Real-Time Intelligent Prediction Method of Cable’s Fundamental Frequency for Intelligent Maintenance of Cable-Stayed Bridges

Abstract

Cable’s fundamental frequency (CFF) is an important characteristic of the working state of long-span cable-stayed bridges. The change in the bridge’s temperature field will influence CFF by altering the cable’s tension and the cables’ sags. An accurate regression model between the temperature-induced variation of CFF and the real-time changing temperature field should be established. Then, the reference value of the temperature-induced variation of CFF can be obtained after inputting the real-time temperature data. In this study, an intelligent real-time prediction model for CFF is proposed based on the full-bridge temperature field, including the average temperature of the main beam, the vertical temperature difference of the main beam, and the temperature of the cable tower. Besides, a machine learning method named the long short-term memory (LSTM) network is exploited to ensure the nonlinear fitting performance of the model, and a paradigm for optimal hyperparameter selection and training strategy selection is provided. To verify the superiority of the LSTM-based model, the output accuracy of the linear regression, BP network, and LSTM network was tested and compared using the monitoring data collected from cable sensors in the main span and side span, which provides an important basis for the intelligent maintenance and sustainable operation of the bridge cables.

1. Introduction

The fundamental frequency of the cable is a comprehensive reflection of the cable health state and the working performance of the cable-stayed bridge [1,2]. Sensing the value of cable’s fundamental frequency (CFF) of a long-span cable-stayed bridge is not only an important part of ensuring the safety and comfort of the traffic flow but also a crucial part of ensuring the healthy service of each component of the bridge [3]. In the design and operation of long-span cable-stayed bridges, it is necessary to ensure that the first-order vibrational frequency (fundamental frequency) of the cable fluctuates within the operating limits, and to judge the service behavior of the cables, a reference value is needed for comparison with the measured value [4,5]. Therefore, during bridge service, the CFF should be accurately predicted in real-time as the benchmark value to facilitate the management and maintenance of cables.

It is known that in engineering practice, the variations in the environmental temperature also exert a non-negligible influence on the static and dynamic behaviors of cable structures [6]. Previous investigations have shown that natural frequencies, mode shapes, damping ratios, elastic modulus, and even boundary conditions are all relevant to temperature variations, more or less [7]. In recent years, the thermal effects on the vibration behavior of cable-stayed structures have attracted extensive attention from scholars [8,9,10,11]. Treyssède [8] studied the effect of thermal stress on free linear vibrations of cables and proposed an analytical model extending Irvine’s theory to thermoelasticity. Montassar et al. [9] developed a nonlinear analytical approach to calculate variations of stay cables’ tensions under uniform temperature changes based on the elastic catenary cable theory. Zhao et al. [10] investigated the free and forced nonlinear vibrations of the suspended cable with thermal effects. It was found that some qualitative and quantitative changes in the vibration behaviors were induced by temperature variations. Ma et al. [11] further studied the influences of environmental temperature variation on the static and dynamic behaviors of cables and developed a very accurate method for identifying the tension of a cable while considering environmental temperature fluctuation. These studies discussed the temperature effect on the nonlinear dynamics of cables deeply and pointed out that the effects of environmental temperature on the cables’ frequencies are because of both the alteration of the cables’ tension and the change of the cables’ sags [10,11], which contributed significantly towards the understanding of the effect of thermal effects on the static and non-linear vibration behavior of cables.

For the cable-stayed bridge, the operating condition of cables is jointly determined by the behavior of beams and towers. Although the temperature of the main girder is the most important factor affecting the CFF [12], the temperature-induced variation of CFF is likely caused by the time-varying temperature field acting by the main girders, towers, and cables. The temperature field of the tower is relatively simple, while the temperature distribution of the main girder, especially the steel box girder, is relatively complex due to numerous members, and the evaluation of the main girder temperature has a direct impact on the modeling accuracy of the dynamic performance of the cable [13]. Based on the monitoring data of the full-bridge temperature field, scholars have studied the relationship between the temperature characteristics and the thermo-induced effects of cable-stayed bridges [14,15,16]. It was found that there was an obvious linear correlation between the thermo-induced effect and the main girder temperature. Besides, several regression models between the average temperature of the main girder and the temperature-induced variation of CFF were established, and regression values of CFF were used as the benchmark to evaluate the cable state. However, considering the complexity of temperature features and the time lag between the temperature change and temperature-induced variation of CFF, the correlation shows a strong high-order nonlinearity. The output result of the model obtained by linear regression has great amplitude error and phase error [17]. Therefore, there is still much space for improving the output accuracy of the existing model to meet the strict requirements for cable management and maintenance. Recently, based on the massive monitoring data available from structural health monitoring (SHM) systems, machine learning, and deep learning have been gradually applied to civil engineering as novel data-driven paradigms. Neural networks have been proven to be a powerful fitting tool that can express highly nonlinear relationships, by which complex input-output models between variables can be built [18,19]. Deep learning can undoubtedly act as a powerful tool to solve nonlinear modeling in this study.

