Bridge temperature prediction method based on long short-term memory neural networks and shared meteorological data

Abstract

Temperature is an important load factor affecting the structural performances of bridges. The rapid acquisition of bridge temperature data is significant for bridge temperature effect analysis and assessment. On the bases of ground meteorological shared big data, a bridge temperature prediction method based on long short-term memory (LSTM) neural network is proposed. The proposed method is used to investigate the key issues of data preprocessing, model input feature selection, time-length determination, and hyper-parameter preference. Moreover, the proposed method relies on the maximum information coefficient to quantify the strongly correlated features and uses a two-layer deep LSTM neural network to improve the model’s time series information utilization and prediction capability. The constructed neural grid model is experimentally studied and verified based on the long-term measured data of the scaled bridge model in an outdoor environment. Comparative assessment with other typical time series models, such as NARX, RNN, and GRU, demonstrate that the LSTM neural network model exhibits the best generalization ability and highest temperature prediction accuracy, with a maximum absolute error of approximately 2°C and relative error below 5%. The engineering applicability and effectiveness of LSTM for bridge temperature prediction are verified.

Keywords

Structural health monitoring bridge temperature effect long and short-term memory neural networks maximum information coefficient shared meteorological data

Introduction

Temperature is one of the main loads on bridges during service life. Temperature significantly affects the static and dynamic performances of bridges and changes the boundary conditions (Li et al., 2023; Xia et al., 2012; Zhou et al., 2020; Zhou and Yi 2013, 2014). It is also the main cause of damage and performance degradation of bridges (Liu 1991; Xia et al., 2020). The basis of structural temperature effect analysis is to obtain the temperature load that can reflect the real temperature distribution of bridges (Shan et al., 2023a, 2023b). At present, bridge temperatures are mainly obtained from field measurements (Xia et al., 2022; Xu et al., 2010) and finite element numerical calculations based on environmental meteorological parameters (Xia et al., 2013; Zhou et al., 2016). However, field measurement encounters such problems as limited measurement points, high cost, time consumption, and low efficiency. However, the calculation results are relatively different because of the uncertainty in the finite element numerical calculation results and environmental meteorological parameters. Therefore, realizing an accurate, efficient, and low-cost acquisition method for the three-dimensional temperature field of bridges is significant but difficult and challenging as well.

Artificial neural network and the most representative deep learning technology have the advantages of highly nonlinear mapping, self-organizing structure, and black box modeling. Therefore, the complex relationship between system input and output can be approximated substantially. This relationship has been thoroughly studied and widely used. Given the development of digital image and time series processing algorithms in recent years, the theory and application of artificial neural networks in the field of bridge structure structural health monitoring have been studied by many scholars, with main focus on structural damage identification and response prediction. Yang et al. (2020) proposed a method for bridge structural damage identification by combining the convolutional neural network (CNN) and long short-term memory (LSTM) recurrent neural network models. In this method, structural damage mode is classified by the input acceleration vibration response. AlexNet convolutional neural networks were trained by Han et al. (2019) using transfer learning techniques. This method is also used for the rapid identification of cracks, rust, defects, and other diseases on the surface of bridge structures. Gu et al. (2016) proposed a damage detection method combining self-associative neural network (AANN) and singularity detection technology. By taking the modal frequency of the structure under the influence of temperature change as damage-sensitive feature, the Euclidean distance between the network and desired outputs was constructed and used as the damage identification index to detect structural damage. Zhou et al. (2020) proposed an ARIMA model combining a NAR dynamic neural network and classical sequences. In this method, strain signals were decomposed by CEEMDAN and PE. Thereafter, different types of neural networks were used to predict the decomposed signal values, and the predicted bridge strain values of the model were obtained by superposition of the results.

Bridge structure temperature and distribution have evident time-varying characteristics, cumulative characteristics of the time course, and spatial nonlinear distribution characteristics. These characteristics prevent the traditional neural network methods of point-to-point mapping from being used for bridge temperature learning prediction. Whether the existing temporal neural network method can achieve accurate bridge temperature prediction should also be studied in depth, which is difficult and challenging.

This study investigates the high time-series characteristics of bridge temperature and external meteorological factors, as well as the nonlinear relationship. An LSTM method is designed to predict bridge temperature based on ground meteorological shared data. Using the data-driven method, the historical temperature data of the bridge measuring points and the corresponding site meteorological conditions are taken as the model training data. The optimized LSTM model can be trained to achieve long-term real-time prediction of bridge temperature.

