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Structural damage identification via physics-guided machine learning: a methodology integrating pattern recognition with finite element model updating
通过物理引导的机器学习识别结构损伤：模式识别与有限元模型更新相结合的方法学

Zhiming Zhang and Chao Sun
张志明和孙超

Abstract 摘要

Structural health monitoring methods are broadly classified into two categories: data-driven methods via statistical pattern recognition and physics-based methods through finite elementmodel updating.
结构健康监测方法大致分为两类：通过统计模式识别的数据驱动方法和通过有限元模型更新的物理方法。
Data-driven structural health monitoring faces the challenge of data insufficiency that renders the learned model limited in identifying damage scenarios that are not contained in the training data.
数据驱动的结构健康监测面临着数据不足的挑战，这使得所学模型在识别训练数据中未包含的损坏情况时受到限制。
Model-based methods are susceptible to modeling error due to model idealizations and simplifications that make the finite element model updating results deviate from the truth.
基于模型的方法容易因模型理想化和简化而产生建模误差，从而使有限元模型更新结果偏离事实。
This study attempts to combine the merits of data-driven and physics-based structural health monitoring methods via physicsguided machine learning, expecting that the damage identification performance can be improved.
本研究试图通过物理引导的机器学习，将数据驱动和基于物理的结构健康监测方法的优点结合起来，期望能提高损伤识别性能。
Physics-guided machine learning uses observed feature data with correct labels as well as the physical model output of unlabeled instances. In this study, physics-guided machine learning is realized with a physics-guided neural network.
物理引导机器学习使用带有正确标签的观测特征数据以及未标签实例的物理模型输出。在本研究中，物理引导机器学习是通过物理引导神经网络实现的。
The original modal-property based features are extended with the damage identification result of finite element model updating.
通过更新有限元模型的损伤识别结果，扩展了原有的基于模态属性的特征。
A physics-based loss function is designed to evaluate the discrepancy between the neural network model output and that of finite element model updating.
设计了一个基于物理学的损失函数，用于评估神经网络模型输出与有限元模型更新输出之间的差异。
With the guidance from the scientific knowledge contained in finite element model updating, the learned neural network model has the potential to improve the generality and scientific consistency of the damage detection results.
在有限元模型更新所包含的科学知识的指导下，学习的神经网络模型有可能提高损伤检测结果的通用性和科学一致性。
The proposed methodology is validated by a numerical case study on a steel pedestrian bridge model and an experimental study on a three-story building model.
通过对钢制人行天桥模型的数值案例研究和对三层建筑模型的实验研究，对所提出的方法进行了验证。

Keywords 关键词

Structural health monitoring, damage detection, pattern recognition, model updating, data insufficiency, modeling error, physics-guided learning
结构健康监测、损伤检测、模式识别、模型更新、数据不足、建模误差、物理引导学习

