Deep learning-driven forecasting of bridge thermal-induced structural response with feature engineering

Abstract

In order to forecast temperature-induced strains and deflections of bridges, this study introduces a deep learning framework enhanced with feature engineering. Utilizing 1 year of field-measured temperature (input), strain and deflection data (output) from the Queensferry Crossing cable-stayed bridge, an LSTM-Attention model is developed as the core predictive architecture, with its performance evaluated both with and without feature engineering, which comprises lag feature, difference feature, rolling standard deviation and moving average. Meanwhile, comparative analyses against LSTM-Attention model and other proposed models are conducted to assess the contribution of feature engineering and attention mechanisms. Results reveal that feature engineering significantly improves predictive accuracy, particularly when integrated with the LSTM-Attention framework. For both strain and deflection forecasting, the feature-engineered LSTM-Attention model achieves up to an 84.32% reduction in $M A E$ and a 98.39% decrease in $M S E$ , alongside notable improvements in R² and EVS (more than 28.57% and 25.32% respectively). Additionally, the attention mechanism also presents considerable function in improving the predictive accuracy by decreasing more than 38.98% in $M A E$ and 60.38% in $M S E$ for strains and deflections, increasing over 3.13% in $R^{2}$ and 2.06% in $E V S$ for strains, while showing no change in $R^{2}$ and $E V S$ for deflection, due to its constrained potential for enhancement. Overall, integrating feature engineering with LSTM-attention model provides a robust approach for accurate structural response prediction and offers valuable insights for future structural health monitoring applications.

Keywords

bridge engineering thermal effects deep learning feature engineering attention mechanism

Introduction

During the service life of a bridge, structural damage and deterioration are often inevitable under the influence of various loads, mainly arising from vehicles, winds, and temperatures (AASHTO, 2020; Li et al., 2020a). Among these, thermal loads often govern the change of structural response, attracting large attention since 1960s (Cornwell et al., 1999; Huang et al., 2022; Jin et al., 2025; Tomé et al., 2018; Xia et al., 2017, 2022; Yang et al., 2018; Zhang et al., 2025; Zhou et al., 2018). For instance, the Tsing Ma Bridge in Hong Kong showcases monthly thermal effects of the main cables and bridge decks, with a maximum displacement of over 500 mm (Xu et al., 2010; Xia et al., 2013; Zhang et al., 2024a). Also, the cable forces of the third Nanjing Yangtze River Bridge in Jiangsu province, China, exhibits seasonal variation, with an amplitude of approximately 80 kN (Ren et al., 2019). Similarly, a prestressed concrete continuous box girder bridge in the G3 expressway in China experiences a longitudinal temperature-induced strain of 200 με (Gong et al., 2023). Furthermore, a suspension bridge located over the Yangtze River in Jiangsu Province, China, is subjected to a temperature-induced stress of 25 MPa on the upper deck (Xia et al., 2018). Therefore, the investigation of the relationships between thermal effect and structural response is indispensable.

Currently, the data-driven approach serves as one of the primary methods to investigate the relationship between temperature and structural response, which are broadly categorized into three types: statistical methods, signal processing techniques and deep learning models. First, in terms of statistical methods, typically a specific distribution (e.g., normal distribution) or variables are assumed to exhibit certain relationships, such as linearity, offering capabilities in statistical inference and providing high interpretability (Gong et al., 2024; Jiang et al., 2024; Xia et al., 2020b; Yan et al., 2025). However, statistical methods are limited by their dependence on assumptions and may struggle with highly nonlinear systems and intricate feature interactions (Xie, 2011). Second, signal processing techniques employ mathematical tools to extract features, denoise signals, or identify features from raw monitoring data, such as temperature, deflection, and strain, without requiring a physical model of the structure, which are particularly effective for multi-scale behavior interpretation (Ding and Li, 2007; Fallahian et al., 2022; Liu et al., 2011; Xia et al., 2020a; Xu et al., 2020). Nevertheless, one major limitation of this approach is its inability to provide causal insights, along with the risk of generating misleading features or data distortions (Smith, 2003; Vaseghi, 2008). Third, deep learning models, characterized by deep neural network architectures, exhibit superior capabilities in feature extraction and handling data variability, significantly outperforming traditional machine learning methods, and thus, it has been widely adopted in practical engineering applications (Liu et al., 2017; Meng et al., 2025; Zhang et al., 2024b). Despite these advantages, deep learning models still have certain limitations, including high data requirements and significant demands on computational resources (Ahmed et al., 2023; Thompson et al., 2020). Consequently, deep learning models possess substantial potential for future engineering applications.

