Reconstructing Missing Data Using a Bi-LSTM Model Based on VMD and SSA for Structural Health Monitoring

Abstract

For structural health monitoring (SHM), a complete dataset is crucial for further modal identification analysis and risk warning. Unfortunately, data loss can occur due to sensor failure, transmission system interruption, or hardware failure, which can lead to missing data. Therefore, this study proposes a bidirectional long short-term memory neural network (Bi-LSTM) response recovery method based on variational mode decomposition (VMD) and sparrow search algorithm (SSA) optimization that utilizes the structural response data between multiple sensors and can simultaneously consider temporal and spatial correlations. A dataset containing approximately half a month of monitoring data was collected from a certain project for training, validation, and testing. A publicly available dataset was also referenced to validate the proposed method in this paper. Using the public dataset, under 13 different data loss rates, the VMD + SSA + Bi-LSTM model reduced the RMSE of data reconstruction by an average of 65.01% and 45.35% compared to the Bi-LSTM model and the VMD + Bi-LSTM models, respectively, while the coefficient of determination increased by 62.21% and 11.19%. The data reconstruction method proposed in this paper can accurately reconstruct the variation trends of missing data without the manual optimization of hyperparameters, and the reconstruction results are close to the real data.

1. Introduction

With the continuous development of China’s construction industry, large-span space structures have been widely used. However, due to the influence of factors such as hot and cold alternation, rain and snow loads, wind loads, environmental erosion, and initial defects during the construction and operation of large-span space structures, strength degradation and other damages will inevitably occur, forming safety hazards. To ensure the structural safety and reliability of large-span space structures, it is very important to analyze and monitor structural stress and deformation during the construction process and long-term operation. Therefore, SHM has made significant progress in China. Various sensors are used to monitor the response of structures in the field of SHM, including but not limited to vibration, displacement, and strain sensors, to accurately and effectively evaluate engineering structure status. Accurate assessment results rely on a large amount of reliable monitoring data. However, the presence of unexpected factors such as equipment failure, accidental damage to data transmission equipment or lines, and unstable power supply during the long-term operation of the monitoring system will inevitably lead to abnormal monitoring data and data loss [1]. Therefore, some scholars have mentioned and paid attention to the problem of data repair in engineering discussions on health monitoring. In the early days, the more commonly used data repair algorithms were mainly linear regression interpolation, support vector machines (SVMs), compressed sensing, principal component analysis, Bayesian methods, etc. For example, Ye [2] proposed a method for reconstructing bridge health monitoring data based on wavelet multi-resolution analysis and SVM combination. Comparison with the traditional autoregressive moving average method (ARMA) verified that wavelet-based SVMs have better effectiveness and accuracy. Dong [3] proposed a two-stage method based on SVMs to simulate and predict the nonlinear dynamic response of structures, achieving good accuracy. Jalet [4] proposed a regression method based on least-squares-optimized SVMs to obtain displacement time series from acceleration data in order to reconstruct displacement data when displacement sensor data corresponding to acceleration sensors are missing. Bao [5] proposed a novel compressed sampling technique for wireless sensor networks. The accuracy of the data loss recovery method was verified using an acceleration time series collected from the Jinzhou West Bridge field test and the SHM system of the China National Swimming Center in Beijing. The results indicate that the recovery method performs well in reconstructing missing data when the original data do not have sparse features in a certain orthogonal basis. Fereidoun [6] proposed a distributed compressive sensing method for the data loss recovery of multi-signal data under a variable loss rate. Unlike single-signal methods, this method utilizes the correlation between vibration signals to improve reconstruction accuracy and has higher robustness. Li [7] proposed a data reconstruction method based on probabilistic principal component analysis. Compared with principal component analysis, this method has higher accuracy, especially in the case of continuous missing data. Wan [8,9] proposed a Bayesian modeling method that employs Gaussian processes and a mobile window strategy to significantly reduce the size of training data, thereby effectively reducing computational costs. Later, Wan [8,9] proposed a novel multi-task learning method that utilizes the correlation between tasks to enhance reconstruction ability and achieve good reconstruction performance. The method employs a multi-dimensional Gaussian process prior to support-modeling a series of tasks simultaneously. Zhang [10] believes that modeling the correlation between sensors can treat data recovery as a regression task. Based on this, a Bayesian dynamic regression method was proposed that can simultaneously reconstruct missing data from different sensors. The regression variables of this method can change according to changes in the data. Ren [11] proposed an incremental Bayesian/tensor learning scheme to address the problem of low efficiency of Bayesian tensor decomposition models when dealing with large datasets. The scheme first constructs a spatiotemporal tensor then performs Bayesian tensor decomposition to extract the potential features of missing data. Ren also developed an incremental learning scheme to update the Bayesian tensor decomposition model effectively. The results show that even in the presence of a large number of random and structured missing data, relatively accurate reconstruction results can still be obtained. Zhang [12] proposed an interpolation method based on point correlation for missing stress data repair, using long-term monitoring data of the steel structure of the Hangzhou Olympic Center Stadium as the research object. The method was used to repair different data loss situations. The results showed that the error of linear regression interpolation was 5% when the correlation coefficient of a single point was 0.9 or higher. Compared to continuous missing data, the interpolation error of discrete missing data is slightly smaller, and the data loss rate should be less than 30%.