In this paper, we use the long-term monitoring data collected by a bridge health monitoring system to analyze the changing state of CFF of a long-span cable-stayed bridge in the time-varying temperature field. Besides, an optimal data compression pattern of the bridge temperature is proposed, and high-precision modeling of CFF is conducted based on deep learning, which provides an important basis for the real-time condition assessment of the bridge. The paper is organized as follows. Section 2 introduces the engineering background and the data sources, and briefly analyzes the correlation between the bridge temperature field and CFF. The intelligent real-time prediction model of CFF is provided in Section 3. Finally, conclusions drawn from this study are given in Section 4. The proposed method can provide references for the intelligent maintenance and sustainable operation of bridge cables.

2. Correlation of Temperature-Fundamental Frequency for Cable-Stayed Bridges

2.1. Case Study Bridge and Its Monitoring Details

The research data in this paper are obtained from the health monitoring system of a long-span cable-stayed bridge in 2020. Figure 1a shows the elevation of the bridge. The bridge health system has thermal sensor T1 on the cable towers, temperature measurement points S1–S12 on the main girders in the main span; and cable accelerometers P1 and P2 in the side span and the main span, respectively. Figure 1b shows the cross-section of the main girder in the mid-span with sensor deployment.

As shown in Figure 1b, there are 12 thermal sensors in the mid span of the bridge, which are named S1–S12. S1–S8 are located in the top plate of the box girder, and S9–S12 are located in the bottom plate of the box girder. It can be speculated that S1–S8 are subjected to the strongest effect of solar radiation, and S9–S12 are subjected to the weaker effect of solar radiation.

The sampling frequency of the cable tower temperature sensor T1 is 1 Hz, the sampling frequency of the main beam temperature sensors S1–S12 is 1 Hz, and the sampling frequency of the cable accelerometers P1 and P2 is 100 Hz.

2.2. Temperature Data Compression

For the main beam temperature, there are 12 measurement points, and using all the data from multiple sensors would make modeling difficult. As shown in Figure 2, on a given day, for example, the temperature magnitudes of the two sensors S4 and S5 in the top plate are essentially the same, which indicates that the temperature sensor data at the same height will be strongly correlated. Retaining many temperature variables will cause the input of the thermo-deflection model to become extremely complex and make it difficult to train the model for learning [20,21], so we should exploit correlations to compress the temperature field data.

Subject to the light conditions, the temperature of the top plate of the box girder differs greatly from that of the bottom plate. In summary, sensors located at the same height have mutual substitutability, while the bottom and top plate temperatures are not interchangeable. Therefore, we keep sensor S4 representing the top plate temperature and sensor S10 for the bottom plate temperature. According to Ref. [22], the main beam temperature field influences the thermochromic effect on bridge members mainly through the average temperature and the vertical temperature difference. Firstly, temperature data collected by S4 and S10 should be preprocessed to repair data anomalies, such as outliers and baseline shifts [23]. Then, two sets of temperature variables are obtained by using the average value of S4 data and S10 data to represent the average main beam temperature and the difference between S4 and S10 to represent the vertical temperature difference of the main beam. The temperature variables are resampled at ten-minute intervals to compress the data while avoiding local information loss [17]. The main beam average temperature variable is expressed as ${\mathit{W}}_{\mathbf{1}}$ (as indicated in Figure 3) and the main beam vertical temperature difference variable is expressed as ${\mathit{W}}_{\mathbf{2}}$ (as indicated in Figure 4), and ${\mathit{W}}_{\mathbf{1}}$ has 432,000 data points throughout the year.