Furthermore, deep learning models such as long short-term memory (LSTM), bidirectional long short-term memory (BiLSTM), and Convolutional Neural Network (CNN) have been widely applied to the prediction of thermally induced strain and deflection in recent years, achieving notable enhancement in prediction accuracy (Liao et al., 2025; Qu et al., 2024; Wang and Peng, 2024; Zhou et al., 2024). For instance, at the aspect of strain, Chen et al. (2023) apply a hybrid deep learning and autoregressive model with attention to predict the temperature-induced strain, indicating a root mean square error (RMSE) of 4.07. Besides, Zhao et al. (2023) employ a BiLSTM regression model with wavelet transform to forecast temperature-induced strain, demonstrating a $1 - T_{r}$ of 0.9813, which refers to a mean square error (MSE) of approximately 106. Similarly, Zhao et al. (2020) utilize the BiLSTM regression model to predict temperature-induced strain, which shows a normalized MSE of 0.05. Mariani et al. (2024) propose a WaveNet model considering causal dilated convolution layers to forecast temperature-induced strain, presenting a maximum coefficient of determination ( $R^{2}$ ) of 93.26% (values approaching 1 represent optimal outcomes). With regard to deflection, Yue et al. (2021b) employ an Independently Recurrent Neural Network in temperature-induced deflection prediction, which demonstrates an average error of 9.69%. Furthermore, Yue et al. (2021a) adopt an LSTM model in minute-scale temperature-induced deflection forecasting with an $M S E$ of 0.34 and a mean absolute error of 0.45. Additionally, Yue et al. (2022) also apply LSTM to forecast temperature-induced deflection at mid-span, which presents an MSE of 17.14 and a mean absolute error ( $M A E$ ) of 3.58. However, considering the above predictive performance of temperature-induced strain and deflection, there are still opportunities for further enhancement.

To delve into the details, the aforementioned studies on the prediction of thermally induced structural response primarily focus on the design and optimization of deep learning model architectures, such as neural network layer configurations. As a result, one limitation of the model lies in the absence of feature engineering, such as lag features or moving averages, which could potentially improve the model’s performance. When combined with feature engineering techniques, these approaches exhibit significant advantages in capturing temporal patterns, periodic trends, and local variations of thermally induced response. This integration holds great promise for enhancing the sensitivity and recognition capabilities of deep models in identifying low-frequency responses, such as thermal strain and deflection. Taking the moving average as an example, it is capable of improving the robustness of the CNN-BiLSTM-Attention model by reducing random noise interference (Pei et al., 2025). Accordingly, the integration of a deep learning model and feature engineering presents a promising direction for future research.

In this study, a deep learning network, specifically the LSTM-Attention network, is constructed to forecast temperature-induced strains and deflections for the Queensferry Crossing, located in Scotland, UK. Initially, the data preprocessing is introduced, with the model development illustrated, including feature engineering and the deep learning model. Subsequently, a case study is conducted to predict temperature-induced strains and deflections, illustrating the accuracy through four different metrics. Ultimately, a comprehensive comparison analysis is performed to evaluate the performance of feature engineering across multiple existing models.

Methodology

In this section, the data preparations are introduced, including the input temperature field and the output strain and deflection data. Subsequently, feature engineering, particularly lag features, difference features, rolling standard deviation, and moving average, is described to illustrate how it contributes to the predictive performance of the model. Finally, the architecture of the LSTM-Attention model is presented, preparing for the following case study.