In recent years, deep learning, proposed by LeCun [13], has been the most rapidly developing technology in the field of artificial intelligence. It enables a calculation model consisting of multiple processing layers to learn data representations with multiple abstract levels, as opposed to traditional machine learning methods. These methods have significantly enhanced technical proficiency in various fields, including speech recognition, visual object recognition, object detection, drug discovery, and genomics. Deep learning discovers complex structures in large datasets and changes the internal parameters of a model by using the backpropagation algorithm. Deep convolutional networks have an advantage in processing images, videos, speech, and audio, while recurrent networks have an advantage in processing sequential data such as text and speech [14]. Bao [1] proposed a data anomaly detection method based on computer vision and deep learning, leveraging the advantages of deep learning in image processing. The method consists of two steps: Data visualization is used to transform data into image vectors and establish a manually labeled dataset. Then, a deep neural network is constructed and trained for anomaly classification. The method has demonstrated the ability to automatically detect multiple pattern anomalies in data with high accuracy. Jeong [15] proposed a data reconstruction algorithm based on bidirectional recurrent neural networks (BRNNs) to better handle the temporal correlation between sensor data. Jeong believed that there is a certain spatiotemporal correlation between sensor data. The results show that the BRNN-based method is superior to traditional algorithms in accurately reconstructing sensor data. Zhao [16] proposed a wind-induced power response prediction algorithm based on long short-term memory (LSTM) neural networks. Harrou [17] applied LSTM to the prediction and detection of traffic congestion and achieved high accuracy. Hittawe [18] proposed a method based on LSTM and Gaussian process regression to predict the sea surface temperature of the Red Sea. The method has good prediction accuracy. This algorithm can effectively solve the problem of gradient vanishing or explosion faced by traditional recurrent neural networks when predicting long sequences with long-term dependencies. Li [19] combined empirical mode decomposition (EMD) with LSTM neural networks to convert the missing data input task into a time series prediction task. EMD helps to model the irregular periodic changes in measurement signal data, while LSTM neural networks can remember more subsequent long-term correlations. This method outperforms the three commonly used prediction models of autoregressive integrated moving average, support vector regression, and artificial neural network models. Lu [20] proposed a structural acceleration data reconstruction method based on Bi-LSTM neural networks. This method can better explore the complex spatiotemporal correlation between sensor data. Through research and verification of the measured monitoring data of a large-span cable-stayed bridge, the results show that this method can effectively extract the spatiotemporal correlation of acceleration data and achieve a high-precision reconstruction of long-term continuous acceleration data of a structure under limited sensors. Liu [21] proposed a Bi-LSTM model based on the whale optimization algorithm and wavelet filtering technology to predict potential mud loss in drilling. Compared with the previous version, the improved model achieved a significant increase in accuracy. Li [22] proposed a method that first uses a convolutional neural network to classify the surface wear state of a workpiece, and then inputs the classification results into a Bi-LSTM regression model to predict the remaining service life of the tool. The prediction error of this method is within 5%. Du [23] proposed a heterogeneous structure response recovery method based on a multi-modal fusion autoencoder. This method can simultaneously consider temporal correlation, spatial correlation, and correlation between heterogeneous structure responses. The results show that this method can achieve greater performance advantages compared to other methods in the case of high missing rates. Tian [24] utilized the advantages of Bi-LSTM neural networks for spatiotemporal correlation between sensors to propose a method for establishing the relationship between the beam vertical deflection and cable tension of a cable-stayed bridge. The test results show that this method has robustness to different noise levels and traffic volumes under normal conditions.

