© F. Sun et al., Published by EDP Sciences 2026

1 Introduction

As an important infrastructure, bridges are widely used in constructing roads, railways, and other transportation systems. Ensuring their stability and safety under extreme wind loads is the key to ensuring traffic safety [1,2]. Flutter is the self-excited vibration that may occur in bridges under wind loads. If a flutter occurs, it may cause structural damage or catastrophic consequences [3,4]. Therefore, accurate assessment of bridge flutter characteristics, especially the calculation of critical flutter wind speed, is crucial for bridge design and safety monitoring. The critical flutter wind speed is the wind speed limit at which a bridge encounters flutter [5,6]. The precise calculation of critical wind speed has important guiding significance for the wind-resistant design of bridges. However, traditional flutter derivative identification methods, such as wind tunnel experiments [7,8] and CFD simulations [9,10], have high costs, low efficiency, and dependence on experimental conditions, which leads to limitations in practical applications. These problems urgently need to be overcome through new methods to improve the precision of flutter derivative identification and the efficiency of critical wind speed calculation.

In recent years, as computing power improves and the artificial intelligence technology develops rapidly, more and more scholars have begun to explore the application of machine learning [11,12], deep learning [13,14], and other methods in bridge flutter characteristics analysis. Machine learning methods, due to their powerful data learning capabilities, have shown great potential in aerodynamic characteristics identification. Some studies have begun to attempt to model bridge aerodynamic data through deep neural networks [15,16] to achieve automatic identification of flutter derivatives and calculate the critical flutter wind speed on this basis. These studies have shown that deep learning can effectively overcome the limitations of traditional methods and improve identification precision and calculation efficiency. However, existing deep learning applications still face some challenges, including dealing with complex dependencies in aerodynamic time series data and improving the model's generalization precision.

Currently, the identification of bridge flutter derivatives mainly relies on wind tunnel tests and computational fluid dynamics simulations. These methods have problems such as high cost, long time consumption, and sensitivity to parameter settings. They cannot meet the needs of efficient and accurate aerodynamic stability assessment in the wind-resistant design of large-span bridges. To solve these problems, this study constructs a hybrid architecture of 1D-CNN and Bi-LSTM, in which 1D-CNN precisely extracts turbulence features through local convolution kernels and Bi-LSTM uses bidirectional time series modeling to capture cross-cycle aerodynamic responses. The two work together to solve the problems of complex aerodynamic dependency modeling and multi-condition generalization. This paper uses CFD simulation to generate aerodynamic data of different bridge sections under different incoming flow conditions, and extracts flutter derivatives as training labels through the modal decomposition method. Then, 1D-CNN and Bi-LSTM are combined to extract local features and long-term dependencies from aerodynamic time series data, thereby achieving high-precision identification of flutter derivatives. This combined model can effectively handle aerodynamic characteristics under various bridge sections and wind speed conditions while improving identification precision. The model training process uses the mean square error loss function and optimizes the model parameters through the Adam optimization algorithm. At the same time, this paper further optimizes the calculation process of the critical flutter wind speed by combining the numerical iteration method with the deep learning model. After identifying the flutter derivative, the improved parabola fitting method is used for interpolation to obtain the aerodynamic parameter curve under continuous wind speed. Then, the numerical iteration method is used in conjunction with the improved Scanlan-Tomko flutter criterion to calculate the critical flutter wind speed. This study provides an efficient and precise method for flutter derivative identification and critical wind speed calculation, which provides a new solution for bridge wind-resistant design and other similar fluid-structure interaction problems.

Major Contributions:

• A 1D-CNN-BiLSTM hybrid architecture is constructed, deeply integrating local convolution and bidirectional temporal memory to achieve end-to-end accurate identification of bridge flutter derivatives, eliminating the strong dependence on wind tunnels and CFD post-processing.

• A full-process method of improving the Scanlan-Tomko criterion coupled with Newton iteration is proposed. Through continuous interpolation and fast iterative solution, the calculation cost of critical flutter wind speed is significantly reduced while maintaining engineering-level accuracy.

• A CFD-data closed-loop system is established for diverse bridge types and complex wind fields. Combined with data augmentation strategies, the robustness and generalization ability of the model are systematically verified in the face of changes in structural form, Reynolds number, and environmental disturbances.