The major innovations and contributions of this study are as follows.

(1) The maximal information coefficient (MIC)-based advection search method is used to quantify the time delay and nonlinear correlation of different meteorological environmental factors with respect to the structural temperature to achieve bridge temperature prediction based on the deep learning method.

(2) On the basis of having the measured temperature data of box-section bridges as training data, the neural network method can compensate for the missing data caused by the failure of the monitoring system and achieve long-term stable temperature acquisition of bridge measuring points.

(3) If the structural temperature obtained through finite element calculation is used as training data of the neural network to obtain the temperature of the measuring points for a long time, then the cost of setting up the temperature monitoring system can be saved. Moreover, the calculation cost and efficiency can be reduced and improved, respectively, compared with the traditional method of calculating the temperature field based on finite element and programming software interactive program.

Basic theory of LSTM neural network method

LSTM neural network, as a special recurrent neural network (RNN), can better deal with the long-term dependence relationship of time series data because it effectively solves the problem of gradient explosion and gradient disappearance generated by RNN in the face of long sequence (Hochreiter and Schmidhuber 2010). The LSTM neural network method is widely used in text translation (Ren et al., 2020), emotion recognition (Li et al., 2017), speech recognition (Huang et al., 2017), and other applications. LSTM relies on three unique types of memory units (i.e., oblivion, input, and output gates) to complete the information flow control in the unit. Oblivion gate controls the amount of information removed from the previous time step cell state $c_{t - 1}$ . Input gate controls the amount of information introduced into the current time step cell state $c_{t}$ . Lastly, output gate determines the output content of the current time step hidden state $h_{t}$ . The formula for calculating the cell and hidden states of each gated input and output of the LSTM neuron node is as follows. The internal structure of the cell is shown in Figure 1.

Figure 1.

Internal structure of the LSTM cell.

Oblivion Gate:

f_{t} = σ (W_{f} x_{t} + U_{f} h_{t - 1} + b_{f})

(1)

Input Gate:

i_{t} = σ (W_{i} x_{t} + U_{i} h_{t - 1} + b_{i})

Output Gate:

o_{t} = σ (W_{o} x_{t} + U_{o} h_{t - 1} + b_{o})

(3)

Current time step input:

Z_{t} = \tanh (W_{Z} x_{t} + U_{Z} h_{t - 1} + b_{Z})

(4)

Current time step cell state:

c_{t} = f_{t} ⊙ c_{t - 1} + i_{t} ⊙ Z_{t - 1}

(5)

Current time step hidden state:

h_{t} = \tanh (c_{t}) ⊙ o_{t}

(6)

Gated function sigmoid:

σ (x) = \frac{1}{1 + e^{- x}}

(7)

Activation function:

\tanh (x) = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}

(8)

Where $⊙$ is the Hadamard product of matrix; $\oplus$ stands for performing matrix addition; W, U, and b are the corresponding weight coefficient matrix and bias terms, respectively; $σ$ and $\tanh$ are the gated function sigmoid and hyperbolic tangent activation function, respectively.

The LSTM model training process uses the back-propagation through time (BPTT) algorithm, which can be mainly divided into four steps: (1) calculate the output values of LSTM cells according to the forward calculation method ((1)–(8)); (2) determine the error term of each LSTM cell by reverse calculation, including two back-propagation directions at the time and network levels; (3) calculate the gradient of each weight according to the corresponding error terms; and (4) use the gradient-based optimization algorithm to update the weights.

The wide variety of gradient-based optimization algorithms include the stochastic gradient descent (SGD) method (Duchi et al., 2011), adaptive moment estimation (Adam) method (KingaD, 2015), and root mean square prop (RMSProp) method (Hinton et al., 2006). This study selects Nadam (Dozat 2016), an adaptive moment estimation algorithm for momentum accumulation acceleration, for optimization calculation. The Nadam algorithm adds the accumulation of first-order momentum terms based on the Adam method. This method has the advantage of being good at handling sparse gradients and smooth objectives and is widely used in non-convex optimization problems.

Bridge temperature prediction based on the LSTM model

This section discusses the training data acquisition, model input feature selection, and hyper-parameter optimization methods based on the characteristics of bridge structural temperature and site meteorological time series data, combined with the relevant theories and methods introduced in previous section. The overall research flowchart is shown in Figure 2.

Figure 2.