In summary, there is currently little research on data reconstruction for SHM systems, and existing results mainly focus on the dynamic response of structures, with insufficient attention paid to the stress–strain part of structures. Although the data reconstruction method based on the Bi-LSTM model proposed by previous researchers has considered the spatiotemporal relationship between sensors and achieved good data reconstruction results, it requires multiple sets of hyperparameters to be pre-set and tested to find the optimal solution for the hyperparameters, which incurs a large time cost. Some people have used EMD to decompose the existing data of the target sensor into modal components and residuals, which improves the model’s ability to obtain the spatiotemporal relationship between sensors. However, EMD has strict restrictions on data noise and sampling sensitivity. Therefore, this paper proposes a method based on Bi-LSTM neural networks that fully considers the spatiotemporal correlation between sensors. It also introduces VMD to decompose the irregular periodicity of data signals. This method can effectively solve EMD’s strict limitations on data noise and sampling sensitivity. It has stronger robustness while maintaining good performance [25]. When training a model, introducing the SSA [26] to optimize the hyperparameters of Bi-LSTM neural networks can reduce the time and effort required for manual hyperparameter optimization to some extent. The combination of VMD, SSA, and Bi-LSTM neural networks can be used to achieve strain monitoring data reconstruction by utilizing the spatiotemporal correlation between multiple sensors.

2. Basic Methods

2.1. Variational Mode Decomposition

VMD is a novel method for signal decomposition and estimation. It is based on a variational problem that minimizes the sum of the estimated bandwidths of each mode. The method assumes that each mode has a finite bandwidth with different center frequencies. To solve this variational problem, the alternating direction multiplier method is employed to iteratively update each mode and its center frequency. Gradually, each mode is demodulated to its corresponding fundamental frequency band, and ultimately, each mode and its corresponding center frequency are extracted together. Compared with the recursive “screening” mode of EMD and local mean decomposition (LMD), VMD transforms the signal decomposition into a non-recursive, VMD mode, and has a solid theoretical foundation. The core of this method is composed of multiple adaptive Wiener filter groups, which demonstrate superior noise robustness. By controlling the convergence conditions reasonably, VMD exhibits a much smaller sampling effect than EMD and LMD. In terms of mode separation, VMD is capable of successfully separating two pure harmonic signals with similar frequencies. Singular value decomposition can effectively extract matrix features and has good stability. When the matrix elements undergo slight changes, the singular values of the matrix change only slightly. Additionally, the singular values of the matrix exhibit proportion invariance and rotation invariance. Therefore, it can stably characterize the characteristics of each mode [25].

2.2. Sparrow Search Algorithm

SSA is an optimization algorithm proposed by Xue [26] in 2020, mainly inspired by the foraging and anti-predator behavior of sparrows. SSA is a novel algorithm with strong optimization ability and fast convergence speed. It simulates the foraging and anti-predator behavior of sparrows, and the population members include finders, followers, and scouts. The finder searches for food and determines the foraging area and direction of the entire population; the follower searches for food based on the information provided by the finder. The identities of the two can be transformed into each other under certain conditions. At the same time, the scout discovers danger and decides whether to evacuate the danger zone. At each stage, the finder and scout account for 10% to 20% of the entire population.

2.3. Bidirectional Long Short-Term Memory Networks

2.3.1. Long Short-Term Memory Networks

The LSTM neural network is a deep learning technique developed by Hochreiter [27] based on the recurrent neural network (RNN). It is the basic layer of the bidirectional LSTM neural network. The RNN can be regarded as a type of neural network with short-term memory capability, and it is commonly used to process and predict sequence data. It has been widely used in fields such as speech recognition, text classification, machine translation, and image analysis. The most significant feature of the RNN is that neurons can not only receive information from other neurons, but also return their own output as input to neurons, forming a loop. This network structure is very suitable for time-series data. In addition, the RNN has a repetitive structure and the network parameters are shared, which greatly reduces the number of neural network parameters required for training. The shared parameters also make the network model applicable to data of different lengths. Although the RNN has memory capability when processing time-series data, a pure RNN will encounter gradient explosion or disappearance when the number of cycles increases, making it difficult to obtain long-term intrinsic features of data and difficult to train and converge. Therefore, using an LSTM neural network can effectively solve the gradient disappearance and explosion problems in the traditional RNN training process [28,29].

