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.

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Compared with physicsbased methods of SHM, a data-driven approach avoids building and validating a numerical model

^{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}

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}

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

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，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}

此外，它还能自动
accommodates the uncertainty that originates from measuring variability.

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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。

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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.

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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。

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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.

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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。

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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。

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Compared with data-driven approaches, a physics-based approach provides a calibrated physics-based numerical model that can be used for damage prognosis.

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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,

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structural damage detection,

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and predicting structural features and responses.

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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、

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结构损伤检测、

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以及预测结构特征和响应带来挑战。

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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.

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PGML has been broadly applied in many areas such as climate pattern discovery,

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turbulence modeling,

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material science,

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quantum chemistry,

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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。

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PGML已被广泛应用于许多领域，如气候模式发现、

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湍流建模、

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材料科学、

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量子化学、

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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)

model, that is physics-guided neural networks (PGNN),

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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)

模型实现，即物理引导神经网络（PGNN），

^{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,

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and experimentally using measured data from a three-story building model.

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利用钢结构人行天桥模型

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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。

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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

can be formulated as
对于受监控的结构，刚度矩阵

K

可表示为

K = K_{0} + \sum_{i = 1}^{n_{α}} α_{i} K_{i}

in which

K_{0}

is the sum of known substructural stiffness matrices prior to model updating;

K_{i}

is the nominal stiffness matrix of substructure

i

with unknown
其中，

K_{0}

是模型更新前已知下部结构刚度矩阵的总和；

K_{i}

是下部结构

i

的标称刚度矩阵，其中包含未知的
stiffness;

α_{i}

is the coefficient corresponding to

K_{i}; n_{α}

is the number of substructures with unknown stiffness. Hence,

α = [α_{1}, α_{2}, \dots, α_{n_{α}}]

containing all unknown stiffness coefficients is the target of FE model updating.
的系数；

α_{i}

是与

K_{i}; n_{α}

相对应的系数，

K_{i}; n_{α}

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}, \dots, α_{n_{α}}]

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.

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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。

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即

\begin{aligned} L (α) = \\ \sum_{i = 1}^{n_{m}} {{(\frac{λ_{i}^{e} - λ_{i} (α)}{λ_{i}^{e}} \cdot w_{λ_{i}})}^{2} + {‖ Q_{i} (Φ_{i}^{e} - Φ_{i}^{m} (α)) \cdot w_{Φ_{i}} ‖}_{2}^{2}} \end{aligned}

in which

n_{m}

is the number of measured modes in dynamic tests;

λ_{i}^{e}

is the experimentally measured eigenfrequency of the

i^{th}

mode;

λ_{i} (α)

is the evaluated value of

λ_{i}

from the FE model using a certain value of

α; Φ_{i}^{e}

is the measured mode shape of the

i^{th}

mode;

Φ_{i}^{m} (α)

is the evaluated

Φ_{i}

at the measured degrees of freedom (DOFs) using

α . Q_{i}

is the selection matrix;

w_{λ_{i}}

and

w_{Φ_{i}}

are the weighting factors of the eigen-frequency and mode shape, respectively.
其中，

n_{m}

是动态测试中测量到的模态数；

λ_{i}^{e}

是实验测量到的

i^{th}

模态的特征频率；

λ_{i} (α)

是有限元模型中

λ_{i}

的评估值，使用一定的

α; Φ_{i}^{e}

值；

i^{th}

是测量到的

i^{th}

模态的模态振型；

Φ_{i}^{m} (α)

是使用选择矩阵

α . Q_{i}

在测量的自由度 (DOF) 上求得的

Φ_{i}

值；

w_{λ_{i}}

和

w_{Φ_{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

of the first several modes as features

(X)

for damage detection.

^{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

变化作为损伤检测的特征

(X)

。

^{20}

即

X = {NFCR; d Φ}

where

d Φ

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

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} = \frac{{FFC}_{i}}{\sum_{j = 1}^{N} {FFC}_{j}}

in which

N

is the number of modes selected for SHM purpose. FFC for the

i^{th}

mode is expressed as
其中

N

是为 SHM 目的而选择的模式数。

i^{th}

模式的 FFC 表示为

{FFC}_{i} = \frac{f_{u i} - f_{d i}}{f_{u i}}

in which

f_{u i}

and

f_{d i}

are the

i^{th}

mode frequencies of the structure in undamaged and damaged states, respectively.
其中

f_{u i}

和

f_{d i}

分别是结构在未损坏和损坏状态下的

i^{th}

模态频率。

After preparing data with features

X

and labels

y

, a standard NN model would be

f_{NN} : X \to y

that yields estimated labels

\hat{y}

. Alternatively, FE model updating using the measured modal properties will yield damage severities (i.e.

z_{mu}

) at interested sites and thus recommend the most probable damage locations. In this study,

y_{mu}

is set as the location with the most severe damage in

z_{mu}

. It is noted that

y_{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

特征和

y

标签的数据后，标准的 NN 模型将是

f_{NN} : X \to y

，产生估计标签

\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}

），从而推荐最可能的损坏位置。在本研究中，

y_{mu}

被设定为

z_{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}

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.

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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。

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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}

will be used as input of the PGNN model in this study.

X_{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}

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}

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)

in which

L_{d}

denotes the data loss;

z_{f}

is the output scores of the NN model

f; y

denotes the target labels of the labeled data;

R (f)

measures the model’s complexity or structural loss;

λ_{R}

is the regularization parameter. For a certain input data

x_{i}

the output of an NN model

z_{f} (x_{i})

contains the scores of each possible class, that is

z_{f} (x_{i}) = [z_{f}^{1} (x_{i}), z_{f}^{2} (x_{i}), \dots, z_{f}^{c} (x_{i})]

in which

c

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}

Then the data loss of

x_{i}

is
其中

L_{d}

表示数据损失；

z_{f}

是 NN 模型的输出分数

f; y

表示标注数据的目标标签；

R (f)

衡量模型的复杂度或结构损失；

λ_{R}

是正则化参数。对于某个输入数据

x_{i}

，NN 模型的输出

z_{f} (x_{i})

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}), \dots, z_{f}^{c} (x_{i})]

，其中

c

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}

则

x_{i}

的数据损失为

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})))

The

ℓ_{1}

and

ℓ_{2}

norms of the network weights

W

are regularized to control the model complexity. That is
网络权重

W

的

ℓ_{1}

和

ℓ_{2}

规范被正则化，以控制模型的复杂性。即

λ_{R} R (W) = λ_{1} ‖ W ‖_{1} + λ_{2} ‖ W ‖_{2}

in which

λ_{1}

and

λ_{2}

are the regularization parameters.
其中

λ_{1}

和

λ_{2}

为正则化参数。
Considering the limitations of standard machine learning procedures for the SHM problem with limited labeled data, physics-based loss functions or scientific inconsistency

L_{p}

are introduced into the PGML to guide the learning process for physically consistent solutions. Then the loss function becomes
考虑到标准机器学习程序在处理标注数据有限的 SHM 问题时存在局限性，我们在 PGML 中引入了基于物理的损失函数或科学不一致性

L_{p}

，以指导物理一致解的学习过程。那么损失函数变为

Loss = L_{d} (z_{f}, y) + λ_{R} R (W) + λ_{p} L_{p}

in which

λ_{p}

is the regularization parameter of

L_{p}

.
其中

λ_{p}

是

L_{p}

的正则化参数。
The physics-based loss function in data-driven SHM is derived using the FE model updating outputs of the damage instances with unknown labels. For a certain damage case with measured modal data, FE model updating yields the damage severities at all interested locations, that is

z_{mu} = [z_{mu}^{1}, z_{mu}^{2}, \dots, z_{mu}^{c}]

in which

c

is the number of damage classes defined above. On one hand, the location with the largest damage severity can be regarded as the most probable damage location and set as the pseudo-target labels of the corresponding case, that is.,

y_{mu}

. Then a cross-entropy loss can be formed over all available unlabeled structural data, which is termed as