The influence of the tower on the bridge mainly lies in its overall temperature variation, so the repaired temperature data from sensor T1 is directly used to represent the tower temperature and resampled once every ten minutes, which is named ${\mathit{W}}_{\mathbf{3}}$. There are also 432,000 data points available in ${\mathit{W}}_{\mathbf{3}}$ throughout the year, as shown in Figure 5.

Besides, according to the monitoring data, the temperature of the sensor located at the top plate of the box girder is higher, which is due to the direct sunlight. The temperature of the top plate in a single day is nearly 10 °C higher than that of the bottom plate with distinct differences.

2.3. Extraction for the Temperature-Induced Variation of CFF

As presented in Figure 1a, the cable sensor P1 in the side span and the cable sensor P2 in the main span were selected for analysis and calculation. To keep the data specifications consistent, the acceleration data of the two cable sensors P1 and P2 are analyzed by power spectrum density (PSD) for a ten-minute interval, which is used to extract the vibration characteristics of the cable [24]. For the power spectrum data of the two sensors as in Figure 6, we take the first-order frequency (fundamental frequency) for analysis.

Figure 7 shows the fundamental frequency curves of the two sensors in a single day, including 144 data points. It can be observed that the trend of fundamental frequency variation in a day has similar characteristics with the trend of temperature variation in a day, but there are still residual effects caused by other disturbing factors. The low-frequency information induced by temperature is further purified using the method of wavelet packet decomposition [25]. The selected wavelet base type is db5 with the decomposition at level 5. As shown in Figure 7, the wavelet-extracted temperature-induced effects are consistent with the temperature variation trend.

The fundamental frequency obtained from sensor P1 is named variable ${\mathit{F}}_{\mathbf{1}}$, and the fundamental frequency obtained from sensor P2 is named variable ${\mathit{F}}_{\mathbf{2}}$. The time series curves of ${\mathit{F}}_{\mathbf{1}}$ and ${\mathit{F}}_{\mathbf{2}}$ in one year are shown in Figure 8, and each of the variables has 432,000 data points in the whole year.

2.4. Temperature-CFF Correlation

We first perform linear regression analysis of the temperature-induced variation of CFF data from sensors P1 and P2 with the main beam temperature data. As shown in Figure 3, Figure 4, Figure 5 and Figure 8, the time-series data for the whole year are divided into 75% training set and 25% test set, respectively. The training set is used to build the regression model and the test set is used to test the model. Figure 9 illustrates the correlation scatter plots and linear regression analysis results of the two data sets ${\mathit{W}}_{\mathbf{1}}$-${\mathit{F}}_{\mathbf{1}}$ and ${\mathit{W}}_{\mathbf{2}}$-${\mathit{F}}_{\mathbf{2}}$.

As in Figure 9a, the average temperature of the main beam is negatively correlated with CFF in the side span, and the goodness of fit of the linear regression is only 0.436; as in Figure 9b, the average temperature of the main beam is positively correlated with CFF in the main span, and the goodness of fit of the linear regression is only 0.611. Both show significant non-linear characteristics in the correlation with the average temperature of the main beam, and the linear regression results are not ideal. Moreover, we bring the average main beam temperature of the test set into the two regression equations in Figure 9, and the regression values are compared with the true values as shown in Figure 10.

As shown in Figure 10, the temperature-induced variation of CFF obtained from the linear regression model based on single-sensor data is greatly different from the measured values. The regression values have obvious defects, which cannot meet the engineering requirements. Therefore, it is necessary to explore a better regression scheme to predict the temperature-induced variation of CFF, and compare and analyze the error values of different methods in a uniform manner.

3. Intelligent Real-Time Prediction Method for CFF

3.1. LSTM-Based Real-Time Prediction Model

If the complexity and volume of the data set are high, the non-linear relationship among variables will be strong, and the traditional regression models cannot achieve high-quality learning tasks [26]. Therefore, we need more powerful tools and richer input information.

The richer input information refers to the import of the vertical temperature difference of the main beam and the tower temperature in addition to the average temperature of the main beam. As shown in Figure 11, the LSTM network was chosen as the technical tool for this chapter because of its hidden units with temporal memory capability, leading to its powerful function in time series modeling [27,28]. LSTM network is a special recurrent neural network (RNN) designed to solve the problem of long-term dependency of RNN and alleviate the vanishing gradient problem caused by the back-propagation during training. Its unit is usually composed of a cell, an input gate, an output gate, and a forget gate [29]. The cell remembers values at arbitrary intervals and three gates regulate the inflow and outflow of information. During deep learning, weights are updated and some information can be forgotten outright, while others can be preserved [30]. The LSTM network unit in Figure 11 can be described by Equations (1)–(6).