Research methodology diagram.

Training data acquisition

The proposed model is trained to achieve sequence-to-sequence prediction. Multiple time steps are output by inputting site meteorological data with multiple time steps. The training data acquisition method is described as follows.

Model input: Site meteorological data

According to the knowledge of heat transfer, the actual bridge temperature boundary conditions are mainly controlled by radiation and convection heat transfers. The influence of random factors, such as cloud amount, atmospheric conditions, and weather conditions, is crucial, given the random characteristics of solar radiation intensity change (e.g., dynamic, time-varying, multi-disturbance quantity, and uncertainty). Therefore, six meteorological parameters, namely, ambient air temperature, zenith angle, rainfall, relative humidity, cloud cover, and site wind speed, are proposed to be selected as reflection factors of weather type and radiation amount. Meteorological data can be obtained using the following two methods.

① Shared data from meteorological websites. According to the distance between the site and the nearest meteorological station for which meteorological data are available, site meteorological data can be obtained according to the range of values provided in the literature (Xue 2020). The nearest meteorological station observation is directly selected to replace the site meteorological data or spatial interpolation is used to obtain the site meteorological data.

② Site installation with temperature and humidity sensors, anemometer, wide-angle camera, and other meteorological sensing elements of the small weather station for field measurements.

Model output: Bridge structural temperature

Finite element method

An ANSYS-MATLAB interaction program is written to establish a two-dimensional thermal analysis model of a concrete box girder bridge. Moreover, the hourly structural temperature field data under the corresponding weather conditions are cyclically calculated for later training of the neural network model.

Considering the two-dimensional conduction process of bridge temperature, the finite element model adopts a 4-node PLANE55 plane thermal element, which has one temperature degree of freedom per node and can be used for steady-state or transient thermal analysis problems. Simultaneously, an additional layer of radiation element LINK32 coupled with the thermal analysis element nodes is also attached to the inner surface to consider the influence of thermal radiation between the inner surfaces of the box. The calculation flow of the finite element method is shown in Figure 3.

Figure 3.

Flowchart for calculating bridge temperature using the finite element method.

Field measurement

Temperature acquisition at measurement points is performed by arranging temperature sensors on the bridge structure using automated instruments. Temperature data time interval should be the same as that of the meteorological series data. Service life of the sensor elements and data acquisition equipment of the bridge SHM system often does not match the service life of bridges. By using the temperature data already collected by the SHM system to build a neural network prediction model, bridge temperature can be effectively obtained during or after the failure of the SHM system.

Determination of input characteristics and delay length of the proposed model based on MIC

A suitable selection of input features can reduce the dimensionality of intermediate parameters and increase the interpretability and computational efficiency of the model. The MIC method (Reshef et al., 2011) was used to conduct multi-variable correlation analysis for the global sample data, and the meteorological factors that had a strong correlation with the bridge temperature value were quantitatively selected.

MIC is the quantification of the relative entropy between the joint and marginal distributions of two random variables. This method is used to measure the degree of linear or nonlinear correlation between two variables x and y. The basic idea is as follows: discretize the two variables in a two-dimensional space and represent them with a scatter plot, divide this two-dimensional space into certain interval numbers in the x and y directions, observe the dispersion of the scatter points in each lattice to simplify the calculation of the mutual information coefficient, and divide it by $l o g_{2} (\min (X, Y))$ for normalization. MIC is calculated using the following equations:

I (x; y) = \int p (x, y) \log_{2} \frac{p (x, y)}{p (x) p (y)} d x d y

(9)

MIC (x; y) = \frac{\max_{a * b < B} I (x; y)}{\log_{2} \min (a, b)}

(10)

where

I (x; y)

is the information coefficient,

p (x, y)

is the joint probability between variables x and y, a and b denote the number of division lattices in the x and y directions, respectively, set a*b<B, and set the size of B to approximately 0.6 times the number of data samples.

The setting of one batch time step of the model is related to the lag relationship between the dependent and independent variables. Moreover, and the preceding MIC can be used to solve the delay length. The variable x or y is translated forward or backward by setting an appropriate flat step length, and MIC is calculated for each translation. The number of steps with the maximum MIC value is the time delay between x and y.