L_{p 1}

. That is, for a certain instance

x_{i}

, the loss is
在数据驱动的 SHM 中，基于物理的损失函数是利用未知标签的损伤实例的 FE 模型更新输出得出的。对于某个具有实测模态数据的损坏案例，FE 模型更新可得出所有相关位置的损坏严重程度，即

z_{mu} = [z_{mu}^{1}, z_{mu}^{2}, \dots, z_{mu}^{c}]

，其中

c

是上文定义的损坏等级数。一方面，可以将损坏严重程度最大的位置视为最可能损坏的位置，并将其设置为相应案例的伪目标标签，即

y_{mu}

。然后可以在所有可用的未标记结构数据上形成一个交叉熵损失，称为

L_{p 1}

。也就是说，对于某个实例

x_{i}

，损失为

L_{p 1} (z_{f} (x_{i}), y_{mu}^{i}) = - z_{f}^{y_{mu}^{i}} (x_{i}) + \log (\sum_{j = 1}^{c} \exp (z_{f}^{j} (x_{i})))

On the other hand, a softmax function transforms

z_{mu}

to normalized damage probabilities,

p_{mu},^{23}

that is
另一方面，软最大函数将

z_{mu}

转换为归一化损伤概率

p_{mu},^{23}

，即

p_{mu}^{i} = \frac{\exp (z_{mu}^{i})}{\sum_{j = 1}^{c} \exp (z_{mu}^{j})}

for

x_{i}

. The same operation can be implemented on the output scores of NN model

z_{f} (x)

, which yields the predicted damage probabilities at each location, that is.,

p_{f}

as shown in Figure 1. Then a mean-squared-error loss can be defined between

p_{mu}

and

p_{f}

, which is termed as

L_{p 2}

. That is, for a certain instance

x_{i}

,
为

x_{i}

。可以对 NN 模型

z_{f} (x)

的输出分数执行相同的操作，从而得到每个位置的预测损坏概率，即

p_{f}

，如图 1 所示。然后，可以在

p_{mu}

和

p_{f}

之间定义均方误差损失，即

L_{p 2}

。也就是说，对于某个实例

x_{i}

、

L_{p 2} (z_{f} (x_{i}), z_{mu} (x_{i})) = \frac{1}{c} \sum_{j = 1}^{c} {(p_{f}^{j} (x_{i}) - p_{mu}^{j} (x_{i}))}^{2}

Figure I. Normalizing the NN model output using the softmax function.

x

is the input of the NN model;

z_{f}

is the output score;

p_{f}

is the probability of each class after normalization. NN : neural network.
图 I：使用 softmax 函数对 NN 模型输出进行归一化处理。

x

是 NN 模型的输入；

z_{f}

是输出得分；

p_{f}

是归一化后每个类别的概率。NN：神经网络。

In terms of equations (8), (9), (11), and (13), the loss function of PGNN in equation (10) can be written as
根据等式 (8)、(9)、(11) 和 (13)，等式 (10) 中 PGNN 的损失函数可写成

\begin{aligned} Loss & = L_{d} + λ_{R} R (W) + λ_{p} L_{p} \\ = \frac{1}{n_{l}} \sum_{i = 1}^{n_{l}} [- z_{f}^{y_{i}} (x_{i}) + \log (\sum_{j = 1}^{c} \exp (z_{f}^{j} (x_{i})))] \\ + λ_{1} ‖ W ‖_{1} + λ_{2} ‖ W ‖_{2} \\ + λ_{p 1} \frac{1}{n_{u}} \sum_{i = n_{l} + 1}^{n_{l} + n_{u}} [- z_{f}^{y_{mu}^{i}} (x_{i}) + \log (\sum_{j = 1}^{c} \exp (z_{f}^{j_{f}} (x_{i})))] \\ + λ_{p 2} \frac{1}{c n_{u}} \sum_{i = n_{l} + 1}^{n_{l} + n_{u}} \sum_{j = 1}^{c} {(p_{f}^{j} (x_{i}) - p_{mu}^{j} (x_{i}))}^{2} \end{aligned}

in which

n_{l}

and

n_{u}

denote the number of available labeled and unlabeled data respectively;

λ_{p 1}

and

λ_{p 2}

are the regularization parameters of

L_{p 1}

and

L_{p 2}

respectively.
其中，

n_{l}

和

n_{u}

分别表示可用的标注数据和未标注数据的数量；

λ_{p 1}

和

λ_{p 2}

分别是

L_{p 1}

和

L_{p 2}

的正则化参数。

Implementation in PyTorch
用 PyTorch 实现

Figure 2 illustrates the framework of the proposed PGML realized with an NN model, of which the details have been introduced in preceding subsections.
图 2 展示了利用 NN 模型实现的 PGML 框架。
The two dashed lines indicate physics guidance using the results of model updating on unknown damage cases, which correspond to the two physics-based loss functions

L_{p 1}

and

L_{p 2}

, respectively.
两条虚线表示利用未知损坏情况下的模型更新结果进行的物理指导，分别对应两个基于物理的损失函数

L_{p 1}

和

L_{p 2}

。

In this study, the PGML is implemented in PyTorch, a popular deep-learning framework based on the Torch library.

^{24}

An MLP-NN model designed with fully connected layers is built for each case study.

{ReLU}^{25}

is selected as the activation function for each hidden layer. Adam algorithm

^{26}

is used to perform stochastic
在本研究中，PGML 是在 PyTorch 中实现的，PyTorch 是一个基于 Torch 库的流行深度学习框架。

^{24}

为每个案例研究设计了一个全连接层的 MLP-NN 模型。

{ReLU}^{25}

被选为每个隐藏层的激活函数。亚当算法

^{26}

用于执行随机

Figure 2. A schematic illustration of the PGML framework.

R

denotes the measured structural responses containing displacement

u

, velocity

\dot{u}

, and acceleration

\ddot{u}

; OMA stands for operational modal analysis;

D

is the vector containing the obtained natural frequencies

f

and mode shapes

Φ; X

denotes the feature vector extracted from D containing the normalized frequency change ratio (NFCR) and the change of mode shapes

d Φ

of the first several modes;

z_{mu}

denotes the output of model updating that indicates the damage severities at each interested location;

y_{mu}

is the most probable damage location
图 2.PGML 框架示意图。

R

表示包含位移

u

、速度

\dot{u}

和加速度

\ddot{u}

的测量结构响应；OMA 表示运行模态分析；

D

是包含所获得的自然频率

f

和模态振型的向量

Φ; X

表示从 D 中提取的特征向量，其中包含归一化频率变化比 (NFCR) 和前几个模态的模态振型变化

d Φ

；

z_{mu}

表示模型更新的输出，其中显示了每个相关位置的损坏严重程度；

y_{mu}

是最可能损坏的位置
recommended by model updating;

X_{ext}

is the extended feature containing

X

and

y_{mu}; p_{mu}

contains damage probabilities of each location calculated from

z_{m u}

using the softmax function;

\hat{y}

denotes the predicted labels by PGNN with the guidance of FE model updating through

y_{mu}

and

p_{mu}

X_{ext}

是包含

X

的扩展特征，

y_{mu}; p_{mu}

包含使用 softmax 函数从

z_{m u}

计算出的每个位置的损坏概率；

\hat{y}

表示 PGNN 在 FE 模型更新的指导下通过

y_{mu}

和

p_{mu}

预测的标签。
PGML: physics-guided machine learning, PGNN, physics-guided neural network.
PGML：物理引导的机器学习，PGNN：物理引导的神经网络。
gradient descent on the model parameters. An early stopping strategy

^{27}

is adopted to avoid overfitting. The datasets are normalized via min-max normalization

^{28}

before being input into the NN model. Fifty runs are implemented for each model training process with random initialization, and the model yielding the least loss is selected as the final solution.
对模型参数进行梯度下降。为避免过度拟合，采用了早期停止策略

^{27}

。在将数据集输入 NN 模型之前，先通过最小最大归一化

^{28}

对其进行归一化。在随机初始化的情况下，每个模型训练过程运行 50 次，并选择损失最小的模型作为最终解决方案。

Numerical study 数值研究

To validate the methodology of damage identification proposed in this paper, this section implements a numerical case study using an FE model of a steel pedestrian bridge.