An LSTM cell is a fundamental building block of the LSTM layer. The data at each time step are fed into different LSTM cells that contain weights and biases shared across the entire temporal space. Each LSTM cell comprises memory cells and gate units. The memory cells selectively retain information from the previous and current time steps to memorize the cell state, capturing the long-term time-dependent dependencies of sequential data. The gate units consist of an input gate, a forget gate, and an output gate. As shown in Figure 1, the cell state is updated by the input gate, which controls the information added, and the forget gate, which regulates the information discarded.

The output of an LSTM cell is determined by the output gate. The calculation formula that each cell follows is shown in Equation (1).

$$\{\begin{array}{c}{f}_t=\sigma (W_{xf}{x}_t+W_{hf}{h}_{t-1}+b_f)\\ \sigma \left(x\right)=\frac{e^x}{e^x-1}\\ i_t=\sigma (W_{xi}{x}_t+W_{hi}{h}_{t-1}+b_i)\\ g_t=\tanh (W_{xg}{x}_t+W_{hg}{h}_{t-1}+b_g)\\ \tanh \left(x\right)=\frac{e^x-e^{-x}}{e^x+e^{-x}}\\ c_t=f_t\times c_{t-1}+i_t\times g_t\\ o_t=\sigma (W_{xo}{x}_t+W_{ho}{h}_{t-1}+b_o)\\ h_t=o_t\times \tanh \left(c_t\right)\end{array}$$

(1)

where $c_t$ represents the tuple state at time $t$, with an initial hidden state of 0; $i_t$ refers to the input gate; $f_t$ represents the forget gate; $g_t$ corresponds to the selection gate; and $o_t$ represents the output gate.

2.3.2. Bidirectional Long Short-Term Memory Networks

When processing time-series problems, LSTM networks only consider past information. The use of Bi-LSTM networks can overcome this limitation and utilize both past and future information. Bi-LSTM networks consist of forward LSTM layers and backward LSTM layers, which iteratively process past and future information from the forward and backward directions, respectively. This allows for better capture of the correlation between time-series data.

The Bi-LSTM network architecture used in this paper is shown in Figure 2.

The input can be expressed as

$$X={\left\{x_{t-T+1},x_{t-T+2},\dots ,x_t\right\}}^T$$

(2)

where $x_i={\left[a_{1,i},a_{2,i},\dots ,a_{m,i}\right]}^T$ represents the response data of the available sensors at time step $i$, $m$ represents the number of available sensors, and $T$ is the length of the response time series. The input matrix contains time-series responses of multiple sensors involving spatiotemporal relationships. The network model only uses one Bi-LSTM layer to capture spatiotemporal relationships. Finally, the fully connected layer is used to reduce the dimensionality of the data to meet the network output requirements. The output of the network can be formulated as

$$Y_j={\left\{y_{t-T+1},y_{t-T+2},\dots ,y_t\right\}}^T\left(j=1,2,\dots ,n\right)$$

(3)

where $y_i$ represents the response data of the target sensor at time step $i$, and $n$ represents the number of IMF components and residuals obtained from VMD.

3. Structural Health Monitoring Data Reconstruction Method Based on VMD + SSA + Bi-LSTM

Due to the correlation between the responses of various measurement points on the structure, Bi-LSTM networks have strong adaptability in solving such problems. VMD can decompose the original strain response signal to make it more regular relative to the original data. Then, SSA is used to optimize the hyperparameters of the Bi-LSTM network, which can further improve the effectiveness of solving this problem. Therefore, this paper adopts a Bi-LSTM-based VMD- and SSA-optimized method to repair missing strain data. The specific process is shown in Figure 3.