(1)$\mathrm{Forget} \mathrm{gate}: {\mathit{f}}_{\mathit{t}}=\mathit{\sigma }\left({\mathit{W}}_{\mathit{f}}\left[{\mathit{h}}_{\mathit{t}-1},{\mathit{x}}_{\mathit{t}}\right]+{\mathit{b}}_{\mathit{f}}\right)$

(2)$\mathrm{Input} \mathrm{gate}: {\mathit{i}}_{\mathit{t}}=\mathit{\sigma }\left({\mathit{W}}_{\mathit{i}}\left[{\mathit{h}}_{\mathit{t}-1},{\mathit{x}}_{\mathit{t}}\right]+{\mathit{b}}_{\mathit{i}}\right)$

(3)$\mathrm{Candidate} \mathrm{memory} \mathrm{unit}: {\tilde{\mathit{C}}}_{\mathit{t}}=\tanh \left({\mathit{W}}_{\mathit{c}}\left[{\mathit{h}}_{\mathit{t}-1},{\mathit{x}}_{\mathit{t}}\right]+{\mathit{b}}_{\mathit{c}}\right)$

(4)$\mathrm{Current} \mathrm{time} \mathrm{memory} \mathrm{unit}: {\mathit{C}}_{\mathit{t}}={\mathit{f}}_{\mathit{t}}·{\mathit{C}}_{\mathit{t}-1}+{\mathit{i}}_{\mathit{t}}·{\tilde{\mathit{C}}}_{\mathit{t}}$

(5)$\mathrm{Output} \mathrm{gate}: {\mathit{o}}_{\mathit{t}}=\mathit{\sigma }\left({\mathit{W}}_{\mathit{o}}\left[{\mathit{h}}_{\mathit{t}-1},{\mathit{x}}_{\mathit{t}}\right]+{\mathit{b}}_{\mathit{o}}\right)$

(6)$\mathrm{Output}: {\mathit{h}}_{\mathit{t}}={\mathit{o}}_{\mathit{t}}·\tanh \left({\mathit{C}}_{\mathit{t}}\right)$

where $\mathit{\sigma }$ denotes the Sigmoid function, $·$ denotes the dot product. ${\mathit{f}}_{\mathit{t}}$, ${\mathit{i}}_{\mathit{t}}$, ${\mathit{o}}_{\mathit{t}}$ and ${\mathit{C}}_{\mathit{t}}$ denote the activation vectors of the gates and the memory unit at time $\mathit{t}$, respectively. ${\mathit{W}}_{\mathit{i}}$, ${\mathit{W}}_{\mathit{f}}$, ${\mathit{W}}_{\mathit{o}}$, ${\mathit{W}}_{\mathit{c}}$, ${\mathit{h}}_{\mathit{t}}$ and ${\mathit{b}}_{\mathit{i}}$, ${\mathit{b}}_{\mathit{f}}$, ${\mathit{b}}_{\mathit{o}}$, ${\mathit{b}}_{\mathit{c}}$ denote the model parameters of the input vector and the hidden layer vector for the gates and the memory unit, respectively. This paper introduces the LSTM regression network to the scenario of CFF real-time prediction, and offers a paradigm for optimal hyperparameter selection and training strategy selection (e.g., the setting of hidden layers, and the data input-output strategy), which provides a novel perspective on the application of LSTM in practical engineering.

Figure 12 presents the CFF real-time prediction model based on the LSTM neural network. The dataset is first converted to the supervised learning mode. Then the data is normalized and fed into a network with two LSTM hidden layers [31]. Finally, the regression prediction values of CFF are obtained after inputting individual values through the fully connected layer and performing the inverse normalization. It should be noted that, before the neural network is formally tested, iterative training of the model is first performed using the training set. There are 30 cells in the input layer of the neural network, so the number of cells in the hidden layer is 64 cells, which is twice the input layer and according to 2/4/8/…/32/64. We tested the networks to compare whether different hidden layers can meet the requirements of modeling accuracy. Set the accuracy of that with two hidden layers as 1, the normalization accuracy of different networks is shown in Table 1.