^{19}

Figure 3 shows the bridge model and its sensor instrumentation. The bridge model is divided into six substructures. The elastic modulus of the frame members (

E_{1}

E_{6}

) are set as the target of FE model updating in this numerical study, assuming known mass matrix and other stiffness components prior to model updating. The bridge model is instrumented with seven uniaxial and seven biaxial accelerometers that measure 21 out of the 274 DOFs.
为了验证本文提出的损伤识别方法，本节使用钢结构人行天桥的 FE 模型进行了一次数值案例研究。

^{19}

图 3 显示了桥梁模型及其传感器仪器。桥梁模型分为六个子结构。在本数值研究中，假设模型更新前质量矩阵和其他刚度分量已知，则框架构件的弹性模量（

E_{1}

至

E_{6}

）被设定为 FE 模型更新的目标。桥梁模型配备了 7 个单轴和 7 个双轴加速度计，可测量 274 个 DOF 中的 21 个。
More details about the bridge model can be found in Dong and Wang

^{19}

有关桥梁模型的更多详情，请参阅 Dong 和 Wang

^{19}

。

Figure 3. Schematic model of a steel pedestrian bridge and its sensor instrumentation.

^{19}

The bridge structure is divided into six substructures: substructure #I to #6.

E_{i} (i = 1, 2, \dots, 6)

represents the elastic modulus of the frame members of substructure #i;

E_{t i} (i = 2, 3, \dots, 6)

denotes the elastic modulus of the truss members of substructure

# i; k_{y 1}

and

k_{y 2}

are the stiffness values of transverse support springs at the two bridge ends;

k_{z 1}

and

k_{z 2}

are the stiffness values of vertical support springs.
图 3.钢结构人行天桥及其传感器仪器的示意模型。

^{19}

桥梁结构分为六个子结构：子结构 #I 至 #6。

E_{i} (i = 1, 2, \dots, 6)

表示下部结构 #i 框架构件的弹性模量；

E_{t i} (i = 2, 3, \dots, 6)

表示下部结构桁架构件的弹性模量；

# i; k_{y 1}

和

k_{y 2}

是桥梁两端横向支撑弹簧的刚度值；

k_{z 1}

和

k_{z 2}

是竖向支撑弹簧的刚度值。

Table I. Damage cases and conditions of the steel pedestrian bridge.
表 I.钢制人行天桥的损坏情况和条件。

Damage class 伤害等级	Case ID 案例编号	Damage severity 损坏严重程度
D1	$I - 100$	-0.1 to -0.9 on $E_{1}$ $E_{1}$ 上的-0.1至-0.9
D2	$101 - 200$	-0.1 to -0.9 on $E_{2}$ $E_{2}$ 上的-0.1至-0.9
D3	$201 - 300$	-0.1 to -0.9 on $E_{3}$ $E_{3}$ 上的-0.1至-0.9
D4	$301 - 400$	-0.1 to -0.9 on $E_{4}$ $E_{4}$ 上的-0.1至-0.9
D5	$401 - 500$	-0.1 to -0.9 on $E_{5}$ $E_{5}$ 上的-0.1至-0.9
D6	$501 - 600$	-0.1 to -0.9 on $E_{6}$ $E_{6}$ 上的-0.1至-0.9

Stiffness reduction is introduced into a certain substructure to simulate single-site structural damage. Multi-site damage detection is more complex using pattern recognition methods in data-driven SHM and will be studied in future research.
在某个下部结构中引入刚度降低来模拟单点结构损伤。使用数据驱动 SHM 中的模式识别方法进行多部位损伤检测更为复杂，将在今后的研究中加以探讨。
A total of 100 damage cases with a random damage severity ranging from

10 %

90 %

are simulated for each damage location. Table 1 shows the damage classes and conditions simulated in this numerical study. Modal properties of the first three modes are used to formulate the objective function in equation (2) with all the weighting factors

w_{λ_{i}}

and

w_{Φ_{i}} (i = 1, 2

, and 3

)

set as 1.0 . Without modeling error introduced in the established FE model, it is found in the present study that model updating yields accurate damage severities for all cases with relative error below

1 %

.
每个损坏位置共模拟了 100 个损坏案例，损坏严重程度从

10 %

到

90 %

不等。表 1 显示了本次数值研究中模拟的损坏等级和情况。前三种模态的模态特性用于制定方程 (2) 中的目标函数，所有加权系数

w_{λ_{i}}

和

w_{Φ_{i}} (i = 1, 2

以及 3

)

设置为 1.0。本研究发现，在不引入已建立的 FE 模型中的建模误差的情况下，模型更新可对所有情况产生准确的损坏严重程度，相对误差低于

1 %

。

However, modeling error always exists when establishing a numerical model for a real structure, which may significantly affect model updating results and the following damage evaluation quality.

^{9}

Hence, this study focuses on damage identification considering modeling error, which is modeled via increasing the support stiffness at bridge end

1 (k_{y 1}

and

k_{z 1})

20 %

and
然而，在为实际结构建立数值模型时，建模误差总是存在的，这可能会严重影响模型更新结果和后续的损伤评估质量。

^{9}

因此，本研究将重点放在考虑建模误差的损伤识别上，建模误差通过增加桥端

1 (k_{y 1}

和

k_{z 1})

的支撑刚度

20 %

和来实现。

Figure 4. Model updating results of example cases with slight damage using the FE model with introduced modeling errors.

δ_{α_{i}}

denotes the identified damage severity at location

i;

(a) to (f) show the cases with damage at locations I-6, respectively. Blue bars denote damage severities evaluated from model updating, and white bars with red edge denote target damage locations and corresponding severities.
图 4.使用引入建模误差的 FE 模型对轻微损坏示例进行模型更新的结果。

δ_{α_{i}}

表示位置

i;

（a）至（f）分别显示了位置 I-6 的损坏情况。蓝条表示根据模型更新评估出的损坏严重程度，白条红边表示目标损坏位置和相应的严重程度。
FE : finite element.
FE：有限元。
decreasing the support stiffness at bridge end

2 (k_{y 2}

and

k_{z 2}

) by

20 %

. The mass matrix elements are also varied by changing the value of each element by a random percentage from

- 20 %

20 %

to simulate the inaccurate measurement of structural masses. Figures 4 and 5 show the results of model updating of example cases with slight (below 20%) and severe damage (above

75 %

), respectively. Figure 4 shows that the introduction of modeling error causes the model updating results significantly off from the correct values when slight damage happens.
通过

20 %

降低桥端支座刚度

2 (k_{y 2}

和

k_{z 2}

。质量矩阵元素也有变化，每个元素的值按随机百分比从

- 20 %

变为

20 %

，以模拟结构质量的不精确测量。图 4 和图 5 分别显示了轻微损坏（低于 20%）和严重损坏（高于

75 %

）示例的模型更新结果。图 4 显示，当发生轻微损坏时，建模误差的引入会导致模型更新结果与正确值有明显偏差。
Incorrect damage severities are identified at target damage site and false damage is predicted at locations without damage, which makes it challenging for accurate damage quantification and localization.
在目标损坏点会识别出不正确的损坏严重程度，在没有损坏的位置会预测出错误的损坏，这给准确的损坏量化和定位带来了挑战。
For example, in Figure 4(a), when slight damage happens to substructure 1 , the identified damage severity at

E_{1}

, that is

δ_{α_{1}}

, is much smaller than that at

E_{2}

, that is.,

δ_{α_{2}}; δ_{α_{3}}

indicates about

40 %

stiffness increase, which is misleading in damage detection. Other incorrect and false damage identification results can be observed in Figures 4(d)-(f).
例如，在图 4(a)中，当下部结构 1 发生轻微损坏时，

E_{1}

（即

δ_{α_{1}}

）处识别出的损坏严重程度远小于

E_{2}

（即

δ_{α_{2}}; δ_{α_{3}}

）处识别出的损坏严重程度，

δ_{α_{2}}; δ_{α_{3}}

表示约

40 %

的刚度增加，这对损坏检测具有误导作用。从图 4(d)-(f) 中还可以观察到其他错误的损伤识别结果。

Figure 5. Model updating results of example cases with severe damage using the FE model with introduced modeling errors.