Introduction 导言

Structural health monitoring (SHM) approaches mainly fall into two categories: data-driven and physics-based approaches.
结构健康监测（SHM）方法主要分为两类：数据驱动和基于物理的方法。
Data-driven approaches detect damage occurrence and identifies its location and severity through pattern recognition and machine learning methods using damage-sensitive features extracted from collected structural responses. 1 1 ^(1){ }^{1} Compared with physicsbased methods of SHM, a data-driven approach avoids building and validating a numerical model 2 , 3 2 , 3 ^(2,3){ }^{2,3} and has the potential of identifying structural damage despite the operational and environmental influence such as traffic loading across bridges, temperature variations, and wind and moisture effects. 4 , 5 4 , 5 ^(4,5){ }^{4,5} Moreover, it automatically
数据驱动方法通过模式识别和机器学习方法，利用从收集的结构响应中提取的损伤敏感特征，检测损伤的发生并确定其位置和严重程度。 1 1 ^(1){ }^{1} 与基于物理的 SHM 方法相比，数据驱动方法避免了建立和验证数值模型 2 , 3 2 , 3 ^(2,3){ }^{2,3} ，并有可能识别结构损伤，而不受运营和环境的影响，如跨桥交通荷载、温度变化、风力和湿度影响。 4 , 5 4 , 5 ^(4,5){ }^{4,5} 此外，它还能自动
accommodates the uncertainty that originates from measuring variability. 3 3 ^(3){ }^{3} While data-driven methods have aforementioned advantages over physics-based methods, a big challenge of data-driven SHM is the availability of sufficient training data with correct labels for learning a statistical model with satisfactory accuracy and generality.
数据驱动的 SHM 方法可适应测量变异性带来的不确定性。 3 3 ^(3){ }^{3} 虽然数据驱动方法与基于物理的方法相比具有上述优势，但数据驱动 SHM 面临的一大挑战是如何获得足够的具有正确标签的训练数据，以学习具有令人满意的准确性和通用性的统计模型。
Specifically, damage localization in data-driven
具体来说，数据驱动的损伤定位
SHM is a supervised learning problem setting the potential damage locations as the target class labels of a machine learning classifier. 6 6 ^(6){ }^{6} This learning process requires training data from both undamaged and damaged conditions.
SHM 是一个有监督的学习问题，它将潜在的损坏位置设置为机器学习分类器的目标类标签。 6 6 ^(6){ }^{6} 这一学习过程需要未损坏和损坏情况下的训练数据。
However, such data especially that of the damaged cases will always be lacked for large and valuable structures, for example., long span bridges and offshore wind turbines, which enlarges the over-fitting probability of the learned diagnostic model.
然而，对于大型和有价值的结构，如大跨度桥梁和海上风力涡轮机，总是缺乏此类数据，尤其是受损情况的数据，这就增加了所学诊断模型的过拟合概率。
The lack of data is probably the greatest challenge in applying pattern recognition and machine learning methods in SHM. 3 , 7 3 , 7 ^(3,7){ }^{3,7}
缺乏数据可能是将模式识别和机器学习方法应用于 SHM 的最大挑战。 3 , 7 3 , 7 ^(3,7){ }^{3,7}
Differing from data-driven SHM approaches, physics-based approaches evaluate structural condition through updating a representative physics-based model of the target structure, such as a finite element (FE) model, by minimizing the discrepancy of its predictions from the measured data.
与数据驱动的 SHM 方法不同，基于物理的方法通过更新目标结构的代表性基于物理的模型（如有限元模型）来评估结构状况，最大限度地减少其预测值与测量数据之间的差异。
6 , 7 6 , 7 ^(6,7){ }^{6,7} Compared with data-driven approaches, a physics-based approach provides a calibrated physics-based numerical model that can be used for damage prognosis.
6 , 7 6 , 7 ^(6,7){ }^{6,7} 与数据驱动方法相比，基于物理的方法提供了一个经过校准的基于物理的数值模型，可用于损伤预报。
However, a critical barrier limiting in-practice application of FE model updating is the modeling error that originates from model simplification and omission.
然而，限制有限元模型更新实际应用的一个关键障碍是模型简化和遗漏造成的建模误差。
In physics-based SHM, modeling error renders the updated model biased from the real structure, which leads to challenge in structural parameter estimation, 8 8 ^(8){ }^{8} structural damage detection, 9 9 ^(9){ }^{9} and predicting structural features and responses. 10 10 ^(10){ }^{10}
在基于物理的 SHM 中，建模误差会使更新后的模型与真实结构产生偏差，从而给结构参数估计、 8 8 ^(8){ }^{8} 结构损伤检测、 9 9 ^(9){ }^{9} 以及预测结构特征和响应带来挑战。 10 10 ^(10){ }^{10}