Data preparations

For the input dataset, the field-measured temperature data is collected over a 1-year period between July 2020 and June 2021 from multiple sections of the Queensferry Crossing, comprising soffits, bottom chords, deck chords, stay cables, concrete decks, and tower shell sections. Through the investigation of the temperature field, it was observed that the bridge exhibited consistent overall thermal trends, while the tower shells experienced the highest rate of data loss, amounting to approximately 63.79%. Besides, the concrete sections of the bridge, including the deck and tower, experienced notable thermal gradients across their thickness—around 6°C and 4°C respectively. Meanwhile, the temperature variation in steel components remained below 1°C, owing to their superior thermal conductivity and thinner cross-sections, resulting in more homogeneous temperature fields (Xu et al., 2025).

For the output dataset, strain and deflection are processed using separate strategies to ensure that only the temperature-induced components are retained for model training. The strain data use 10-min mean values, which effectively represent the thermal effects because they are less sensitive to short-term dynamic loads. However, for the deflection data, the 10-min mean is insufficient to suppress high-frequency noise caused by traffic and environmental disturbances. Therefore, following the procedure proposed in our previous study (Xu et al., 2020), a multi-resolution wavelet-based method is applied to decompose the measured deflection signals into multiple frequency bands. The low-frequency component corresponding to the diurnal temperature variation is extracted and used as the temperature-induced deflection, while the high-frequency components dominated by vehicle loads and measurement noise are filtered out. This wavelet-based separation ensures that the LSTM network is trained on structural response data primarily governed by thermal actions rather than transient load effects.

Feature engineering

To effectively capture long-term sequence characteristics, feature engineering is incorporated to enhance overall model performance, which consists of lag feature, difference feature, rolling standard deviation and moving average. A summary and comparison of these four features is presented in Table 1. These features are selected due to their general applicability and effectiveness in capturing the temporal dependency, trend variation, and local fluctuation characteristics of structural monitoring time-series data. In particular, the lag feature captures the time-lag relationship between temperature and response, which is a key characteristic of bridge thermal behavior; the difference feature quantifies the rate of temperature change, which directly influences short-term strain or deflection variations; and the rolling standard deviation together with the moving average describe local fluctuations and smoothed thermal trends, reflecting both transient and steady-state thermal effects. Collectively, these engineered features enable the model to learn the dynamic coupling between temperature variation and structural response, thereby improving the accuracy and robustness of prediction.

Table 1.

Summary and comparison of feature engineering.

Feature	Objective	Description
Lag feature	Introduce historical data; capture temporal dependencies	The influence of the past on the present
Difference feature	Mitigate seasonal components; enhance stationarity of the model	The variation of value
Rolling standard deviation	Capture fluctuation patterns; assess instability	The local variability
Moving average	Smooth the data; extract underlying trends	The long-term trends

Lag feature represents one of the most fundamental techniques in time-series feature engineering, which involves introducing past observations (e.g., at time $t - 1$ , $t - 2$ , etc.) as additional input variables for the current time step $t$ . This is based on the principle that current values in a time series are often influenced by their historical counterparts, a phenomenon known in statistics as autocorrelation or lag correlation (Faraone and Dorfman, 1987). At each time step $t$ , the lag feature of order $k$ corresponds to the observed value at time $t - k$ , and is computed as

{L a g}_{k} (t) = y_{t - k}

(1)

where

y_{t - k}

refers to the value of the target variable at time step

t - k

Difference feature is a time-series transformation technique that involves computing the difference between the current value and a lagged value, typically to quantify the magnitude of change over time, contributing to revealing dynamic variations in the series. The $k$ th-order difference is presented as

{Δ y}_{t}^{(k)} = y_{t} - y_{t - k}

(2)

The rolling standard deviation is calculated by computing the standard deviation of values within a fixed-length moving window, and assigning the result to the last time step of each window, capturing the local variability of the time series, with the “rolling” mechanism enabling the analysis of dynamic fluctuations over time. However, this feature only quantifies the magnitude of fluctuations and does not distinguish between upward and downward trends. Generally, the rolling standard deviation at time step $t$ is outlined as