δ_{α_{i}}

denotes the identified damage severity at location

i

; (a) to (f) show the cases with damage at locations I-6, respectively. Blue bars denote damage severities evaluated from model updating, and white bars with red edge denote target damage locations and corresponding severities.
图 5.使用引入建模误差的 FE 模型对损坏严重的示例案例进行模型更新的结果。

δ_{α_{i}}

表示位置

i

处已确定的损坏严重程度；（a）至（f）分别表示位置 I-6 处损坏的案例。蓝条表示根据模型更新评估出的损坏严重程度，带红色边缘的白条表示目标损坏位置和相应的严重程度。
FE : finite element.
FE：有限元。

When severe damage happens, modeling error has less adverse influence on the model updating results than when slight damage happens, as shown in Figure 5. The target damage location always has the highest damage severity that is very close to the real quantity.
如图 5 所示，当发生严重损坏时，建模误差对模型更新结果的不利影响比发生轻微损坏时要小。目标损坏位置的损坏严重程度总是最高的，与实际数量非常接近。
However, considerable misleading damage is still evaluated at locations without damage. Therefore, the existence of modeling error will inevitably cause incorrect/false identification results using FE model updating.
但是，在没有损坏的位置，仍然会评估出相当大的误导性损坏。因此，建模误差的存在必然会导致使用 FE 模型更新得出错误/虚假的识别结果。
Perhaps in the case of severe damage, FE model updating can identify the damage location and quantity, yet the results are still misleading for overall damage evaluation. Moreover, in practice, accurate damage detection is necessary before severe damage happens.
也许在严重损坏的情况下，更新有限元模型可以确定损坏的位置和数量，但其结果仍然会对整体损坏评估产生误导。此外，在实际应用中，必须在严重损坏发生之前进行准确的损坏检测。
In a word, existence of modeling error poses a critical challenge for damage identification using deterministic FE model updating.
总之，建模误差的存在对使用确定性 FE 模型更新进行损伤识别提出了严峻的挑战。
Figure 6 shows the histograms of identified damage severities at correct/ wrong locations obtained using model updating, in which one can observe the overall adverse effects caused by modeling errors during model updating. In
图 6 显示了利用模型更新获得的正确/错误位置上已识别的损坏严重程度直方图，从中可以观察到模型更新过程中的建模错误所造成的总体不利影响。在

Figure 6. Histograms of identified damage severities from FE model updating.

i = I, 2, \dots, 6

denotes the correct damage location of the plotted cases;

δ_{α_{i}}

denotes the damage severity detected at the correct location

i; δ_{α_{j}}

denotes the damage severity detected at wrong locations

j (j = 1, \dots, i - I, i + I, \dots 6)

; (a) to (f) show the cases with damage at locations I-6, respectively.
图 6.根据 FE 模型更新确定的损坏严重程度直方图。

i = I, 2, \dots, 6

表示图中案例的正确损坏位置；

δ_{α_{i}}

表示在正确位置检测到的损坏严重程度

i; δ_{α_{j}}

表示在错误位置检测到的损坏严重程度

j (j = 1, \dots, i - I, i + I, \dots 6)

；（a）至（f）分别表示在 I-6 位置损坏的案例。
addition, model updating yields positive damage severity at target locations for some damage cases, which can give spurious recommendations about structural conditions.
此外，在某些损坏情况下，模型更新会在目标位置产生正的损坏严重程度，从而对结构状况提出错误的建议。

Differing from FE modeling updating that identify potential damage locations and severities simultaneously, data-driven SHM first identifies damage locations through machine learning classification; then damage severities can be evaluated through statistical regression.
与同时确定潜在损坏位置和严重程度的 FE 建模更新不同，数据驱动的 SHM 首先通过机器学习分类确定损坏位置，然后通过统计回归评估损坏严重程度。
The present study implements damage localization through classification and evaluates the corresponding damage severities via FE model updating on the detected damaged sites. A three-hidden-layer MLPNN model is designed with the number of neurons set as [50 10050 ].
本研究通过分类实现损伤定位，并通过对检测到的损伤部位进行 FE 模型更新来评估相应的损伤严重程度。本研究设计了一个三隐层 MLPNN 模型，神经元数量设置为 [50 10050 ]。
The data assignment is

5 %

for training,

5 %

for validation, and

90 %

for testing to simulate the lack of data with correct labels in reality. In addition, slight damage is assigned to all training data, as labeled data collected in reality are usually from slight or medium damage severity scenarios.
数据分配为：

5 %

用于训练，

5 %

用于验证，

90 %

用于测试，以模拟现实中缺乏具有正确标签的数据的情况。此外，所有训练数据都分配为轻微损坏，因为在现实中收集到的标注数据通常来自轻微或中等损坏严重程度的场景。
It is worth noting that this design can efficiently test the generality of the
值得注意的是，这种设计可以有效地检验

Figure 7. Results of damage localization on the steel pedestrian bridge model using an NN model without physics guidance. Two features are selected to visualize the data in the 2D feature space: (I) NFCR

_{I}

(

x

axis); (2)

d Φ

of the

I^{st}

measured degree of freedom (

y

axis).

E_{i}

denotes the damage class corresponding to damage at location

i

. (a) illustration of all the data and their labels, and (b)-(d): classification results on the training data, validation data, and testing data, respectively; solid circles with different colors denote the predicted labels and the black cross indicates wrong labeling.
图 7.使用无物理学指导的 NN 模型对钢人行天桥模型进行损伤定位的结果。在二维特征空间中选择了两个特征来直观显示数据：(I) NFCR

_{I}

（

x

轴）；(2)

I^{st}

测量自由度的

d Φ

（

y

轴）。

E_{i}

表示位置

i

损坏对应的损坏类别。 (a) 所有数据及其标签的图示，(b)-(d)：分别对训练数据、验证数据和测试数据的分类结果；不同颜色的实心圆圈表示预测标签，黑叉表示错误标签。
NN: neural network; 2D: two-dimensional; NFCR: normalized frequency change ratio.
NN：神经网络；2D：二维；NFCR：归一化频率变化比。
learned model in PGML in detecting future damage, while it makes pattern recognition challenging due to the lack of representative data.
在 PGML 中学习到的模型在检测未来损害方面具有重要作用，但由于缺乏代表性数据，模式识别具有挑战性。

For comparison, an NN model is first built without physics guidance by setting the corresponding regularization parameters (

λ_{p 1}

and

λ_{p 2}

in equation (14)) as zero. Figure 7 shows the results of damage localization on different datasets using the trained NN classifier. The data are plotted in a two-dimensional (2D) feature space.
为了进行比较，首先通过将相应的正则化参数（等式 (14) 中的

λ_{p 1}

和

λ_{p 2}

）设置为零，在没有物理指导的情况下建立一个 NN 模型。图 7 显示了使用训练有素的 NN 分类器对不同数据集进行损伤定位的结果。数据在二维（2D）特征空间中绘制。
Damage classes corresponding to the damage locations are labeled with solid circles with different colors as indicated in the legend of Figure 7(a).
图 7（a）的图例中用不同颜色的实心圆圈标出了与损坏位置相对应的损坏等级。
Figure 7(b) shows that all the training data are distributed in the top edge region of the feature space as a result of their limited quantity and damage severity. Hyperparameters tuning in validation fails to improve the generality of the learned model.
图 7(b) 显示，所有训练数据都分布在特征空间的顶部边缘区域，这是由于这些数据的数量和损坏严重程度有限。验证中的超参数调整无法提高所学模型的通用性。
As a result, it yields a low accuracy in model testing, as can be seen from the testing results in Figure 7(d). Examining Figures 7(b) through (d), one can find that correct labeling only occurs to data lying close to the
因此，从图 7(d) 中的测试结果可以看出，该方法在模型测试中的准确率较低。通过观察图 7(b) 至 (d)，我们可以发现，正确的标注只发生在接近于 "阈值 "的数据上。

Figure 8. Results of damage localization on the steel pedestrian bridge model using a PGNN model with physics guidance. Two features are selected to visualize the data in the 2D feature space: (I) NFCR

_{I}

(

x

axis); (2)

d Φ

of the

I^{st}

measured degree of freedom (

y

axis).