Considering that both data-driven and physicsbased approaches have critical shortcomings and that their merits are complementary, it would be attractive if they can be synergistically integrated in SHM so that their merits get preserved and their shortcomings become less critical.
考虑到数据驱动方法和基于物理的方法都有严重的缺陷，而它们的优点又是互补的，如果能将它们协同整合到 SHM 中，使它们的优点得到保留，缺点变得不那么严重，那将会很有吸引力。
To this end, the present study proposes integrating pattern recognition with FE model updating in SHM via physics-guided machine learning (PGML).
为此，本研究提出通过物理引导的机器学习（PGML），在 SHM 中整合模式识别与 FE 模型更新。
PGML leverages measured data with correct labels as well as the scientific knowledge contained in the physics-based model (FE model in SHM), so that the model predictions are consistent with the scientific principles behind the physics-based model and maintain sufficient accuracy on the labeled data.
PGML 利用带有正确标签的测量数据以及基于物理的模型（SHM 中的 FE 模型）所包含的科学知识，使模型预测与基于物理的模型背后的科学原理保持一致，并在标签数据上保持足够的准确性。
It has the potential of improving the generality of the learned model and scientific consistency of its predictions even when representative labeled samples are very limited. 11 11 ^(11){ }^{11} PGML has been broadly applied in many areas such as climate pattern discovery, 12 12 ^(12){ }^{12} turbulence modeling, 13 13 ^(13){ }^{13} material science, 14 14 ^(14){ }^{14} quantum chemistry, 15 15 ^(15){ }^{15} and so on
即使在有代表性的标注样本非常有限的情况下，它也有可能提高所学模型的通用性及其预测的科学一致性。 11 11 ^(11){ }^{11} PGML已被广泛应用于许多领域，如气候模式发现、 12 12 ^(12){ }^{12} 湍流建模、 13 13 ^(13){ }^{13} 材料科学、 14 14 ^(14){ }^{14} 量子化学、 15 15 ^(15){ }^{15} 等。
Based on the literature review, a new method using PGML for structural damage identification is proposed and evaluated in the present study. PGML is realized with a multi-layer perceptron (MLP) neural network ( NN ) ( NN ) (NN)(\mathrm{NN}) model, that is physics-guided neural networks (PGNN), 16 16 ^(16){ }^{16} through extending the original modal-prop-erty-based feature with the damage localization output
根据文献综述，本研究提出并评估了一种使用 PGML 进行结构损伤识别的新方法。PGML 采用多层感知器（MLP）神经网络 ( NN ) ( NN ) (NN)(\mathrm{NN}) 模型实现，即物理引导神经网络（PGNN）， 16 16 ^(16){ }^{16} 通过将原有的基于模态-属性的特征与损伤定位输出扩展在一起
of FE model updating and incorporating a physicsbased loss function. The physics-based loss function evaluates the discrepancy between the output of the NN model and that of FE model updating.
的 FE 模型更新，并加入了基于物理的损失函数。基于物理的损失函数用于评估 NN 模型输出与 FE 模型更新输出之间的差异。
With physics guidance from the updated FE model, the learned NN model generalizes well to the unseen test data.
在更新的 FE 模型的物理指导下，学习到的 NN 模型能很好地概括未见过的测试数据。
Moreover, it is shown that errors in damage locations and severities can be significantly reduced by integrating the results of damage localization with PGNN into FE model updating.
此外，研究还表明，将使用 PGNN 进行损伤定位的结果整合到 FE 模型更新中，可以显著减少损伤位置和严重程度的误差。
The efficiency of the proposed methodology in structural damage localization is validated numerically using a steel pedestrian bridge model, 17 17 ^(17){ }^{17} and experimentally using measured data from a three-story building model. 18 18 ^(18){ }^{18}
利用钢结构人行天桥模型 17 17 ^(17){ }^{17} 对所提方法在结构损伤定位方面的效率进行了数值验证，并利用三层建筑模型的测量数据进行了实验验证。 18 18 ^(18){ }^{18}
The remaining part of this paper is structured as follows. The section “Methodology” establishes the framework of PGML for structural damage localization. The section “Numerical study” presents a numerical case study with a steel pedestrian bridge model.
本文其余部分的结构如下。方法论 "部分建立了用于结构损伤定位的 PGML 框架。数值研究 "部分介绍了一个钢结构人行天桥模型的数值案例研究。
The section “Experimental validation” presents an experimental study with a three-story building. The section “Conclusions” concludes this study with remarks and recommendations.
实验验证 "部分介绍了对一座三层建筑的实验研究。结论 "部分对本研究进行了总结，并提出了意见和建议。