{S t d}_{t} = \sqrt{\frac{1}{ω} \sum_{i = t - ω + 1}^{t} {(y_{i} - \bar{y_{t}})}^{2}}

(3)

where

ω

denotes the window size and

\bar{y_{t}}

indicates the rolling mean within the window ending at

t

The moving average is calculated by taking the mean of values within a fixed-size time window and assigning the result to the final time step of the window. The moving average at time step $t$ is defined as

{M A}_{t} = \frac{1}{ω} \sum_{i = t - ω + 1}^{t} y_{t - i} = \frac{1}{ω} (y_{t} + y_{t - 1} + \dots + y_{t - ω + 1})

(4)

Deep learning model

In this study, the LSTM-Attention model is constructed to forecast the temperature-induced stain and deflection, with its architecture illustrated in Figure 1.

Figure 1.

Architecture of the LSTM-Attention model.

LSTM

Since its introduction by Hochreiter and Schmidhuber, the LSTM network has become a widely adopted tool in engineering applications (Hochreiter and Schmidhuber, 1997). To enhance memory control, Gers et al. (2000) subsequently incorporated a Forget Gate into the LSTM structure, allowing the cells to regulate memory retention and prevent output saturation. As a specialized structure of Recurrent Neural Network (RNN), the network effectively overcomes the vanishing and exploding gradient problems inherent in traditional RNN, and exhibits strong performance in handling time series data and capturing long-term dependencies (Li et al., 2020b). Normally, the structure of the LSTM cell, which is the core of the LSTM network, is divided into three parts, including the forget gate, the input gate, and the output gate, with the structure of the LSTM cell shown in Figure 2 (Zhou et al., 2024b).

Figure 2.

The structure of LSTM cell.

Additionally, the calculation process of LSTM network is outlined as follows:

F_{t} = (W_{F} x_{t} + U_{F} h_{t - 1} + b_{F})

(5)

c_{t}^{'} = \tanh (W_{c^{'}} x_{t} + U_{c^{'}} h_{t - 1} + b_{c^{'}})

(6)

I_{t} = σ (W_{I} x_{t} + U_{I} h_{t - 1} + b_{I})

(7)

c_{t} = F_{t} c_{t - 1} + I_{t} c_{t}^{'}

(8)

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

(9)

h_{t} = O_{t} \tanh (c_{t})

(10)

y_{t} = W_{y} h_{t} + b_{y}

(11)

where

F_{t}

I_{t}

, and

O_{t}

denote the outputs of the Forget, Input, and Output Gates at time step

t

respectively,

σ

and

\tanh

refer to the activation functions,

x_{t - 1}, x_{t}, x_{t + 1}

c_{t - 1}, c_{t}, c_{t + 1}

h_{t - 1}, h_{t}, h_{t + 1}

and

y_{t - 1}, y_{t}, y_{t + 1}

represent the input vector, cell states, hidden states and output vector at corresponding time steps,

W

and

U

are weight matrix, and

b

is bias vector parameter.

Attention mechanism

The attention mechanism, which enables the model to focus on critical information while filtering out redundant or irrelevant signals, was initially inspired by findings in biological cognitive systems (Wu et al., 2024). Currently, this novel mechanism, which is capable of capturing inherently significant information in the input data, has been successfully utilized in time series prediction by paying attention to input data selectively (Li et al., 2021; Liao et al., 2023). Typically, there are three vectors primarily construct this mechanism, comprising query vector $Q$ , key vector $K,$ and value vector $V$ (Bahdanau et al., 2014). In this case, the architecture of the attention mechanism is outlined in Figures 3 and 4.

Figure 3.

Overview of attention mechanism.

Figure 4.

Attention mechanism expressed with $Q$ , $K$ and $V$ .