It can be observed that when the number of the hidden layer exceeds 2, the accuracy of the model does not enhance significantly as the number increases. Therefore, the LSTM neural network with two hidden layers is the best choice in this study.

3.2. Performance Validation of the LSTM Model

We first use the average temperature of the main beam ${\mathit{W}}_{\mathbf{1}}$ in the LSTM model to verify that the LSTM network has the best performance with the same input data. In addition to comparing the results with linear regression, we also perform the same task using a backpropagation (BP) neural network, which is a traditional machine learning method, to illustrate the superiority of the LSTM [20].

First, the dataset is converted into a supervised learning mode. For the temperature effect, it has been shown that the input with previous 5-h data from the current moment is better, so we use the main beam average temperature ${\mathit{W}}_{\mathbf{1}}$ from 5 h before the current moment (i.e., the first 30 data points) as the input. Table 2 shows the data input-output strategy of the neural network used in this subsection by enumerating from the 30th moment.

The first 75% of the training set of the data is used for network training, and the last 25% of the test set is used for model validation. The LSTM neural input layer is set to 30 units, the input size is 1, and the number of hidden layers is 2, with 64 units in a single hidden layer. After passing through the fully connected layer of the hidden layer, in the output layer, the multidimensional data is regularized into one output data, which is the neural network regression value of the temperature-induced variation of CFF. The LSTM parameters will be continuously optimized using Adam’s algorithm during the backpropagation process. The number of EPOCHs is set to 2000, and the trained LSTM temperature-CFF model will be obtained with good performance [32]. Besides, the BP network model is trained according to similar steps, which will not be discussed in detail in this paper. Figure 13 shows the loss functions of the training process for both models.

After the training process, temperature variables of the test set are brought into the trained BP and LSTM network model. The results of the BP network are shown in Figure 14, and the results of the LSTM network are shown in Figure 15.

Comparing Figure 14 with Figure 10, it can be discovered that results from the BP network are slightly better than those from the binary linear regression model. Further comparing Figure 14 with Figure 15, the effect of the LSTM model is stronger than that of the BP network model, indicating that LSTM has the best regression accuracy when inputting the same scale of temperature information. The errors of different models will be compared in detail in the next section.

3.3. High-Precision CFF Model Based on Temperature Field Data and LSTM

Even with the advanced tool of the LSTM network, the CFF model with the single input of the average temperature ${\mathit{W}}_{\mathbf{1}}$ needs to be modified to enhance its prediction accuracy and meet engineering demands. In this study, it is necessary to import the whole bridge temperature field to the CFF model, including the average main girder temperature ${\mathit{W}}_{\mathbf{1}}$, the vertical temperature difference of the main girder ${\mathit{W}}_{\mathbf{2}}$, and the tower temperature ${\mathit{W}}_{\mathbf{3}}$. The regression results of CFF using the LSTM network with the three temperature variables inputted are shown in Figure 16, respectively.

As shown in Figure 16, the LSTM network can achieve excellent regression results after importing the whole bridge temperature field. Besides, Table 3 lists the regression accuracy of each model with different input temperature variables, including the indicators of average relative error (ARE), the goodness of fit ${\mathit{R}}^{\mathbf{2}}$, and mean squared error (MSE). It can be observed that the LSTM network provides the smallest ARE of the output CFF compared with the linear regression method and the BP network using only ${\mathit{W}}_{\mathbf{1}}$ as input. When inputting ${\mathit{W}}_{\mathbf{1}}+{\mathit{W}}_{\mathbf{2}}+{\mathit{W}}_{\mathbf{3}}$, the prediction accuracy of LSTM can be further improved and ARE of the output ${\mathit{F}}_{\mathbf{1}}$ and ${\mathit{F}}_{\mathbf{2}}$ are reduced to 2.9% and 2.7%, respectively, which is less than one-third of the ARE of the linear regression model with ${\mathit{W}}_{\mathbf{1}}$ as input. Meanwhile, the outcomes of ${\mathit{R}}^{\mathbf{2}}$ and MSE also prove that the deep learning regression model built by the LSTM technique with the full-bridge temperature field as input is the best solution in the study.