Methodology 方法

To incorporate physics into data-driven SHM and realize PGML for structural damage evaluation, the present study uses the FE model as an implicit representation of scientific knowledge underlying the monitored structure and incorporates the output of FE model updating into the NN model setup and learning.
为了将物理学纳入数据驱动的 SHM 并实现用于结构损伤评估的 PGML，本研究将 FE 模型作为受监测结构基础科学知识的隐式表示，并将 FE 模型更新的输出纳入 NN 模型的设置和学习中。
This section establishes the framework of PGML for structural damage localization, which contains two major steps: (1) extending the original feature vector with the output of FE model updating; (2) designing a physics-based loss function that integrates the scientific knowledge underlying the FE model into the NN model learning process.
本节建立了用于结构损伤定位的 PGML 框架，其中包含两个主要步骤：(1) 利用 FE 模型更新的输出扩展原始特征向量；(2) 设计基于物理学的损失函数，将 FE 模型所依据的科学知识整合到 NN 模型学习过程中。
This section first introduces the method of FE model updating used in this study, then describes the two major steps of PGML and introduces how to implement PGML in PyTorch, a framework of deep learning.
本节首先介绍本研究中使用的 FE 模型更新方法，然后描述 PGML 的两个主要步骤，并介绍如何在深度学习框架 PyTorch 中实现 PGML。