Step 1: Initially, determine the data loss rate of the target sensor and the proportion of the validation set to the entire dataset, assumed to be 20% and 10% here, respectively. Then, perform VMD on the initial 70% of the target sensor data and normalize each IMF component, residual, and all complete data from other sensors obtained after the decomposition.

Step 2: Formulate the training, validation, and test sets. Use the initial 70% of the data from other sensors as the input for the training set, the middle 10% as the input for the validation set, and the last 20% as the input for the test set. Use the IMF components and residuals of the target sensor corresponding to the first 70% as the output of the training set, and the corresponding middle 10% as the output of the validation set. Establish a Bi-LSTM model for each IMF component and residual of the target sensor.

Step 3: Initialize the relevant parameters of the SSA. Conduct model training using a subset of the data to ascertain the optimal architecture for adept data fitting and prediction. This involves configuring hyperparameters such as hidden layer units, iteration counts, and learning rates.

Step 4: Train a Bi-LSTM model separately for each IMF component and residual of the target sensor. Subsequently, perform inverse normalization on the model-generated outcomes and aggregate them to derive the ultimate prediction result. Evaluate the data reconstruction efficacy of the model by scrutinizing the error between this result and the initial data from the target sensor.

This study employs various metrics to assess prediction performance, including root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (R²). The definitions for these indicators are provided below.

$$\{\begin{array}{c}RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left(t_i-y_i\right)}^2}\\ MAE=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left|t_i-y_i\right|\\ MAPE=\frac{100\%}{n}{\displaystyle \sum }_{i=1}^n\left|\frac{t_i-y_i}{y_i}\right|\\ R^2=\frac{{\left[{{\displaystyle \sum }}_{i=1}^n\left(t_i-\overline{t}\right)\left(y_i-\overline{y}\right)\right]}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(t_i-\overline{t}\right)}^2{{\displaystyle \sum }}_{i=1}^n{\left(y_i-\overline{y}\right)}^2}\\ \end{array}$$

(4)

where $t$ is the measured value, $y$ is the predicted value, $\overline{t}$ is the mean value of $t$, and $\overline{y}$ is the mean value of $y$.

4. Results

This section utilizes the steel roof of the F hall from a specific project as a case study to validate the precision and efficacy of the response reconstruction network, which is founded on the integration of VMD, SSA, and Bi-LSTM.

4.1. Project Description

The plan of the certain project adopts a “starfish” layout (Figure 4 shows the architectural rendering of the project). The steel roof is divided into six parts: the F hall (Figure 5 shows the structural model of the hall) and the A, B, C, D, and E corridors. The steel roof between each structural unit is separated by a 300 mm wide seismic joint. Among them, the plan size of the steel roof of the F hall is 500 m × 339 m, and the highest point of the metal roof is 42.125 m above sea level. The rigid steel structure of the roof adopts an orthogonal four-corner pyramid grid (Figure 6 shows the structure described here, with a stress–strain sensor located at the point marked by red fabric on the component surface), and the entire roof is equipped with nine skylights. The skylight area adopts a three-dimensional truss. The roof support is made of steel pipe–concrete columns. The steel pipe–concrete column in the middle of the F hall extends downward to the foundation or underground floor and is rigidly connected with the concrete beams of each floor. The steel pipe–concrete column on the periphery of the F hall extends downward to the foundation. The grid nodes adopt welded hollow nodes, the skylight truss nodes adopt welded steel plate nodes, and some adopt cast steel nodes. The column top support mainly adopts finished ball hinge supports. To reduce temperature stress, the steel roof adopts a sliding two-way support along the peripheral column top support. The specific parameters of the vibrating wire strain gauge used in this project are listed in

Table 1. Vibrating wire strain gauge parameters.

Performance	Parameter
Measuring range	3000 $\mu \epsilon$
Accuracy	1% FS
Protection level	IP68
Non-linearity	Straight line: ≤1% FS;
	Polynomial: ≤0.1% FS
Resolution	0.035% FS
Temperature range	−20 °C to +80 °C
Gauge length	150 mm

.

4.2. Data Sample

The data sample in this paper is a complete dataset collected from 8 sensors selected from 55 strain sensors installed on the steel roof of project F hall from 14 June 2023 to 3 July 2023. The data type is strain response, and the data have not been preprocessed by noise reduction, interpolation, or other methods. The missing data are systematically missing data from a single sensor, simulating the situation that the target sensor is damaged or has failed due to accidental reasons. Figure 7 shows the strain–time curves of these eight strain gauges.