Furthermore, considering the diverse information relationships among structural responses, a multi-head attention mechanism is employed in this study, with the detailed architecture described in Figure 5. In addition, the process of transformations is summarized in equations (12) and (13) (Liu et al., 2025; Shi et al., 2024).

{h e a d}_{i} = A t t e n t i o n (Q_{i}, K_{i}, V_{i}) = s o f t m a x (\frac{Q_{i} {K_{i}}^{T}}{\sqrt{d_{k}}}) V_{i}

(12)

M u l t i h e a d (Q, K, V) = C o n c a t ({h e a d}_{1}, {h e a d}_{2}, \dots, {h e a d}_{h}) W_{h}

(13)

where

s o f t m a x (x)

is an approach for data normalization and

W_{h}

denotes a linear transformation matrix.

Figure 5.

Architecture of multi-head attention mechanism.

Case study

This study utilizes 1 year of temperature, strain, and deflection data from the Queensferry Crossing within an LSTM–Attention framework incorporating feature engineering to forecast temperature-induced strains and deflections, aiming to validate the contribution of feature engineering to model performance. The dataset was partitioned into training, validation, and test subsets in an 8:1:1 ratio, with the structure and basic parameters of the model respectively showcased in Table 2 and Table 3.

Table 2.

Structure of LSTM–attention model.

No.	Layer type	Parameters	Total amount
1	LSTM	Units = 128	3
2	CustomMultiHeadAttention	Heads = 4, key_dim = 32	1
3	Dense	Units = 64	2
4	Dense	Units = 32	1
5	Dense	Units = 16	1
6	Dense	Units = 1	1
7	GlobalAveragePooling1D	-	1

Table 3.

Basic parameters of LSTM–attention model.

Parameter	Value	Description
BATCH_SIZE	32	Number of samples per weight update
TIME_STEPS	32	Number of historical time steps per input sequence
Epochs	50	Maximum number of training epochs
NUM_RUNS	20	Number of model runs

Besides, preceding the model training phase, four evaluation metrics are applied, comprising $M A E$ , $M S E$ , $R^{2}$ and Explanatory Variance Score ( $E V S$ ), which are calculated as:

M A E = \frac{1}{n} \sum_{i = 1}^{n} | y_{i} - \hat{y_{i}} |

(14)

M S E = \frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - \hat{y_{i}})}^{2}

(15)

R^{2} = 1 - \frac{\sum_{i = 1}^{n} {(y_{i} - \hat{y_{i}})}^{2}}{\sum_{i = 1}^{n} {(y_{i} - \bar{y_{i}})}^{2}}

(16)

E V S = 1 - \frac{v a r (y - \hat{y})}{v a r (y)}

(17)

where

n

represents the total amount of data,

y_{i}

refers to the ground truth,

\bar{y_{i}}

demonstrates the mean of ground truth,

\hat{y_{i}}

denotes the predicted values and

v a r ()

is the variance function.

First, in terms of strains, the LSTM–Attention model with and without feature engineering is conducted. Also, to further confirm the effectiveness of feature engineering, an LSTM model is implemented, with the results of all models summarized in Figure 6 and Table 4.

Figure 6.

Summary of results for strains.

Table 4.

Optimal results of different models for strains.

Metrics	LSTM-attention model with feature engineering	LSTM-attention model without feature engineering	LSTM model with feature engineering
$M A E$	0.37	2.36	0.89
$M S E$	0.22	13.64	1.52
$R^{2}$	0.99	0.67	0.96
$E V S$	0.99	0.79	0.97

Observed from Figure 6 and Table 4, the LSTM-Attention model with feature engineering exhibits a better predictive performance than that without feature engineering, specifically an 84.32% reduction in $M A E$ , a 98.39% decrease in $M S E$ , a 47.76% increase in $R^{2}$ and a 25.32% increase in $E V S$ . Furthermore, compared to the LSTM model with feature engineering, the results of LSTM-Attention model with feature engineering showcase a 58.43% reduction in $M A E$ , an 85.53% decrease in $M S E$ , a 3.13% increase in $R^{2}$ and a 2.06% increase in $E V S$ , demonstrating the effectiveness of the attention mechanism.