FE model updating FE 模型更新

For a monitored structure, the stiffness matrix K K K\mathbf{K} can be formulated as
对于受监控的结构，刚度矩阵 K K K\mathbf{K} 可表示为
K = K 0 + ∑ i = 1 n α α i K i K = K 0 + ∑ i = 1 n α   α i K i K=K_(0)+sum_(i=1)^(n_(alpha))alpha_(i)K_(i)\mathbf{K}=\mathbf{K}_{0}+\sum_{i=1}^{n_{\alpha}} \alpha_{i} \mathbf{K}_{i}
in which K 0 K 0 K_(0)\mathbf{K}_{0} is the sum of known substructural stiffness matrices prior to model updating; K i K i K_(i)\mathbf{K}_{i} is the nominal stiffness matrix of substructure i i ii with unknown
其中， K 0 K 0 K_(0)\mathbf{K}_{0} 是模型更新前已知下部结构刚度矩阵的总和； K i K i K_(i)\mathbf{K}_{i} 是下部结构 i i ii 的标称刚度矩阵，其中包含未知的
stiffness; α i α i alpha_(i)\alpha_{i} is the coefficient corresponding to K i ; n α K i ; n α K_(i);n_(alpha)\mathbf{K}_{i} ; n_{\alpha} is the number of substructures with unknown stiffness. Hence, α = [ α 1 , α 2 , … , α n α ] α = α 1 , α 2 , … , α n α alpha=[alpha_(1),alpha_(2),dots,alpha_(n_(alpha))]\boldsymbol{\alpha}=\left[\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n_{\alpha}}\right] containing all unknown stiffness coefficients is the target of FE model updating.
的系数； α i α i alpha_(i)\alpha_{i} 是与 K i ; n α K i ; n α K_(i);n_(alpha)\mathbf{K}_{i} ; n_{\alpha} 相对应的系数， K i ; n α K i ; n α K_(i);n_(alpha)\mathbf{K}_{i} ; n_{\alpha} 是具有未知刚度的子结构的数量。因此，包含所有未知刚度系数的 α = [ α 1 , α 2 , … , α n α ] α = α 1 , α 2 , … , α n α alpha=[alpha_(1),alpha_(2),dots,alpha_(n_(alpha))]\boldsymbol{\alpha}=\left[\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n_{\alpha}}\right] 是 FE 模型更新的目标。
Measured modal properties including natural frequencies and mode shapes are usually used to formulate the objective function of model updating. This study adopts the formulation with the eigen-frequency and mode shape differences. 19 19 ^(19){ }^{19} That is
通常使用测量到的模态特性（包括固有频率和模态振型）来制定模型更新的目标函数。本研究采用了特征频率和模态振型差值的方法。 19 19 ^(19){ }^{19} 即
L ( α ) = ∑ i = 1 n m { ( λ i e − λ i ( α ) λ i e ⋅ w λ i ) 2 + ‖ Q i ( Φ i e − Φ i m ( α ) ) ⋅ w Φ i ‖ 2 2 } L ( α ) = ∑ i = 1 n m   λ i e − λ i ( α ) λ i e ⋅ w λ i 2 + Q i Φ i e − Φ i m ( α ) ⋅ w Φ i 2 2 {:[L(alpha)=],[sum_(i=1)^(n_(m)){((lambda_(i)^(e)-lambda_(i)(alpha))/(lambda_(i)^(e))*w_(lambda_(i)))^(2)+||Q_(i)(Phi_(i)^(e)-Phi_(i)^(m)(alpha))*w_(Phi_(i))||_(2)^(2)}]:}\begin{aligned} & L(\boldsymbol{\alpha})= \\ & \sum_{i=1}^{n_{\mathrm{m}}}\left\{\left(\frac{\lambda_{i}^{\mathrm{e}}-\lambda_{i}(\boldsymbol{\alpha})}{\lambda_{i}^{\mathrm{e}}} \cdot w_{\lambda_{i}}\right)^{2}+\left\|\mathbf{Q}_{i}\left(\boldsymbol{\Phi}_{i}^{\mathrm{e}}-\boldsymbol{\Phi}_{i}^{\mathrm{m}}(\boldsymbol{\alpha})\right) \cdot w_{\Phi_{i}}\right\|_{2}^{2}\right\} \end{aligned}