In practical engineering, the LSTM-based model proposed in this paper can achieve high-accuracy prediction of CFF using the temperature field data collected by the SHM system of cable-stayed bridges. The vibration performance of the cables can be dynamically updated during operation, thereby realizing the real-time intelligent monitoring of bridge cables. Meanwhile, the models can provide the reference value for cable sustainable operation and condition assessment. When the differences between the predicted and measured CFF values are too large, the SHM system will quickly alert the staff to the inspection and maintenance of the bridge structure, which can provide insight into early warning of cable failures. This will be detailed in the subsequent study by our group. It should be noted that this study only concentrates on the changing state of CFF in the time-varying temperature field. In fact, the cable-stayed bridge is simultaneously subjected to various loading conditions during service, such as temperature, wind, and vehicles [6,33], which may affect the prediction of CFF to some extent. In future research, our group will further investigate machine learning methods for CFF analysis under the combined loading condition by integrating multiple research methods (e.g., finite element analysis, field tests, and SHM data analysis), thus enhancing the application value of the study in practical engineering.

4. Conclusions

This paper utilizes the long-term monitoring data collected by a bridge health monitoring system to analyze the changing state of CFF of a long-span cable-stayed bridge in the time-varying temperature field. An intelligent real-time prediction model for CFF is proposed using the LSTM network to exploit the non-linear correlation between CFF and the bridge temperature field. The modeling accuracy of the linear regression, BP network, and LSTM network was also tested and compared. Besides, a novel data compression method containing the whole bridge temperature information was proposed to simplify the input dataset. The main conclusions are as follows:

• (1). The nonlinear correlation between the temperature-induced variation of CFF and the temperature field of the cable-stayed bridge was verified. The analysis shows that the conventional linear regression method cannot ensure the accuracy of the mapping model between the main girder temperature and CFF to meet the engineering requirements.

• (2). The LSTM network can achieve better nonlinear fitting performance than the BP network, although it is still difficult to build a high-precision regression model between CFF and a single temperature variable. After further importing the full-bridge temperature field data, the output accuracy of the LSTM network can be greatly enhanced with no more than 3% ARE. The proposed method and presented results can guide real-time monitoring, routine inspection, intelligent maintenance, and sustainable operation for bridge cables.

Author Contributions

Y.-Q.L.: writing—original draft preparation; H.-W.Z.: conceptualization, supervision; Z.-X.Y.: methodology; Y.-W.L.: writing—review and editing; Y.Z.: validation; D.-C.Z.: resources. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1. Sensor placement for bridge health monitoring: (a) Elevation of the bridge; (b) Cross-sectional view of the mid-span of the main girder.

Figure 2. Time-temperature curves of main beam sensors in a single day: (a) S4 (top plate of box girder); (b) S5 (top plate of box girder).

Figure 3. [Forumla omitted. See PDF.]-Average temperature of the main beam in one year.

Figure 4. [Forumla omitted. See PDF.]-Vertical temperature difference of the main beam in one year.

Figure 5. [Forumla omitted. See PDF.]-Temperature data of the bridge tower in one year.

Figure 6. Power spectrum analysis results of 10 min interval data from two acceleration sensors: (a) Sensor P1; (b) Sensor P2.

Figure 7. Extraction of temperature-induced variation of CFF from two sensors within a day: (a) Sensor P1; (b) Sensor P2.

Figure 8. Temperature-induced variation of CFF of the two accelerometers over one year: (a) Sensor P1-[Forumla omitted. See PDF.]; (b) Sensor P2-[Forumla omitted. See PDF.].

Figure 9. Scatter plot of correlation between temperature and temperature-induced variation of CFF with linear regression analysis: (a) [Forumla omitted. See PDF.]-[Forumla omitted. See PDF.]; (b) [Forumla omitted. See PDF.] -[Forumla omitted. See PDF.].

Figure 10. Output CFF from linear regression model versus true CFF values from the test set: (a) Sensor P1-[Forumla omitted. See PDF.]; (b) Sensor P2-[Forumla omitted. See PDF.].

Figure 11. A hidden layer unit of the LSTM neural network.

Figure 12. Real-time prediction model of CFF based on LSTM neural network.