Second, similar models are also executed for deflections, with the results presented in Figure 7 and Table 5.

Figure 7.

Summary of results for deflections.

Table 5.

Optimal results of different models for deflections.

Metrics	LSTM-attention model with feature engineering	LSTM-attention model without feature engineering	LSTM model with feature engineering
$M A E$	0.36	4.14	0.59
$M S E$	0.21	23.76	0.53
$R^{2}$	0.99	0.77	0.99
$E V S$	0.99	0.82	0.99

As shown in Figure 7 and Table 5, compared to that without feature engineering, the LSTM-Attention model with feature engineering achieves superior prediction accuracy, which is a decrease of 91.30% in $M A E$ and 99.12% in $M S E$ , and an increase of 28.57% in $R^{2}$ and 20.73% in $E V S$ . Then, compared to the LSTM model with feature engineering, the LSTM-Attention model with feature engineering demonstrates the decreases of 38.98% in $M A E$ and 60.38% in $M S E$ , and no change both in $R^{2}$ and $E V S$ , also suggesting the effectiveness of the attention mechanism.

These findings suggest that while model selection influences performance to some extent, it is the implementation of feature engineering and attention mechanism that drives the most significant gains in predictive accuracy.

To further evaluate the effectiveness of feature engineering, other research results are summarized in Table 6 and Table 7.

Table 6.

Comparison of temperature-induced strains.

Research reference	This study	Xu et al. (2025)	Zhao et al. (2023)	Zhao et al. (2020)	Chen et al. (2023)	Mariani et al. (2024)
Model	LSTM-Att with feature engineering	LSTM network	BiLSTM regression network	Backpropagation neural network	Deep learning- autoregressive model	1D causal dilated CNNs
$M A E$	0.37	-	1.732	-	-	-
$M S E$	0.22	0.89	-	5.29	7.29	-
$R^{2}$	0.99	0.98	0.94	0.97	-	0.93
$E V S$	0.99	-	-	-	-	-

Table 7.

Comparison of temperature-induced deflections.

Research reference	This study	Xu et al. (2025)	Yue et al., (2021a)	Yue et al. (2022)	Wang et al. (2022)	Meng et al. (2025)
Model	LSTM-Att with feature engineering	LSTM network	LSTM network	Double hidden layer LSTM network	LSTM network	Bidirectional LSTM network
$M A E$	0.36	-	5.85	3.58	-	5.56
$M S E$	0.21	28.4	58.45	27.17	157.5	46.0
$R^{2}$	0.99	0.825	-	0.94	0.81	0.94
$E V S$	0.99	-	-	-	-	-

At the aspect of strains, first, compared to the results from Xu et al. (2025), which utilized the same dataset with an LSTM model, there is only a slight improvement observed in $R^{2}$ under the LSTM-Attention model with feature engineering, where $M S E$ increases about 75.28%, further illustrating the effectiveness of feature engineering. Second, compared to other researchers’ results with different datasets, the LSTM-Attention model with feature engineering achieves a $M A E$ of 0.37, outperforming the BiLSTM regression network approximately 78.64% (Zhao et al., 2023). Also, in terms of a $R^{2}$ of 0.98, the model with feature engineering demonstrates the strongest fit to the observed data, surpassing the 0.97 from Backpropagation Neural Network, 0.94 from BiLSTM regression network and 0.93 from 1D causal dilated CNNs (Mariani et al., 2024; Zhao et al., 2020, 2023). Besides, it also presents a $M S E$ of 0.22, which is significantly better than the values from previous models (5.29 and 7.29), showing better control over large errors (Chen et al., 2023; Zhao et al., 2020).

With regard to deflections, the LSTM-Attention model with feature engineering showcases significant enhancement when comparing with the LSTM network from Xu et al. (2025), specifically a 99.26% decrease in $M S E$ and 20% increase in $R^{2}$ . Then, to further assess the effectiveness of feature engineering, several LSTM-based architectures reported in recent studies are compared. The $M A E$ of the LSTM-Attention model with feature engineering exhibits at least 89.94% decrease, $M S E$ presents more than 99% decrease and $R^{2}$ demonstrates more than 5.31%, showcasing the significance of feature engineering (Meng et al., 2025; Wang et al., 2022; Yue et al., 2021a, 2022).

