nash, V.J. Langston, C. Dyer, N. Heess, D. Wierstra, P. Kohli, M. Botvinick, O. Vinyals, Y. Li, R. Wierstra.Nash, V.J. Langston, C. Dyer, N. Heess, D. Wierstra, P. Kohli, M. Botvinick, O. Vinyals, Y. Li, R. Pascanu, Relational inductive biases, deep learning, and graph networks, 2018, http://dx.doi.org/10.48550/arXiv.1806.01261, arXiv preprint arXiv:1806.01261.
[61] X. Glorot, Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, J. Mach. Learn. Res. 9 (2010) 249-256.
[61] X. Glorot, Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, J. Mach.Learn.9 (2010) 249-256.
[61] X. Glorot, Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, J. Mach.Learn.9 (2010) 249-256.
[62] D. Kingma, J. Ba, Adam: A method for stochastic optimization, in: 3rd International Conference on Learning Representations, Vol. 3, 2015, pp. 1-13.
[62] D. Kingma, J. Ba, Adam: A method for stochastic optimization, in：3rd International Conference on Learning Representations, Vol. 3, 2015, pp.
[62] D. Kingma, J. Ba, Adam: A method for stochastic optimization, in：3rd International Conference on Learning Representations, Vol. 3, 2015, pp.
- Corresponding author 通讯作者
E-mail addresses: matteo.torzoni@polimi.it (M. Torzoni), marco.tezzele@austin.utexas.edu (M. Tezzele), stefano.mariani@polimi.it (S. Mariani), andrea1.manzoni@polimi.it (A. Manzoni), kwillcox@oden.utexas.edu (K.E. Willcox).
电子邮件地址：matteo.torzoni@polimi.it (M. Torzoni)、marco.tezzele@austin.utexas.edu (M. Tezzele)、stefano.mariani@polimi.it (S. Mariani)、andrea1.manzoni@polimi.it (A. Manzoni)、kwillcox@oden.utexas.edu (K.E. Willcox)。Received 2 August 2023; Received in revised form 30 October 2023; Accepted 31 October 2023
2023 年 8 月 2 日收到；2023 年 10 月 30 日收到修订稿；2023 年 10 月 31 日接受Available online 8 November 2023
可于 2023 年 11 月 8 日在线查阅0045-7825/© 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
0045-7825/© 2023 The Author(s).出版商：Elsevier B.V.本文为 CC BY 许可下的开放存取文章 ( http://creativecommons.org/licenses/by/4.0/)。