E_{i}

denotes the damage class corresponding to damage at location

i

. (a) illustration of all the data and their labels, and (b)-(d): classification results on the training data, validation data, and testing data, respectively; solid circles with different colors denote the predicted labels and the black cross indicates wrong labeling.
图 8.使用带有物理指导的 PGNN 模型对钢人行天桥模型进行损伤定位的结果。在二维特征空间中选择了两个特征来直观显示数据：（I）NFCR

_{I}

（

x

轴）；（2）

I^{st}

测量自由度的

d Φ

（

y

轴）。

E_{i}

表示位置

i

损坏对应的损坏类别。 (a) 所有数据及其标签的图示，(b)-(d)：分别对训练数据、验证数据和测试数据的分类结果；不同颜色的实心圆圈表示预测标签，黑叉表示错误标签。
PGNN: physics-guided neural network; 2D: two-dimensional; NFCR: normalized frequency change ratio.
PGNN：物理引导神经网络；2D：二维；NFCR：归一化频率变化比。
training data, which is very limited in representation. Hence, the generality of the learned model needs to be improved for better damage localization performance.
训练数据的代表性非常有限。因此，需要改进所学模型的通用性，以提高损伤定位性能。

To improve the generality of the learned model and its performance when tested on new cases, physics guidance is incorporated in the model learning process by activating the physics loss terms in equation (14). The corresponding regularization parameters

λ_{p 1}

and

λ_{p 2}

are tuned using the validation dataset via a grid search strategy

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with

λ_{1}

and

λ_{2}

kept unchanged from the trained NN model. Figure 8 shows the results of damage evaluation using the trained PGNN model.
为了提高所学模型的通用性及其在新情况下的测试性能，通过激活方程 (14) 中的物理损失项，在模型学习过程中加入了物理指导。相应的正则化参数

λ_{p 1}

和

λ_{p 2}

是通过网格搜索策略

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使用验证数据集进行调整的，而

λ_{1}

和

λ_{2}

与训练的 NN 模型保持不变。图 8 显示了使用训练有素的 PGNN 模型进行损坏评估的结果。
Compared with the results of NN model without physics guidance as shown in Figure 7, PGNN generalizes better to the unseen testing data that has much larger quantity than either the training data or the validation data, as shown in Figure 8(d).
如图 8（d）所示，与图 7 所示没有物理指导的 NN 模型结果相比，PGNN 对未知测试数据的泛化效果更好，因为测试数据的数量远远大于训练数据或验证数据。
Comparing the distributions of data in the 2D feature space as shown in Figure 8(b)-(d), one can find that many of the testing data are located in regions not covered by the training
比较图 8(b)-(d) 所示二维特征空间中的数据分布，可以发现许多测试数据位于训练数据未覆盖的区域。

Figure 9. Model updating results of example cases using the FE model with introduced modeling errors with damage locations identified by the PGML model. (a) example cases with slight damage as shown in Figure 4 and (b) example cases with severe damage as shown in Figure 5.
图 9.使用引入了建模误差的 FE 模型和 PGML 模型确定的损坏位置的示例案例的模型更新结果。(a) 如图 4 所示的轻微损坏示例和 (b) 如图 5 所示的严重损坏示例。
Each damage case is plotted with a single bar as damage has been located.

δ_{α_{i}}

denotes the identified damage severity at location

i (i = 1, 2, \dots, 6)

. Blue bars denote damage severities evaluated from model updating, and white bars with red edge denote target damage locations and corresponding severities.
每种损坏情况在确定损坏位置后都会绘制一个条形图。

δ_{α_{i}}

表示位置

i (i = 1, 2, \dots, 6)

上已确定的损坏严重程度。蓝色条形图表示根据模型更新评估的损坏严重程度，带红色边缘的白色条形图表示目标损坏位置和相应的严重程度。
FE: finite element; PGML: physics-guided machine learning.
FE：有限元；PGML：物理引导的机器学习。
data or the validation data, while their labels are correctly predicted by the learned classifier. This further validates the improved generality or implicit scientific consistency of the learned model with physics guidance incorporated in the model learning process.
而学习到的分类器则能正确预测它们的标签。这进一步验证了在模型学习过程中加入物理学指导后，所学模型的通用性或隐含科学一致性得到了提高。

With most damage locations correctly identified using the learned PGNN classifier, FE model updating is rerun with the damage severity

δ_{α_{i}}

at the identified location

i

set as the target of model updating. Figure 9 shows the results of enhanced model updating with PGML-detected damage locations. Compared with
在使用学习的 PGNN 分类器正确识别出大多数损坏位置后，重新运行 FE 模型更新，并将识别出的位置

i

上的损坏严重程度

δ_{α_{i}}

设置为模型更新的目标。图 9 显示了使用 PGML 检测到的损坏位置进行增强模型更新的结果。与

(a)
(b)

Figure 10. Distribution of model updating errors on the steel pedestrian bridge model (a) without and (b) with PGML regulation.
图 10.钢制人行天桥模型（a）和（b）在没有 PGML 调节的情况下模型更新误差的分布。
PGML: physics-guided machine learning.
PGML：物理引导的机器学习。
results presented in Figures 4 and 5, enhanced model updating yields damage severities closer to the target values with incorrect damage locations excluded, which further highlights the merits of PGML in structural damage evaluation when integrated with FE model updating.
根据图 4 和图 5 所示的结果，在排除不正确的损伤位置后，增强型模型更新产生的损伤严重程度更接近目标值，这进一步凸显了 PGML 与 FE 模型更新相结合在结构损伤评估中的优势。
To further evaluate the improvement in model updating results through PGML, the error of damage severities in each case is summed over all the locations. For example, the real damage severity of case 1 is

[- 0.4336, 0, 0, 0, 0, 0]

, and the results of model updating without PGML is

[- 0.4519, 0.0490, 0.0480

- 0.0594, 0.1563, 0.1060]

, then the sum of error for case 1 is

0.0183 + 0.0490 + 0.0480 + 0.0594 + 0.1563 + 0.1060

= 0.2817

. The distribution of the sum of errors is plotted in Figure 10. It shows that without the regulation on damage locations from PGML, many of the error sums (

22.3 %

) are above 1.0; however, PGML
为了进一步评估 PGML 对模型更新结果的改进，将每个案例中的损坏严重程度误差加总到所有位置。例如，情况 1 的实际损坏严重程度为

[- 0.4336, 0, 0, 0, 0, 0]

，未使用 PGML 的模型更新结果为

[- 0.4519, 0.0490, 0.0480

- 0.0594, 0.1563, 0.1060]

，则情况 1 的误差总和为

0.0183 + 0.0490 + 0.0480 + 0.0594 + 0.1563 + 0.1060

= 0.2817

。误差总和的分布如图 10 所示。从图中可以看出，如果没有 PGML 对损坏位置的规定，许多误差总和 (

22.3 %

) 都高于 1.0；但是，PGML

Figure II. Three-story building model in Figueiredo

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(a) the LANL test structural model and (b) the schematic model.

m_{0}

denotes the mass on each floor,

k_{i} (i = 0, 1, \dots 3

) is the stiffness of the

i^{th}

floor;

f_{0} (t)

is the excitation at the base;

{\ddot{x}}_{i} (t)

is the acceleration measured on the

i^{th}

floor.
图 II.Figueiredo

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中的三层建筑模型 (a) LANL 试验结构模型和 (b) 示意图模型。

m_{0}

表示每层的质量，

k_{i} (i = 0, 1, \dots 3

) 是

i^{th}

层的刚度；

f_{0} (t)

是底部的激励；

{\ddot{x}}_{i} (t)

是在

i^{th}

层测得的加速度。
reduces the error sums of most cases significantly such that most of them

(97.5 %)

are below 0.2 . Hence, it is indicated that PGML can largely improve the damage identification performance of both data-driven and FE model updating methods.
在大多数情况下，PGML 都能显著降低误差和，使大多数误差和

(97.5 %)

低于 0.2。因此，PGML 可以在很大程度上改善数据驱动法和 FE 模型更新法的损伤识别性能。

Experimental validation 实验验证

In addition to the numerical study, an experimental study is implemented in this section using the experimental data of a three-story frame structure published by Los Alamos National Laboratory.