Data Z-score standardized processing

Before entering the neural network, the time series data need to be processed with Z-score standardization and converted into dimensionless data with a mean of 0 and variance of 1. The objective is to ensure the comparability of data and improve the stability of the neural network. The Z-score standardized formula is as follows:

x^{'} = \frac{x - μ}{σ}

(11)

where μ is the mean value of the overall data, σ is the standard deviation of the overall data, x is the pre-treatment sample data, and x^’ is the post-treatment sample data.

Hyper-parametric optimization method based on grid search

Many parameters are involved in the construction of the LSTM prediction model, among which the number of neurons in the hidden layer n, batch size, and dropout rate of neurons are the most critical (Priyadarshini and Cotton. 2021). To achieve superior prediction effect, this study adopts the multilayer grid search (Lavalle et al., 2004) for parameter optimization. Grid search is a simple and practical method for parameter optimization that is easy to compute in parallel and has controllable computational time. The combination of the minimum value of the objective function is the optimal parameter based on the exhaustive traversal of the objective parameters. The calculation process of this method is shown in

Figure 4. The specific implementation steps are as follows. (1) Fix the number of random seeds to ensure that the model is reproducible. (2) Preset the value range of three parameters based on experience (Dernoncourt et al., 2016). (3) Iterate over the values of each parameter and save the corresponding model parameters and model accuracy. (4) Sort all results in descending order according to prediction accuracy; the one with the highest accuracy is the preferred model parameter.

Figure 4.

Flowchart of the grid search parameter preference method.

The objective function is selected using the mean absolute error (MAE), which can be expressed as follows:

MAE = \frac{1}{n} \sum_{i = 1}^{n} | y_{i} - {\hat{y}}_{i} |

(12)

where

y_{i}

is the real value of the structure temperature,

{\hat{y}}_{i}

is the prediction result of the neural network, and n is the number of samples.

Experimental verification

This section tests and verifies the proposed neural network model and parameter optimization method based on the case of a box-section bridge. The configuration of the computer used for the experiments is as follows: processor is Intel CORE i5-8265U with CPU frequencies of 1.60 GHz and 1.80 GHz, running memory is 8.00 GB, operating system is Windows 10 (64-bit), and programming language is Python 3.8.6 (64-bit).

Training data sources

Structural temperature data were obtained from a scaled model test system of a concrete box girder bridge at the Key Laboratory of Subtropical Architecture, South China University of Technology, Guangzhou. The box girder bridge model, as shown in Figure 5, has a span of 1500 mm, with a northeast-southwest orientation, and a 5° angle between the longitudinal axis and the north-south direction. To simulate the convection state of a real bridge box chamber, the two ends of the scaled model box are sealed with polystyrene foam boards. The cross-sectional dimensions and locations of the temperature measurement points are shown in Figure 6. The temperature measurement points were arranged with K-type thermocouples, and data were collected using a multiplexed automatic temperature acquisition instrument.

Figure 5.

Concrete bridge scale model for collecting temperature data.

Figure 6.

Structure size and temperature measurement point location.

Meteorological data were obtained from Meteoblue, a product of the Swiss Meteorological Company. The website provides meteorological parameters, such as ambient temperature and wind speed, from meteorological stations over the past 30 years. For areas distant from meteorological stations, spatial interpolation processing was used to ensure data acquisition with a spatial resolution of 4–30 km. In this experiment, the observed data from the Wushan meteorological station, which is 500 m away from the site, were used instead of the actual meteorological data of the site. A total of 4104 time steps of meteorological and structural temperature synchronous series data from June to November 2021 with a data interval of 1 h were obtained after processing the singular and missing values.

Neural network input feature selection

A correlation diagram of the variables is plotted on the bases of the preceding data, as shown in Figure 7. The diagonal subgraph shows the density distribution of each variable, and the other subgraphs show the correlation between any two variables. Moreover, Figure 7 shows that the correlation scattering matrix of structural temperature and other variables mostly shows a regular point cloud shape; the more regular the shape, the stronger the correlation. The MIC method was used to quantify the degree of correlation and select the network input features. The translation search window size was set to 6 time steps and step length to 1 h. Taking model temperature measurement point 1 as an example, the calculation results are shown in Table 1.

Figure 7.

Correlation dispersion matrix of each variable.

Table 1.

MIC calculation results.