in which n m n m n_(m)n_{\mathrm{m}} is the number of measured modes in dynamic tests; λ i e λ i e lambda_(i)^(e)\lambda_{i}^{\mathrm{e}} is the experimentally measured eigenfrequency of the i th i th  i^("th ")i^{\text {th }} mode; λ i ( α ) λ i ( α ) lambda_(i)(alpha)\lambda_{i}(\boldsymbol{\alpha}) is the evaluated value of λ i λ i lambda_(i)\lambda_{i} from the FE model using a certain value of α ; Φ i e α ; Φ i e alpha;Phi_(i)^(e)\boldsymbol{\alpha} ; \boldsymbol{\Phi}_{i}^{\mathrm{e}} is the measured mode shape of the i th i th  i^("th ")i^{\text {th }} mode; Φ i m ( α ) Φ i m ( α ) Phi_(i)^(m)(alpha)\boldsymbol{\Phi}_{i}^{\mathrm{m}}(\boldsymbol{\alpha}) is the evaluated Φ i Φ i Phi_(i)\boldsymbol{\Phi}_{i} at the measured degrees of freedom (DOFs) using α . Q i α . Q i alpha.Q_(i)\boldsymbol{\alpha} . \mathbf{Q}_{i} is the selection matrix; w λ i w λ i w_(lambda_(i))w_{\lambda_{i}} and w Φ i w Φ i w_(Phi_(i))w_{\Phi_{i}} are the weighting factors of the eigen-frequency and mode shape, respectively.
其中， n m n m n_(m)n_{\mathrm{m}} 是动态测试中测量到的模态数； λ i e λ i e lambda_(i)^(e)\lambda_{i}^{\mathrm{e}} 是实验测量到的 i th i th  i^("th ")i^{\text {th }} 模态的特征频率； λ i ( α ) λ i ( α ) lambda_(i)(alpha)\lambda_{i}(\boldsymbol{\alpha}) 是有限元模型中 λ i λ i lambda_(i)\lambda_{i} 的评估值，使用一定的 α ; Φ i e α ; Φ i e alpha;Phi_(i)^(e)\boldsymbol{\alpha} ; \boldsymbol{\Phi}_{i}^{\mathrm{e}} 值； i th i th  i^("th ")i^{\text {th }} 是测量到的 i th i th  i^("th ")i^{\text {th }} 模态的模态振型； Φ i m ( α ) Φ i m ( α ) Phi_(i)^(m)(alpha)\boldsymbol{\Phi}_{i}^{\mathrm{m}}(\boldsymbol{\alpha}) 是使用选择矩阵 α . Q i α . Q i alpha.Q_(i)\boldsymbol{\alpha} . \mathbf{Q}_{i} 在测量的自由度 (DOF) 上求得的 Φ i Φ i Phi_(i)\boldsymbol{\Phi}_{i} 值； w λ i w λ i w_(lambda_(i))w_{\lambda_{i}} 和 w Φ i w Φ i w_(Phi_(i))w_{\Phi_{i}} 分别是特征频率和模态振型的加权系数。
The Levenberg-Marquardt algorithm is selected for optimization and is implemented using the “Isqnonlin” solver in MATLAB. The Jacobian derivative is used to determine the local search direction at each iteration.
选择 Levenberg-Marquardt 算法进行优化，并使用 MATLAB 中的 "Isqnonlin "求解器实现。雅各布导数用于确定每次迭代的局部搜索方向。
A number of runs, for example 50, are implemented with random starting points, and the solution yielding the least objective function is selected as the final solution.
以随机起点执行若干次运行，例如 50 次，然后选择目标函数最小的解作为最终解。