Assuming that the last 10% to 70% (interval of 5%) of the ST1-8 sensor data is missing, corresponding training, validation, and test sets are constructed. The missing data are repaired using a pure Bi-LSTM neural network, VMD + Bi-LSTM neural network, and VMD + SSA + Bi-LSTM neural network. The reconstruction results are evaluated for each of the above datasets.

4.3. Model Settings

The information derived from the target sensors (ST1-8) underwent VMD, resulting in decomposition into five intrinsic mode function (IMF) components and one residual component, as illustrated in Figure 8. The Bi-LSTM model was initialized with a single layer and underwent training employing the Adam optimizer. The training process consisted of a maximum of 100 iterations, with an initial learning rate set at 0.01. Following 25 training iterations, a learning rate adjustment factor of 0.2 was applied. The regularization parameter was fine-tuned to 0.01, and the model employed 70 hidden units with the rectified linear unit (ReLU) function serving as the activation function. A subset of the data was utilized for pre-training, and subsequent to the completion of training, inference evaluation was conducted. The SSA was employed to fine-tune the three hyperparameters associated with model configurations: the quantity of units, the number of training iterations, and the learning rate. This optimization process was guided by the analysis of results in conjunction with their corresponding model parameters.

4.4. Reconstruction Results

Table 2. Hyperparameters of the Bi-LSTM model optimized using SSA for target sensor (ST1-8).

Missing Data Rate	Number of Units	Max Epochs	Learning Rate
10	275	270	0.00248
15	352	165	0.00321
20	420	157	0.00311
25	465	137	0.00473
30	372	282	0.00551
35	521	164	0.00667
40	565	248	0.00199
45	398	194	0.00126
50	591	248	0.00238
55	839	132	0.0053
60	698	293	0.00364
65	679	199	0.00316
70	631	181	0.00722

shows the Bi-LSTM model parameters optimized by SSA under different missing rates. The data in the table indicate that as the missing rate increases, the number of units in the Bi-LSTM model parameters optimized by SSA shows an upward trend, while the maximum training count and learning rate remain relatively stable.

Table 3. The time (s) required for model training under different loss rates.

Missing Data Rate	Bi-LSTM	VMD + Bi-LSTM	VMD + SSA + Bi-LSTM
10	5.2	32.7	163.7
15	5.1	31	112.7
20	4.1	24.5	96.1
25	4.1	25.3	119.1
30	3.9	24.8	114.3
35	3.5	16.9	62
40	3	16.9	105.5
45	2.8	16.4	71.9
50	2.9	17.1	70.8
55	2.5	13.5	52.9
60	2.2	11.6	61.7
65	1.7	8.9	41.8
70	1.5	8.5	27.4

shows the time required for different models to train under different loss rates. From the table, it can be seen that the training time of the Bi-LSTM and VMD + Bi-LSTM models, without adjusting hyperparameters, decreases as the loss rate increases, and the training time of the VMD + Bi-LSTM model is about six times that of the Bi-LSTM model under the same loss rate. The time consumed by the VMD + SSA + Bi-LSTM model fluctuates greatly, mainly because the model optimization time of the SSA fluctuates, and the different optimized model hyperparameters also cause changes in the model training time. Compared with the VMD + Bi-LSTM model, the VMD + SSA + Bi-LSTM model requires three to six times more time, but it still significantly reduces the time required for the manual setting of multiple groups of hyperparameters to debug the model.

From Figure 9, it can be observed that the reconstructed data obtained using the three methods exhibit similar trends to the original data. However, there are some errors present, which are mainly concentrated around the peak values of the signal mutation. In addition, the data reconstructed using the single Bi-LSTM method exhibit an obvious pinching phenomenon (the reconstructed data near the local maximum and minimum values are closer to the local mean value), while the data reconstructed using the VMD + SSA + Bi-LSTM method are closer to the actual data and have a smaller error.