Steps	Zenith angles	Temperature	Rainfall	Relative humidity	Wind speed	Cloud coverage
0	1	0.703	0.207	0.740	0.519	0.325
1	1	0.636	0.201	0.686	0.425	0.283
2	1	0.529	0.201	0.527	0.31	0.235
3	0.962	0.468	0.208	0.399	0.244	0.249
4	0.825	0.379	0.212	0.337	0.245	0.249
5	0.638	0.329	0.223	0.342	0.304	0.254

The calculated maximum value of MIC is mostly concentrated when the time delay step is 0. That is, there is no hysteresis effect of the temperature of the measurement point relative to the meteorological environment change. The reason is that the measurement point is arranged on the surface of the structure without the internal heat conduction process and can respond immediately to environmental change. Owing to the large fluctuation of wind speed, MIC of this variable with respect to the structure temperature shows a fluctuating trend when the time delay step increases. Referring to the literature (Xu et al., 2016), the top five features with the highest correlation with the class labels were selected as model input features: zenith angle, air temperature, relative humidity, wind speed, and cloud coverage.

Stationary time series processing based on the sliding average method

For time series models, the use of stationary time series data for model training and prediction is an important prerequisite for the applicability of deep learning methods. The two types of time series, air temperature and structural temperature, have significant seasonal trend components, and the non-stationary time series can be pretreated using the simple moving average method for trend removal. The calculation is as follows:

\bar{f_{t}} = (X_{t - w + 1} + X_{t - w + 2} + \dots + X_{t}) / w

(13)

f_{t}^{'} = f_{t} - {\bar{f}}_{t}

(14)

where

{\bar{f}}_{t}

is the seasonal component of the series value at time t,

f_{t}

is the original series value at time t, and

f_{t}^{'}

is the stationary time series data value at time t with the seasonal component removed.

Considering the temperature fluctuation and calculation efficiency, the sliding window size w is set at 48 time steps. After the neural network outputs the prediction results, the results must be added to the corresponding seasonal components to restore the real non-stationary series data of the structural temperature. Owing to the unknown structural temperature in the prediction stage and the lack of statistical annual trend change data, the seasonal components of the structural temperature prediction values are difficult to obtain. Moreover, the prediction results of the neural network cannot be reduced to a non-stationary time series by adding the seasonal components obtained with its own moving average. Therefore, the stabilization of the structural temperature series is performed by the difference between the seasonal component calculated by the sliding average of the temperature data and ambient temperature. Non-stationary reduction is also obtained by adding the forecast results to the seasonal component of the ambient temperature in that period. The effect before and after smoothing of the time series is shown in Figure 8.

Figure 8.

Example of time series smoothing process. (a) Ambient air temperature data.(b) Temperature data of measurement point 1.

Lastly, the ADF method is used to test the stationarity of the time series. Taking the ambient temperature and temperature series data at measuring point 1 as an example, the calculated statistical values t = 10.42 and 8.19 > $t_{0.005, 4104}$ = 3.960, both of which have p = 0, indicate that the null hypothesis of the unit root is rejected at the significance level of 1%. In addition, the processed time series meets the stationary condition, which can be used for model training.

Hyper-parameter selection and network training results

After standardized processing of the sample data, the sample time series data with stationarity and homoscedasticity are divided into 80% training data and 20% test data under the condition of no disorder. According to the parameter dimension requirements of the input layer, the weather sequence data is processed as a three-dimensional tensor of the shape (None, 24, 5). In particular, “24” is the number of time steps within a batch of time series data, which must be larger than the time delay calculated by MIC. Furthermore, “5” is the selected feature number. Given that the input and output time steps are the same size, the LSTM layer is configured to return one value per input time step, with the network parameter return_sequence = True.

The hyper-parameters are selected according to the method described in Section 2.4. Table 2 shows the range of the hyper-parameters used in the multilayer grid search. Taking the sub-model of measurement point 1 as an example, the optimal parameter combination is calculated as follows. The number of neurons in the hidden layer is set to 9, batch size is 5, and neuron discard rate is 0.0, when the model accuracy is the highest and the model validation set loss function is 0.030734.

Table 2.

Grid search parameter range.