Feature extension 功能扩展

Modal properties, including the natural frequencies and mode shapes, are widely used in data-driven SHM in deriving damage-sensitive features and designing objective function in FE model updating.
模态特性，包括固有频率和模态振型，在数据驱动的 SHM 中被广泛应用于推导损伤敏感特征和设计 FE 模型更新的目标函数。
This study uses the normalized frequency change ratio (NFCR) and the change of mode shapes d Φ d Φ dPhi\mathrm{d} \Phi of the first several modes as features ( X ) ( X ) (X)(X) for damage detection. 20 20 ^(20){ }^{20} That is
本研究使用归一化频率变化比（NFCR）和前几个模态的模态振型 d Φ d Φ dPhi\mathrm{d} \Phi 变化作为损伤检测的特征 ( X ) ( X ) (X)(X) 。 20 20 ^(20){ }^{20} 即
X = { NFCR ; d Φ } X = { NFCR ; d Φ } X={NFCR;dPhi}X=\{\mathrm{NFCR} ; \mathrm{d} \Phi\}
where d Φ d Φ dPhi\mathrm{d} \Phi is the difference of mode shapes between damaged and intact cases. NFCR of a certain mode is calculated as the normalized fractional frequency change ( FFC ), that is
其中， d Φ d Φ dPhi\mathrm{d} \Phi 是受损和完好情况下的模态振型差。某个模态的 NFCR 用归一化的分数频率变化（FFC）计算，即
NFCR i = FFC i ∑ j = 1 N FFC j NFCR i = FFC i ∑ j = 1 N   FFC j NFCR_(i)=(FFC_(i))/(sum_(j=1)^(N)FFC_(j))\mathrm{NFCR}_{i}=\frac{\mathrm{FFC}_{i}}{\sum_{j=1}^{N} \mathrm{FFC}_{j}}
in which N N NN is the number of modes selected for SHM purpose. FFC for the i th i th  i^("th ")i^{\text {th }} mode is expressed as
其中 N N NN 是为 SHM 目的而选择的模式数。 i th i th  i^("th ")i^{\text {th }} 模式的 FFC 表示为
FFC i = f u i − f d i f u i FFC i = f u i − f d i f u i FFC_(i)=(f_(ui)-f_(di))/(f_(ui))\mathrm{FFC}_{i}=\frac{f_{\mathrm{u} i}-f_{\mathrm{d} i}}{f_{\mathrm{u} i}}
in which f u i f u i f_(ui)f_{\mathrm{u} i} and f d i f d i f_(di)f_{\mathrm{d} i} are the i th i th  i^("th ")i^{\text {th }} mode frequencies of the structure in undamaged and damaged states, respectively.
其中 f u i f u i f_(ui)f_{\mathrm{u} i} 和 f d i f d i f_(di)f_{\mathrm{d} i} 分别是结构在未损坏和损坏状态下的 i th i th  i^("th ")i^{\text {th }} 模态频率。
After preparing data with features X X XX and labels y y yy, a standard NN model would be f NN : X → y f NN : X → y f_(NN):X rarr yf_{\mathrm{NN}}: X \rightarrow y that yields estimated labels y ^ y ^ hat(y)\hat{y}. Alternatively, FE model updating using the measured modal properties will yield damage severities (i.e. z mu z mu z_(mu)z_{\mathrm{mu}} ) at interested sites and thus recommend the most probable damage locations. In this study, y mu y mu y_(mu)y_{\mathrm{mu}} is set as the location with the most severe damage in z mu z mu z_(mu)z_{\mathrm{mu}}. It is noted that y mu y mu y_(mu)y_{\mathrm{mu}} may not be a correct representation of the structural damage distribution due to simplification and/or idealization in FE model establishment and updating.
在准备了具有 X X XX 特征和 y y yy 标签的数据后，标准的 NN 模型将是 f NN : X → y f NN : X → y f_(NN):X rarr yf_{\mathrm{NN}}: X \rightarrow y ，产生估计标签 y ^ y ^ hat(y)\hat{y} 。另外，使用测量的模态属性更新 FE 模型将得出相关站点的损坏严重程度（即 z mu z mu z_(mu)z_{\mathrm{mu}} ），从而推荐最可能的损坏位置。在本研究中， y mu y mu y_(mu)y_{\mathrm{mu}} 被设定为 z mu z mu z_(mu)z_{\mathrm{mu}} 中损坏最严重的位置。需要注意的是，由于在建立和更新 FE 模型时进行了简化和/或理想化， y mu y mu y_(mu)y_{\mathrm{mu}} 可能无法正确反映结构损伤分布。
To address this issue, PGML integrates the output of a physics-based model into the original feature data, so that both information from measured data and physics can be leveraged in learning a model. 21 21 ^(21){ }^{21} For example, in an NN model, the hidden layers can extract complex features from the extended feature input so that the insufficiency of the physics-based model can be complemented. Then we have the extended feature for damage detection in SHM as follows
为了解决这个问题，PGML 将基于物理模型的输出集成到原始特征数据中，这样在学习模型时就可以同时利用测量数据和物理信息。 21 21 ^(21){ }^{21} 例如，在 NN 模型中，隐藏层可以从扩展特征输入中提取复杂特征，从而补充基于物理模型的不足。因此，用于 SHM 损伤检测的扩展特征如下
X ext = { X ; y mu } X ext = X ; y mu X_(ext)={X;y_(mu)}X_{\mathrm{ext}}=\left\{X ; y_{\mathrm{mu}}\right\}
X ext X ext  X_("ext ")X_{\text {ext }} will be used as input of the PGNN model in this study.
X ext X ext  X_("ext ")X_{\text {ext }} 将作为本研究中 PGNN 模型的输入。