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Figure 11(a) depicts the three-story frame structure where damage was introduced by stiffness reduction of floor columns and mass addition on the first floor to simulate the operational and environmental influence, and bumper impacts on the third floor to simulate the nonlinear behaviors of fatigue cracks.
除数值研究外，本节还利用洛斯阿拉莫斯国家实验室公布的三层框架结构的实验数据进行了实验研究。

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图 11(a) 描述了三层框架结构，其中第一层通过降低楼层支柱刚度和增加质量来模拟运行和环境影响，第三层通过保险杠撞击来模拟疲劳裂缝的非线性行为。
An electrodynamic shaker was used to excite the frame structure with various damage conditions with Gaussian white noise laterally on the base floor along the structural centerline.
使用电动振动台沿结构中心线在底层横向用高斯白噪声激励各种损坏条件下的框架结构。
The excitation force applied from the shaker to the structure was recorded with a load cell mounted on the stringer and the structural responses were measured using four accelerometers attached at the center line of each floor as shown in Figure 11(a).
如图 11（a）所示，振动器施加到结构上的激振力由安装在支柱上的称重传感器记录，结构响应则由连接在每层中心线上的四个加速度计测量。
The data were collected and processed at a sampling frequency of 320 Hz with a data acquisition system. For each structural damage state, 10 shaking tests were conducted considering the variability of excitations and structural properties.
数据采集系统以 320 Hz 的采样频率采集和处理数据。考虑到激励和结构特性的可变性，对每种结构损伤状态进行了 10 次振动试验。
The measured data including excitations and responses can be used to extract damage-sensitive features for the SHM purposes.

^{18}

包括激励和响应在内的测量数据可用于提取损伤敏感特征，以实现 SHM 目的。

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Figure 12. Illustration of first three mode shapes extracted from measured responses of the three-story building. degrees of freedom I-4 correspond to the horizontal motions of floors.

Φ_{i} (i = 1, 2

, and 3

)

is the shape of the

i^{th}

mode.

D i

in the legend denotes the damage class defined in Table 2 and the value after the colon is the identified natural frequency.
图 12.从三层楼实测响应中提取的前三个模态振型示意图。自由度 I-4 对应楼层的水平运动。

Φ_{i} (i = 1, 2

和 3

)

是

i^{th}

模式的形状。图例中的

D i

表示表 2 中定义的损坏等级，冒号后的值为确定的固有频率。

As shown in Figure 11(b), this study uses a fourDOF lumped-mass structural model to represent the three-story building for damage detection. In FE model updating, the mass of each floor

(m_{0})

is calculated in terms of the plate dimension

(30.5 \times 30.5 \times 2.5 {cm}^{3})

and the density of aluminum

(2.7 g / {cm}^{3})

. In addition, the sliding friction between the base floor and the rails is represented by a spring with small stiffness

k_{0}

following the simplification by Sun and Betti.

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k_{0}

is mainly involved in the rigid body motion, its value is first evaluated in model updating using the 10 intact cases, which will be used subsequently for updating stiffness parameters (i.e.

k_{1}, k_{2}

, and

k_{3}

) with possible damage.
如图 11(b)所示，本研究使用一个四自由度（fourDOF）的总重结构模型来表示三层建筑，以进行损坏检测。在更新 FE 模型时，每层楼板的质量

(m_{0})

是根据板尺寸

(30.5 \times 30.5 \times 2.5 {cm}^{3})

和铝密度

(2.7 g / {cm}^{3})

计算得出的。此外，根据 Sun 和 Betti 的简化方法，底层地板和导轨之间的滑动摩擦由刚度较小的弹簧

k_{0}

表示。

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由于

k_{0}

主要参与刚体运动，因此在模型更新时首先使用 10 个完好无损的案例对其值进行评估，随后将用于更新可能出现损坏的刚度参数（即

k_{1}, k_{2}

和

k_{3}

）。

The present study selects four structural conditions of the three-story building from the database available in Figueiredo

^{18}

to examine the effectiveness of PGML in damage localization.
本研究从 Figueiredo

^{18}

提供的数据库中选取了四种三层建筑的结构条件，以检验 PGML 在损伤定位中的有效性。
The selected conditions include the baseline condition without structural damage (termed as D0), structural condition with stiffness reduction on the first floor column(s) (D1), structural condition with stiffness reduction on the second floor column(s) (D2), and structural condition with stiffness reduction on the third floor column(s) (D3).
所选条件包括无结构损坏的基线条件（称为 D0）、一楼支柱刚度降低的结构条件（D1）、二楼支柱刚度降低的结构条件（D2）以及三楼支柱刚度降低的结构条件（D3）。
Each damage class contains two damage severities: (1) moderate damage:

87.5 %

stiffness reduction of one column of a certain floor yielding

22 %

reduction of the floor
每个损坏等级包含两种损坏严重程度：(1) 中等损坏：

87.5 %

某个楼层的一根柱子刚度降低，导致

22 %

楼层降低

Table 2. Damage cases and conditions of the 3-story frame structure.
表 2.3 层框架结构的损坏情况和条件。

Damage class 伤害等级	Case ID 案例编号	Damage severity 损坏严重程度
D1	$1 - 10$	-0.22 on $k_{1}$ $k_{1}$ 上的-0.22
	$1 I - 20$	-0.44 on $k_{1}$ $k_{1}$ 上的-0.44
D2	$2 I - 30$	-0.22 on $k_{2}$ $k_{2}$ 上的-0.22
	$3 I - 40$	-0.44 on $k_{2}$ $k_{2}$ 上的-0.44
D3	$4 I - 50$	-0.22 on $k_{3}$ $k_{3}$ 上的-0.22
	$5 I - 60$	-0.44 on $k_{3}$ $k_{3}$ 上的-0.44

stiffness

k_{i} (i = 1, 2

, or 3 ); (2) severe damage:

87.5 %

stiffness reduction of two columns of a certain floor yielding

44 %

reduction of

k_{i}

. Table 2 lists all the damage cases used in this experimental study.
刚度

k_{i} (i = 1, 2

，或 3 ）；（2）严重损坏：

87.5 %

某层两根柱子刚度降低，

44 %

k_{i}

降低。表 2 列出了本实验研究中使用的所有损坏情况。

The dynamic responses measured at the four DOFs

{\ddot{x}}_{i} (t) (i = 0, 1, \dots 3)

are processed using the frequency domain decomposition (FDD) method

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to obtain the natural frequencies and mode shapes, which will be used to formulate the objective function in FE model updating and derive damage-sensitive features in datadriven SHM.
使用频域分解（FDD）方法

^{31}

对四个 DOF

{\ddot{x}}_{i} (t) (i = 0, 1, \dots 3)

测得的动态响应进行处理，以获得固有频率和模态振型，用于制定 FE 模型更新的目标函数，并在数据驱动的 SHM 中推导出损伤敏感特征。
Figure 12 compares the first three mode shapes of a representative case when damage happens at a certain floor. The results for case D0 are the average of modal properties obtained using the 10 intact
图 12 比较了在某一楼层发生损坏时一个代表性案例的前三个模态振型。案例 D0 的结果是使用 10 个完好无损的模态振型获得的模态特性的平均值。
cases. The mode shapes of each damage class presented in Figure 12 is obtained from an example case with severe damage (

44 %

damage severity). The mode caused by rigid motion is excluded in this study for structural damage detection. Figure 12 shows that damage causes slight-to-obvious frequency reduction of the three modes.
案例。图 12 所示的各损伤等级的模态振型是从一个严重损伤（