Figure 13. Loss function of the training process when only [Forumla omitted. See PDF.] is input: (a) [Forumla omitted. See PDF.]-BP network; (b) [Forumla omitted. See PDF.]-BP network; (c) [Forumla omitted. See PDF.]-LSTM network; (d) [Forumla omitted. See PDF.]-LSTM network.

Figure 14. Prediction values of the BP network with only the average temperature of the main beam input and the measured results: (a) [Forumla omitted. See PDF.] versus side span CFF; (b) [Forumla omitted. See PDF.] versus mid span CFF.

Figure 14. Prediction values of the BP network with only the average temperature of the main beam input and the measured results: (a) [Forumla omitted. See PDF.] versus side span CFF; (b) [Forumla omitted. See PDF.] versus mid span CFF.

Figure 15. Prediction values of the LSTM network with measured results when inputting the average temperature of the main beam: (a) [Forumla omitted. See PDF.] versus side span CFF; (b) [Forumla omitted. See PDF.] versus mid span CFF.

Figure 16. Prediction values of the LSTM network with measured results when inputting bridge temperature field data: (a) [Forumla omitted. See PDF.] versus side span CFF; (b) [Forumla omitted. See PDF.] versus mid span CFF.

Table 1

Accuracy of networks with different hidden layers.

Number of Hidden Layers	Normalization Accuracy
1 layer	0.65
2 layers	1
3 layers	1.01
4 layers	1.015
…	…

Table 2

Inference pattern of the data set over time.

Time Point	Input Data	Output Data
30	${\mathit{W}}_{\mathbf{1}\left(\mathbf{1}\right)}$, ${\mathit{W}}_{\mathbf{1}\left(\mathbf{2}\right)}$, …, ${\mathit{W}}_{\mathbf{1}\left(\mathbf{29}\right)}$, ${\mathit{W}}_{\mathbf{1}\left(\mathbf{30}\right)}$	${\mathit{F}}_{\mathbf{1}\left(\mathbf{30}\right)}$ and ${\mathit{F}}_{\mathbf{2}\left(\mathbf{30}\right)}$
31	${\mathit{W}}_{\mathbf{1}\left(\mathbf{2}\right)}$, ${\mathit{W}}_{\mathbf{1}\left(\mathbf{3}\right)}$, …, ${\mathit{W}}_{\mathbf{1}\left(\mathbf{30}\right)}$, ${\mathit{W}}_{\mathbf{1}\left(\mathbf{31}\right)}$	${\mathit{F}}_{\mathbf{1}\left(\mathbf{31}\right)}$ and ${\mathit{F}}_{\mathbf{2}\left(\mathbf{31}\right)}$
…	…

Table 3

Computational accuracy of different regression models.

Output CFF	Input Temperature	Method	ARE	${\mathit{R}}^{\mathbf{2}}$	MSE
${\mathit{F}}_{\mathbf{1}}$	${\mathit{W}}_{\mathbf{1}}$	Linear Regression	11.0%	0.4343	3.7513 × 10−7
${\mathit{F}}_{\mathbf{2}}$	${\mathit{W}}_{\mathbf{1}}$	Linear Regression	9.7%	0.6273	5.2841 × 10−7
${\mathit{F}}_{\mathbf{1}}$	${\mathit{W}}_{\mathbf{1}}$	BP Network	7.1%	0.6555	2.4029 × 10−7
${\mathit{F}}_{\mathbf{2}}$	${\mathit{W}}_{\mathbf{1}}$	BP Network	6.8%	0.7573	3.4918 × 10−7
${\mathit{F}}_{\mathbf{1}}$	${\mathit{W}}_{\mathbf{1}}$	LSTM Network	5.9%	0.5610	2.9384 × 10−7
${\mathit{F}}_{\mathbf{2}}$	${\mathit{W}}_{\mathbf{1}}$	LSTM Network	5.4%	0.7320	3.7972 × 10−7
${\mathit{F}}_{\mathbf{1}}$	${\mathit{W}}_{\mathbf{1}},{\mathit{W}}_{\mathbf{2}},{\mathit{W}}_{\mathbf{3}}$	LSTM Network	2.9%	0.9002	7.3461 × 10−8
${\mathit{F}}_{\mathbf{2}}$	${\mathit{W}}_{\mathbf{1}},{\mathit{W}}_{\mathbf{2}},{\mathit{W}}_{\mathbf{3}}$	LSTM Network	2.7%	0.9396	9.4930 × 10−8

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