Physics-based loss function
基于物理的损失函数

A standard NN model using the extended feature X ext X ext  X_("ext ")X_{\text {ext }} as input aims to minimize the data loss calculated on the labeled training data as well as the model complexity that is expressed as the regularization terms of the model parameter norms. So the loss function is defined as
使用扩展特征 X ext X ext  X_("ext ")X_{\text {ext }} 作为输入的标准 NN 模型的目标是，最大限度地减少根据标注的训练数据计算得出的数据损失以及模型的复杂性，模型的复杂性用模型参数规范的正则化项来表示。因此，损失函数定义为
Loss = L d ( z f , y ) + λ R R ( f )  Loss  = L d z f , y + λ R R ( f ) " Loss "=L_(d)(z_(f),y)+lambda_(R)R(f)\text { Loss }=L_{\mathrm{d}}\left(z_{\mathrm{f}}, y\right)+\lambda_{\mathrm{R}} R(f)
in which L d L d L_(d)L_{\mathrm{d}} denotes the data loss; z f z f z_(f)z_{\mathrm{f}} is the output scores of the NN model f ; y f ; y f;yf ; y denotes the target labels of the labeled data; R ( f ) R ( f ) R(f)R(f) measures the model’s complexity or structural loss; λ R λ R lambda_(R)\lambda_{\mathrm{R}} is the regularization parameter. For a certain input data x i x i x_(i)x_{i} the output of an NN model z f ( x i ) z f x i z_(f)(x_(i))z_{\mathrm{f}}\left(x_{i}\right) contains the scores of each possible class, that is z f ( x i ) = [ z f 1 ( x i ) , z f 2 ( x i ) , … , z f c ( x i ) ] z f x i = z f 1 x i , z f 2 x i , … , z f c x i z_(f)(x_(i))=[z_(f)^(1)(x_(i)),z_(f)^(2)(x_(i)),dots,z_(f)^(c)(x_(i))]z_{\mathrm{f}}\left(x_{i}\right)=\left[z_{\mathrm{f}}^{1}\left(x_{i}\right), z_{\mathrm{f}}^{2}\left(x_{i}\right), \ldots, z_{\mathrm{f}}^{c}\left(x_{i}\right)\right] in which c c cc is the number of classes or number of locations of interest in SHM. For a multiclass problem in data-driven structural damage localization, the cross-entropy loss is used to evaluate the classification performance. 22 22 ^(22){ }^{22} Then the data loss of x i x i x_(i)x_{i} is
其中 L d L d L_(d)L_{\mathrm{d}} 表示数据损失； z f z f z_(f)z_{\mathrm{f}} 是 NN 模型的输出分数 f ; y f ; y f;yf ; y 表示标注数据的目标标签； R ( f ) R ( f ) R(f)R(f) 衡量模型的复杂度或结构损失； λ R λ R lambda_(R)\lambda_{\mathrm{R}} 是正则化参数。对于某个输入数据 x i x i x_(i)x_{i} ，NN 模型的输出 z f ( x i ) z f x i z_(f)(x_(i))z_{\mathrm{f}}\left(x_{i}\right) 包含每个可能类别的得分，即 z f ( x i ) = [ z f 1 ( x i ) , z f 2 ( x i ) , … , z f c ( x i ) ] z f x i = z f 1 x i , z f 2 x i , … , z f c x i z_(f)(x_(i))=[z_(f)^(1)(x_(i)),z_(f)^(2)(x_(i)),dots,z_(f)^(c)(x_(i))]z_{\mathrm{f}}\left(x_{i}\right)=\left[z_{\mathrm{f}}^{1}\left(x_{i}\right), z_{\mathrm{f}}^{2}\left(x_{i}\right), \ldots, z_{\mathrm{f}}^{c}\left(x_{i}\right)\right] ，其中 c c cc 是 SHM 中感兴趣的类别数或位置数。对于数据驱动的结构损伤定位中的多类问题，使用交叉熵损失来评估分类性能。 22 22 ^(22){ }^{22} 则 x i x i x_(i)x_{i} 的数据损失为
L d ( z f ( x i ) , y i ) = − z f y i ( x i ) + log ⁡ ( ∑ j = 1 c exp ⁡ ( z f j ( x i ) ) ) L d z f x i , y i = − z f y i x i + log ⁡ ∑ j = 1 c   exp ⁡ z f j x i L_(d)(z_(f)(x_(i)),y_(i))=-z_(f)^(y_(i))(x_(i))+log(sum_(j=1)^(c)exp(z_(f)^(j)(x_(i))))L_{\mathrm{d}}\left(z_{\mathrm{f}}\left(x_{i}\right), y_{i}\right)=-z_{\mathrm{f}}^{y_{i}}\left(x_{i}\right)+\log \left(\sum_{j=1}^{c} \exp \left(z_{\mathrm{f}}^{j}\left(x_{i}\right)\right)\right)