After assessing the outcomes shown in Figure 10, derived from employing three distinct approaches for data reconstruction across varying levels of data incompleteness, the results revealed a proportional escalation in loss rates as the data missing rate increased. Notably, the VMD + SSA + Bi-LSTM model exhibited the least susceptibility to elevated missing rates, underscoring the efficacy of employing VMD for decomposing the initial monitoring data. Simultaneously, optimizing the Bi-LSTM model hyperparameters through SSA demonstrated a consequential enhancement in the accuracy of the Bi-LSTM model data reconstruction.

From the analysis of Figure 11, it is evident that the stress data reconstruction outcomes produced by the Bi-LSTM network manifest a distinct inclination to converge toward the mean value of the dataset. Specifically, the reconstructed data exhibit a tendency to be diminished for segments of original data in proximity to the maximum value, while being augmented for portions near the minimum value. This convergence trend towards the center is identified as the primary source of error. Notably, VMD capability enables the dissection of data into intrinsic modal components, facilitating the Bi-LSTM network in leveraging these components to comprehend the inherent spatiotemporal relationships within the data. Consequently, this aids in mitigating the inclination of extreme data points towards convergence. Building upon this foundation, the Bi-LSTM network, optimized through the SSA, further mitigates this convergence trend, leading to superior reconstruction outcomes.

Figure 12 shows the scatter plot and smooth surface of the evaluation indicators of the reconstructed data using data from five sensors (excluding ST1-6 and ST1-7), data from six sensors (excluding ST1-7), and data from seven sensors. From the figure, it is quite clear that the accuracy of the data reconstruction results is affected to some extent by the number of sensors used. As the number of sensors used decreases, the reconstruction accuracy of the data also decreases.

The VMD + SSA + Bi-LSTM method demonstrates significant improvement in data reconstruction across 13 varying rates of data loss. The RMSE is reduced by an average of 37.24% and 25.39% in comparison to the Bi-LSTM and VMD + Bi-LSTM approaches, respectively. Concurrently, the MAE experiences an average reduction of 39.31% and 26.63%, while the MAPE sees an average decrease of 36.77% and 24.68%. Furthermore, the R² exhibits an average increase of 15.80% and 6.95%.

5. Public Dataset Validation

5.1. Dataset Source

The dataset established by the Norwegian University of Science and Technology through a long-term monitoring project on the famous Hardanger Bridge in Norway is now available for open access [30].

5.2. Data Samples

The data used in this paper were obtained from seven acceleration sensors, namely H1 East, H1 Vest, H2 Vest, H3 East, H3 Vest, H4 East, and H4 Vest, along the Z-axis of the Hardanger Bridge under the influence of Storm Ole on 7 February 2015 (as shown in Figure 13). The sampling rate of the data was 200 Hz, and the duration of the data was one minute.

Constructing datasets for training, validation, and testing involves addressing missing sensor data within the range of 10% to 70% (in intervals of 5%) from the H4 Vest sensor data. To rectify this absence, three different approaches were employed: a pure Bi-LSTM neural network, a VMD + Bi-LSTM neural network, and a VMD + SSA + Bi-LSTM neural network. The performance of these methods was assessed through the reconstruction of the missing data across the specified datasets.

5.3. Model Settings

The information from the target sensors underwent vibrational mode decomposition (VMD), resulting in decomposition into five IMF components and one residual component, as depicted in Figure 14, when the missing data rate is 10%. The Bi-LSTM model was initialized with a single layer and trained to utilize the Adam optimizer. The training process employed a maximum count of 100 iterations, an initial learning rate of 0.01, and a learning rate adjustment factor of 0.2 after 25 iterations. A regularization parameter of 0.01 was set, and the model included 70 hidden units, with the ReLU function serving as the activation function. A subset of the data was incorporated for initial training, and following the completion of the training process, an assessment of inference was conducted. The optimization of model parameters employed the SSA, targeting three hyperparameters: unit quantity, training iterations, and learning rate. This optimization was carried out by analyzing the outcomes alongside their respective model parameters.

5.4. Reconstruction Results

Table 4. Hyperparameters of the Bi-LSTM model optimized using SSA for target sensor (H4 Vest).