In summary, when applied to the same dataset, the LSTM-Attention model shows significant improvement in deflection prediction. Furthermore, compared to previous studies using different datasets, the LSTM-Attention model with feature engineering reports substantial improvements in predictive accuracy for both strains and deflections.

Concluding remarks

This study introduced a deep learning model applying feature engineering to forecast the temperature-induced strains and deflections from the Queensferry Crossing cable-stayed bridge. The LSTM-Attention model is initially constructed to serve as the core predictive framework. Subsequently, to evaluate the role of feature engineering, the predictive performance of models with and without its application is first analyzed. Finally, based on the results of recent studies, for the sake of strengthening the findings, a comparative analysis referencing recent studies is also implemented. In summary, the conclusions drawn from this study are as follows:

(1) Feature engineering showcases noticeable function through comparative analysis. Specifically, for strains, the LSTM-Attention model with feature engineering presents an 84.32% reduction in $M A E$ , a 98.39% decrease in $M S E$ , a 47.76% increase in $R^{2}$ and a 25.32% increase in $E V S$ compared to that without feature engineering. In terms of deflections, when benchmarked against the LSTM-Attention model lacking feature engineering, the feature-engineered version exhibits markedly improved predictive capability, achieving a 91.30% reduction in $M A E$ , a 99.12% reduction in $M S E$ , and respective increases of 28.57% and 20.73% in $R^{2}$ and $E V S$ .

(2) The attention mechanism is capable of enhancing the predictive accuracy of temperature-induced strains and deflections. With regard to strains, in comparison with the LSTM model incorporating feature engineering, the feature-engineered LSTM-Attention model yields markedly superior results, exhibiting a 58.43% decrease in $M A E$ , an 85.53% reduction in $M S E$ , and respective improvements of 3.13% in $R^{2}$ and 2.06% in $E V S$ . Then, relative to the LSTM model with feature engineering, the LSTM-Attention model with feature engineering achieves reductions of 38.98% in $M A E$ and 60.38% in $M S E$ , while showing no change in $R^{2}$ and $E V S$ . These findings collectively suggest the effectiveness of the attention mechanism.

(3) According to previous studies, the LSTM-Attention model with feature engineering demonstrates superior predictive performance. At the aspect of strains, the LSTM-Attention model with feature engineering outperforms other models by approximately 78.64% in $M A E$ , more than 75.28% in $M S E$ and more than 6.45% in $R^{2}$ . Concerning deflections, the LSTM-Attention model with feature engineering achieves better accuracy, which is at least 89.94% decrease in $M A E$ , more than 99% decrease in $M S E$ and about 5.31% in $R^{2}$ . These results collectively highlight that the integration of feature engineering with the LSTM-Attention framework substantially enhances predictive accuracy for both strains and deflections

In conclusion, this study underscores the potential of deep learning models incorporating feature engineering and the attention mechanism, indicating that future research may further explore adaptive or domain-informed feature engineering strategies to fully leverage the learning capacity of advanced neural architectures.

Footnotes

ORCID iD

Xiang Xu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project has received funding from the National Natural and Science Foundation of China grant No. 52308150; the Jiangsu Provincial Department of Science and Technology grant No. BZ2024015; China Railway Group Limited Science and Technology Research and Development Program (2025-Major Project-17); the European Union’s Horizon 2020 Research and Innovation Program under the Marie Skłodowska-Curie grant agreement No. 801215 and the University of Edinburgh Data-Driven Innovation Program, part of the Edinburgh and South East Scotland City Region Deal, and Screening Eagle, AG, Zurich, Switzerland. This research work is supported by the Big Data Computing Center of Southeast University.

Declaration of conflicting interests

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

Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.*

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