Hyper-parameter types	Range of grid search parameters
Number of neurons in the hidden layer	6, 7, 8, 9, 10
Batch size	5, 10, 15, 20
Neuron dropout rate	0.0, 0.1, 0.2, 0.3, 0.4, 0.5

Using the test data, the temperature prediction results of the preferred LSTM network for the five measurement points are shown in Figure 9. Note that for the five measurement points, the peak moments of the structure temperature predicted by the LSTM model are consistent with the real situation. Except for the data of individual peak points of air temperature in the roof and box, the absolute error of the model prediction is mostly controlled within 1°C, which can better reflect the trend of structural temperature change. On November 20, when rainfall and temperature decreased sharply, the model also responded immediately with superior prediction accuracy. Overall, the predicted temperature of the model is generally lower than the actual temperature. The reason is that for the time series model, the predicted value of the next moment generally depends on the historical data of the previous moment. That is, the model has a certain prediction inertia. Therefore, when the ambient temperature and wind speed fluctuate significantly, the gradient of the activation function disappears and the overall prediction value of the model will be conservative.

Figure 9.

Comparison between the predicted and measured temperature of measurement points. (a) Measurement point 1 (b) Measurement point 2 (c) Measurement point 3 (d) Measurement point 4 (e) Measurement point 5.

Contrast analysis

In addition to the RNN model mentioned in Section1, this study also compares the LSTM model with two typical time series prediction models. (1) Gated recurrent unit (GRU) model. The GRU model is a variant of the LSTM model. Unlike LSTM, GRU uses only two gating units (i.e., update and reset gates) to process the state information, and the structure and calculation are relatively simple (Dey and Salemt 2017). Among them, the update gate is used to control the state of the previous moment and the extent to which the information is brought into the current state. By contrast, the reset gate is used to control the extent to which the state information of the previous moment is disregarded. (2) Nonlinear auto-regressive with exogenous inputs (NARX) model. The NARX model is a recursive dynamic neural network. It is based on the BP neural network by introducing the output of the neural network as external feedback into the input layer of the neural network and adding s unit delay. Compared with the static neural network, its output considers the information from the beginning of t-s to the time step of t at the same time and can be regarded as a multilayer perceptron with input delay and short-term memory capability.

Calculation accuracy and time consumption were used to evaluate the prediction effect of the model. The absolute errors of the four neural network models for predicting the temperature at measuring point 1 are shown in Figure 10. The errors have certain random fluctuations, among which the RNN and NARX models have the largest errors, the GRU model has the middle error, and LSTM has the highest prediction accuracy; its error is significantly smaller than the other three models. The computational time consumption of the four models using the same computational hardware is listed in Table 3. The computation time of the LSTM and GRU models is longer, among which LSTM has the longest running time, which is related to the number of its gating units. However, the computation time of the LSTM model in this case is only 61.52 s. In summary, the LSTM model exhibited the highest accuracy and better overall prediction performance compared with the other three models.

Figure 10.

Absolute errors of the prediction of the four neural network models for measurement Point 1

Table 3.

Comparison of the Performances of the LSTM, NARX, GRU, and RNN Models.

Neural network types	Running time (s)	MAE	MSE
LSTM	61.52	0.639	0.615
GRU	58.01	0.770	0.917
NARX	30.67	2.584	10.677
RNN	19.91	2.056	8.932

MSE stands for mean square error.

Conclusion

In response to the shortcomings of the traditional methods for obtaining bridge temperature loads based on field monitoring or finite element analysis, a real-time structural temperature prediction method based on the LSTM neural network is proposed in this study. The proposed method can be used for the continuous acquisition of long-term structural temperature using ground meteorological shared data. The main research findings are as follows.

(1) The MIC method is used for model feature selection of LSTM neural network method. Compared with the application of the Pearson and Spearman correlation coefficients, which are limited to linear and simple nonlinear relationships, MIC can identify the nonlinear relationships of parameters and achieve an effective dimensionality reduction of the feature parameters.

(2) The LSTM model can respond better to the trend of structural temperature change. The proposed model accurately predicts the temperature measuring points of the bottom plate, top plate, east web, and west web; and the predicted peak moments of structural temperature are consistent with the real situation. The maximum absolute error of the model occurs at the daily maximum temperature, and the maximum absolute error is about 2°C, and the relative error is 5%, which can meet the engineering requirements.

(3) The maximum information coefficient is used to quantify the strong correlation features, and the two-layer depth LSTM neural network is built. Its unique gate structure is used to improve the information utilization and prediction ability of the LSTM model to the time series, so that the proposed method has a high engineering usability. Compared with the typical time series models, such as RNN, GRU, and NARX, the LSTM model has an overall better prediction performance.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (52078220), the International Science & Technology Cooperation Program of Guangdong Province (2023A05050501 55), and the International and Hong Kong, Macao and Taiwan Talents Communication Program of Guangdong Province.

ORCID iD

Linren Zhou

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