44 %

损伤严重程度）的实例中获得的。本研究在结构损伤检测中排除了由刚性运动引起的模态。图 12 显示，损坏导致三种模态的频率略有降低，但并不明显。
Obvious mode shape variations can be observed among different damage cases of each mode. Figure 12 indicates that natural frequency and mode shape variations can be used to evaluate structural damage.
在每种模态的不同损坏情况下，可以观察到明显的模态振型变化。图 12 表明，固有频率和模态振型变化可用于评估结构损伤。

With the experimental data, FE model updating is implemented using the extracted modal properties to demonstrate the limitation of model updating in the presence of modeling error.
通过实验数据，利用提取的模态特性对 FE 模型进行了更新，以证明在存在建模误差的情况下模型更新的局限性。
Figure 13 shows the identified damage results using model updating and the measured modal properties. It shows that model updating can detect damage occurrence in the 60 cases as indicated by the “correct” bars.
图 13 显示了利用模型更新和测量模态特性确定的损坏结果。从图中可以看出，正如 "正确 "条所示，模型更新可以检测出 60 个案例中发生的损坏。
However, notable errors exist in the damage evaluation results due to the considerable modeling error in establishing the numerical model (e.g. mass measurement and connection modeling), as shown by the “wrong” bars; especially when moderate damage happens on

k_{1}

(case 1-10), significant wrong damage is identified on

k_{2}

, which may cause error in damage localization. It should be noted that the experimental tests in LANL contain no slight damage case, for which model updating results might be influenced by significant modeling error, as indicated in the numerical case study.
然而，由于在建立数值模型时存在相当大的建模误差（例如质量测量和连接建模），损坏评估结果存在显著误差，如 "错误 "条所示；特别是当

k_{1}

上发生中度损坏时（案例 1-10），在

k_{2}

上识别出明显的错误损坏，这可能会导致损坏定位错误。应该注意的是，LANL 的实验测试不包含轻微损坏情况，对于轻微损坏情况，模型更新结果可能会受到重大建模误差的影响，这一点在数值案例研究中已经指出。

The same NN structure as used in the numerical study section is used in this experimental study with all the hyperparameters retuned. The data assignment is

15 %

for training,

15 %

for validation, and

70 %

for testing, so that each dataset contains data from different damage classes. Figure 14 shows the results of classification for damage detection. The data are plotted using the two principal components (PCs) of the features obtained from principal component analysis (PCA).
本实验研究采用了与数值研究部分相同的 NN 结构，并对所有超参数进行了重新调整。数据分配为：

15 %

用于训练，

15 %

用于验证，

70 %

用于测试，因此每个数据集包含来自不同损伤类别的数据。图 14 显示了损伤检测的分类结果。数据使用主成分分析 (PCA) 得出的特征的两个主成分 (PC) 绘制。
Figure 14 indicates that with the hyperparameters

(λ_{1}

and

λ_{2}

in equation (14)) tuned in validation, the accuracy on the training data is 0.78 .
图 14 显示，在验证中调整方程 (14) 中的超参数

(λ_{1}

和

λ_{2}

后，训练数据的准确率为 0.78。
Without physics guidance, the trained NN model yields an accuracy as low as 0.67 on the testing data because of insufficient data with correct labels for learning a model with satisfactory generality.
在没有物理学指导的情况下，训练好的 NN 模型在测试数据上的准确率低至 0.67，这是因为没有足够的具有正确标签的数据来学习具有令人满意的通用性的模型。

Figure 15 shows the results of damage localization with a PGNN model, in which the regularization parameters for the physics loss

(λ_{p 1}

and

λ_{p 2})

are tuned in validation with

λ_{1}

and

λ_{2}

kept unchanged from the tuned NN model above. Comparison between Figure 14 and Figure 15 shows that PGNN improves the damage detection accuracy of all datasets through incorporating physics guidance obtained in model updating in the classifier’s learning process. It should
图 15 显示了使用 PGNN 模型进行损伤定位的结果，其中物理损失的正则化参数

(λ_{p 1}

和

λ_{p 2})

在验证中进行了调整，而

λ_{1}

和

λ_{2}

与上述调整后的 NNN 模型保持不变。图 14 和图 15 之间的比较表明，PGNN 在分类器的学习过程中结合了模型更新中获得的物理指导，从而提高了所有数据集的损伤检测精度。它应该

Figure 13. Model-updating-based damage identification results of the three-story building using the four-degree of freedom numerical model.

δ_{α_{i}}

denotes the damage severities on

k_{i} (i = 1

, 2 , and 3 ) evaluated through comparing the updated stiffness values with that of the intact structure. Green bars labeled as “correct” in the legend indicate evaluated damage at correct locations, and red bars denote identified false damage at locations without damage.
图 13.使用四自由度数值模型对三层建筑进行基于模型更新的损伤识别结果。

δ_{α_{i}}

表示通过比较更新后的刚度值和完整结构的刚度值，评估出的

k_{i} (i = 1

、2 和 3 上的损坏严重程度。图例中标注为 "正确 "的绿条表示在正确位置评估出的损坏，红条表示在无损坏位置识别出的错误损坏。
The two reference lines indicate the damage severities designed in the experimental tests, that is -0.22 for moderate damage and -0.44 for severe damage. See Table 2 for more details about the damage cases.
两条参考线表示实验测试中设计的损坏严重程度，即中度损坏为-0.22，严重损坏为-0.44。有关损坏情况的详细信息，请参见表 2。
be noted that although the correct labels of the validation data is not contained in the training process, and that the labels evaluated from model updating (

y_{mu}

in Figure 2) contain errors due to some of the misleading model updating results, the PGNN can provide consistently accurate damage identification results.
值得注意的是，虽然在训练过程中没有包含验证数据的正确标签，而且由于一些误导性的模型更新结果，从模型更新中评估出的标签（图 2 中的

y_{mu}

）包含误差，但 PGNN 仍能提供持续准确的损伤识别结果。

Figure 14. Results of damage localization on the three-story building using an NN model without physics guidance. The

x

axis is PCI of the data, and the

y

axis is PC 2 .

D i (i = 1, 2

, and 3

)

denotes the damage cases as listed in Table 2.
图 14.使用无物理引导的 NN 模型对三层建筑进行损伤定位的结果。

x

轴为数据的 PCI，

y

轴为 PC 2。

D i (i = 1, 2

和 3

)

表示表 2 中列出的损坏情况。
(a) illustration of all the data and their labels, and (b)-(d) classification results on the training data, validation data, and testing data, respectively; solid circles with different colors denote the predicted labels and the black cross indicates wrong labeling.
(a) 所有数据及其标签示意图，(b)-(d) 分别为训练数据、验证数据和测试数据的分类结果；不同颜色的实心圆圈表示预测标签，黑叉表示错误标签。
The damage detection accuracy of each dataset is shown in the subfigures. NN : neural network; PC, principal component.
各数据集的损伤检测精度见下图。NN：神经网络；PC：主成分。

Figure 16 shows the results of model updating with damage localized via PGML. With incorrect damage locations excluded in model updating, the identified damage severities at correct sites generally get closer to the correct values as indicated by the reference lines.
图 16 显示了通过 PGML 定位损害的模型更新结果。在模型更新中排除了不正确的损坏位置后，正确位置上识别出的损坏严重程度通常更接近参考线所示的正确值。
The distribution of the sum of errors is plotted in Figure 17. It shows that without the regulation on damage locations from PGML, many of the error sums (

20 %

) are above 0.2 ; however, PGML reduces the error sums of most cases significantly such that most of them (

93.3 %

) are below 0.1. Hence, it can be concluded that the PGML can largely improve the damage identification performance.
误差总和的分布如图 17 所示。图 17 显示，如果没有 PGML 对损坏位置的规定，许多误差和（