Missing Data Rate	Number of Units	Max Epochs	Learning Rate
10	111	128	1 × 10−4
15	285	203	1 × 10−4
20	381	287	0.00822
25	710	187	7.07624 × 10−4
30	378	156	0.00156
35	439	109	0.00312
40	865	92	0.00211
45	450	127	2.48469 × 10−4
50	507	184	0.00651
55	497	163	0.00339
60	576	163	0.00145
65	746	132	3.87395 × 10−4
70	791	235	0.00241

presents the hyperparameters of the Bi-LSTM model optimized using SSA across various missing rates. The analysis of the table data reveals an observable upward trend in the hyperparameters of the SSA-optimized Bi-LSTM model as the missing rate increases. Notably, the maximum training times exhibit relative stability, while the learning rate experiences significant fluctuations.

From the analysis of Figure 15, it can be seen that the data reconstruction outcomes from the three methodologies exhibit substantial overlap with the original dataset. Furthermore, the reconstructed data trends consistently align with the actual monitoring data. Notably, errors are primarily concentrated around the signal mutation peaks. In this context, the data reconstruction achieved by the VMD + SSA + Bi-LSTM method demonstrates a higher fidelity to the actual data, displaying smaller errors in comparison to the alternative methods.

From Figure 16, it can be observed that the VMD + SSA + Bi-LSTM model performs well in the reconstruction of acceleration data and maintains a very stable error rate compared to the Bi-LSTM model and VMD + Bi-LSTM model. The impact of the increase in loss rate is relatively low. This may be due to the large amount of data in the dataset; the first 30% of the data is sufficient for the model to learn the time–space relationship between the various reliable sensor data and achieve better data reconstruction results.

From Figure 17, it can be observed that due to the use of acceleration sampling data, the dataset has more mutation points. Although the absolute difference between the upper and lower limits of the data is not significant, the data change more dramatically. In this dataset, the trend of the Bi-LSTM model data reconstruction results towards the mean value is more pronounced. However, after using VMD decomposition of IMF components and residuals (VMD + Bi-LSTM) for reconstruction, this trend was suppressed. The performance of the VMD + SSA + Bi-LSTM model is equally impressive as that of the strain sensor data reconstruction, effectively suppressing the convergence phenomenon and achieving good reconstruction accuracy.

Based on an analysis encompassing 13 distinct rates of missing data, the VMD + SSA + Bi-LSTM method showcases a remarkable improvement in data reconstruction performance. Specifically, it achieves an average reduction of 65.01% and 45.35% in RMSE compared to the data reconstructed using the Bi-LSTM and VMD +Bi-LSTM methods, respectively. Additionally, the MAE experiences an average reduction of 65.72% and 45.64%, while the MAPE demonstrates an average reduction of 66.87% and 47.82%. Notably, the R² registers an average increase of 62.21% and 11.19%.

6. Conclusions

The integrity and accuracy of data in SHM is one of the important factors affecting the reliability of building structural safety assessment results. This paper proposes a method for reconstructing missing sensor data by using other sensor data within the structure to address the problem of missing strain sensor data in large-span space structure health monitoring systems. The method involves introducing VMD and using SSA to optimize the hyperparameters of a Bi-LSTM neural network. The method was validated using acceleration sensor data from the Hardanger Bridge in Norway. The main conclusions are as follows:

• The method overcomes the problem of the manual adjustment of hyperparameters faced by the current single deep learning algorithm model, which can greatly reduce the time and energy spent on optimizing model hyperparameters.
• Through the verification of SHM data from actual engineering projects, it is shown that the missing data reconstruction method proposed in this paper can accurately reconstruct the change law of missing data, and the reconstruction result is close to the real data.
• By comparing with the unoptimized Bi-LSTM model and the method combining VMD with the Bi-LSTM model, the data reconstruction method proposed in this paper has higher accuracy, smaller error, and is less sensitive to the missing rate of data. When a large amount of training data is provided, it can achieve better stability in data reconstruction.
• The method also exhibits excellent reconstruction performance and good generalization ability in the acceleration response of large-span suspension bridges.
• As the number of data source sensors available for data reconstruction decreases, the accuracy of data reconstruction will decrease.
• Finally, it should be noted that the SSA used in this paper is just one of many optimization algorithms, and it also faces the problem of falling into the trap of local optimal solutions, which makes it difficult to achieve better optimization results. In the future, more powerful optimization algorithms can be considered to optimize the hyperparameters of deep learning models, further reducing manual intervention, improving the adaptive ability of models, and enhancing the universality of models.