20 %

）都在 0.2 以上；但是，PGML 大幅降低了大多数情况下的误差和，使大多数误差和（

93.3 %

）都低于 0.1。因此，可以得出结论，PGML 可以在很大程度上提高损伤识别性能。

Conclusion 结论

This study proposes a PGML method to integrate the data-driven SHM with FE model updating, so that the merits of the two categories of SHM approaches can be preserved and the negative influence of their shortcomings can be mitigated.
本研究提出了一种 PGML 方法，将数据驱动 SHM 与 FE 模型更新相结合，从而保留了两类 SHM 方法的优点，并减轻了其缺点的负面影响。
Regarding an established FE model and its updated outputs as an implicit representation of underlying physics of the target structure, this
将已建立的有限元模型及其更新输出作为目标结构基本物理特性的隐含表示，这就需要

Figure 15. Results of damage localization on the three-story building using a PGNN model with physics guidance. The

x

axis is PCI of the data, and the

y

axis is PC2. D

i (i = 1, 2

, and 3 ) denotes the damage cases listed in Table 2.
图 15.使用带物理指导的 PGNN 模型对三层楼进行破坏定位的结果。

x

轴为数据的 PCI，

y

轴为 PC2。D

i (i = 1, 2

和 3 ) 表示表 2 中列出的损坏情况。
(a) illustration of all the data and their labels, and (b)-(d) classification results on the training data, validation data, and testing data, respectively; solid circles with different colors denote the predicted labels and the black cross indicates wrong labeling.
(a) 所有数据及其标签示意图，(b)-(d) 分别为训练数据、验证数据和测试数据的分类结果；不同颜色的实心圆圈表示预测标签，黑叉表示错误标签。
The damage detection accuracy of each dataset is shown in the subfigures. PGNN: physics-guided neural network; PC, principal component.
各数据集的损伤检测精度见下图。PGNN：物理引导神经网络；PC：主成分。
study implements PGML through incorporating the results of FE model updating into a NN model to have a PGNN model.
该研究通过将 FE 模型更新的结果纳入一个 NN 模型来实现 PGML，从而得到一个 PGNN 模型。
On one hand, the original feature vector is extended with output labels predicted from FE model updating; on the other hand, the outputs of FE model updating, including the output scores and output labels, are both incorporated into the objective function of the modified NN model.
一方面，原始特征向量通过 FE 模型更新预测的输出标签得到扩展；另一方面，FE 模型更新的输出，包括输出分数和输出标签，都被纳入修正后的 NN 模型的目标函数中。
These modifications of the NN model with respect to its inputs and objective function formulation are expected to improve the physicsrelated consistency of the learned model.
对 NN 模型的输入和目标函数表述进行这些修改，可望改善所学模型在物理学方面的一致性。
In addition, with the most probable damage location identified from PGML, FE model updating can be rerun with constrained targets, which may largely improve the efficiency of solving the optimization problem and increase its accuracy.
此外，通过 PGML 确定最可能的损坏位置后，就可以利用约束目标重新运行 FE 模型更新，这可以在很大程度上提高优化问题的求解效率和准确性。
Hence, the adverse effects of modeling error can be significantly mitigated.
因此，建模误差的不利影响可以大大降低。

A numerical case study is implemented with an FE model of a steel pedestrian bridge where the elastic modulus of the frame members are set as the target of FE model updating and damage identification. Features are extracted from modal properties following
通过对钢结构人行天桥的有限元模型进行数值案例研究，将框架构件的弹性模量设定为有限元模型更新和损伤识别的目标。从模态特性中提取的特征如下

Figure 16. Model updating results of the three-story building using the four-degree of freedom numerical model with damage location identified by PGML.

δ_{α_{i}}

denotes the damage severities on

k_{i} (i = 1, 2

, and 3 ) evaluated through comparing with the updated stiffness values of the intact structure. The two reference lines indicate the damage severities designed in the experimental tests, that is -0.22 for moderate damage and -0.44 for severe damage.
图 16.使用 PGML 确定损伤位置的四自由度数值模型对三层建筑进行模型更新的结果。

δ_{α_{i}}

表示

k_{i} (i = 1, 2

和 3 ) 上的损坏严重程度，是通过与完整结构的更新刚度值进行比较而评估得出的。两条参考线表示实验测试中设计的损坏严重程度，即中度损坏为-0.22，严重损坏为-0.44。
See Table 2 for more details about the damage cases.
有关损坏情况的详细信息，请参见表 2。
PGML: physics-guided machine learning.
PGML：物理引导的机器学习。
previous studies in literature.
以前的文献研究。
It shows that before implementing PGML, FE model updating yields misleading values of target parameters due to significant modeling error when establishing the FE model, and an NN model learned with the limited data cannot generalize well when applied on new testing data.
结果表明，在实施 PGML 之前，由于在建立 FE 模型时存在很大的建模误差，FE 模型更新会产生误导性的目标参数值，而且利用有限数据学习的 NN 模型在应用于新的测试数据时不能很好地泛化。
In comparison, PGML not only improves the generality of the
相比之下，PGML 不仅提高了

Figure I7. Distribution of model updating errors on the three-story building (a) without and (b) with PGML regulation. PGML: physics-guided machine learning.
图 I7.模型更新误差在三层楼建筑上的分布（a），无 PGML 调节；（b），有 PGML 调节。PGML：物理引导的机器学习。
learned NN model with testing accuracy increased from 0.8 to 0.94 but enhances the performance of FE model updating in damage detection with largely reduced sum of errors.
学习到的 NN 模型的测试精度从 0.8 提高到 0.94，但在损伤检测方面却提高了 FE 模型更新的性能，大大减少了误差总和。
These improved damage identification outcome validates the effectiveness of PGML in integrating data-driven and physics-based SHM for better performance.
这些改进的损伤识别结果验证了 PGML 在整合数据驱动和基于物理的 SHM 方面的有效性。
An experimental study with a three-story frame structure further validates the effectiveness of PGML in structural damage identification with either data-driven or FE model updating method.
通过对三层框架结构的实验研究，进一步验证了 PGML 在结构损伤识别中使用数据驱动或 FE 模型更新方法的有效性。
Therefore, it can be concluded that PGML has the potential of mitigating the challenge of data insufficiency in datadriven SHM and the potential of reducing the effects of modeling error in FE model updating.
因此，可以得出结论，PGML 有可能缓解数据驱动 SHM 中数据不足的挑战，并有可能减少 FE 模型更新中建模误差的影响。
However, it is noted by the authors that FE model updating outputs may not be the best representation of scientific rules for structural damage evaluation, which will be one of the focuses of future research.
不过，作者指出，有限元模型的更新输出可能无法最好地体现结构损伤评估的科学规则，这将是未来研究的重点之一。

Funding 资金筹措

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by Industrial Ties Research Subprogram of Louisiana State Board of Regents (Project No.
作者披露在本文的研究、撰写和/或发表过程中获得了以下资助：本研究得到了路易斯安那州监管委员会工业纽带研究子项目的支持（项目编号：No.
AWD-001515) and the support of Louisiana Transportation Research Center (Project No. 20-3TIRE).
AWD-001515）和路易斯安那州交通研究中心（项目编号：20-3TIRE）的支持。

ORCID iDs

Zhiming Zhang (D) https://orcid.org/0000-0003-2221-9209
Chao Sun (iD) https://orcid.org/0000-0003-3909-0325

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Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA, USA
美国洛杉矶巴吞鲁日路易斯安那州立大学土木与环境工程系

Corresponding author: 通讯作者：

Chao Sun, Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA.
Chao Sun，路易斯安那州立大学土木与环境工程系，美国洛杉矶 70803，巴吞鲁日。
Email: csun@lsu.edu 电子邮件：csun@lsu.edu

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

Abstract 摘要

Keywords 关键词

Introduction 导言

Methodology 方法

FE model updating FE 模型更新

Feature extension 功能扩展

Physics-based loss function基于物理的损失函数

Implementation in PyTorch用 PyTorch 实现

Numerical study 数值研究

Experimental validation 实验验证

Conclusion 结论

Funding 资金筹措

ORCID iDs

References 参考资料

Corresponding author: 通讯作者：

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

Physics-based loss function
基于物理的损失函数

Implementation in PyTorch
用 PyTorch 实现