2 Related work

The identification of bridge flutter derivatives has long relied on two traditional methods: wind tunnel experiments [17,18] and numerical simulations [19,20]. Talaee & Hakimzadeh used a combined CFD-wind tunnel approach to provide a reliable benchmark for modeling the aerodynamic environment [21]. In wind tunnel experiments, flutter derivatives are extracted using methods such as modal decomposition [22,23], which is costly, time-consuming, and difficult to cover all structural forms and working conditions. Numerical simulation combines CFD and structural dynamics to generate derivative data under more parameter combinations [24,25], but its calculation complexity is huge, and it is highly sensitive to meshing and numerical algorithms. It is prone to convergence and precision problems in complex bridge types or high Reynolds number flow fields.

In view of the limitations of traditional methods, machine learning technology has been applied to flutter derivative identification. In the early stage, shallow learning models such as support vector machines [26,27] and random forests [28,29] were used to improve the identification efficiency in small-scale data scenarios. Recurrent neural networks (RNNs) have also been shown to be effective in capturing the time dependency in structural dynamic responses [30], but they relied significantly on feature engineering, and the precision was difficult to improve further. With the development of deep learning, the structure combining CNN and neural networks began to be applied to aerodynamic time series data processing [31,32]. CNN can automatically extract local features from time series signals and reduce manual intervention. LSTM further enhances the model's ability to express complex nonlinear flutter behaviors. Although deep learning methods have shown advantages in feature extraction and pattern recognition, the model's generalization ability and stability still have room for improvement under changes in wind speed and bridge section shape.

The calculation of critical flutter wind speed has evolved from classical criteria to numerical coupling. The Scanlan-Tomko criterion provides a closed-form solution for simplified sections and regular structures by bringing flutter derivatives and structural natural frequencies, damping, and other parameters into the characteristic equation [33,34]. However, it ignores the influence of complex sections and unsteady flow fields, making it difficult to meet the design requirements of long-span bridges. Fluid-structure interaction numerical simulation methods can more precisely capture wind-structure interactions by simultaneously solving the airflow and structural dynamics equations [35]. They are suitable for complex bridge types, but they consume a lot of computational resources and are difficult to apply in real-time engineering design. However, deep learning can efficiently extract complex nonlinear features from aerodynamic time series data, replacing the manual extraction of complex sections and flow field features in traditional methods.

To improve the efficiency and generalization ability of critical wind speed calculation, researchers combined deep learning with traditional criteria and proposed regression or extreme value prediction models based on neural networks [36,37]. These methods can reduce the calculation time from hours to minutes while maintaining the calculation precision by training a large amount of numerical simulation or experimental data. Although this idea has shown significant advantages in preliminary applications, the current models rely heavily on the amount and diversity of data and lack interpretability, and the generalization ability across structural types and complex wind field conditions still needs further verification and optimization.

3 Bridge flutter derivative identification and critical wind speed calculation method

3.1 Dataset construction

During the dataset construction phase, CFD is utilized to simulate the aerodynamic response of different bridge section shapes under various wind conditions. The simulation objects include several common bridge shapes, such as rectangular, circular, trapezoidal, and curved sections. Different sizes and wind speed conditions are selected for each section according to actual application requirements. Table 1 shows examples of different bridge section shapes and their parameter settings.

Table 1 shows the different bridge section shapes used in this study and their corresponding geometric parameters and wind tunnel experimental settings, including rectangular, circular, trapezoidal, and curved sections, and lists the width, height, width-height ratio, wind speed range, wind attack angle, and mesh size used for numerical simulation. Among them, due to the symmetry of the circular section, its height and width-height ratio parameters are not set. Figure 1 shows different cross-sections of bridges. (a) A prestressed concrete bridge with piers features a flat deck and a simple support structure. (b) A steel box girder bridge, with multiple steel beam supports and a complex internal structure, provides stability. (c) A truss bridge uses a cross-section of trusses to provide strong support and is suitable for long-span bridges. (d) A concrete box girder bridge, with a cross-section featuring hollow areas to increase strength and reduce weight.

To ensure the accuracy of the simulation, a turbulence model based on the Reynolds-averaged Navier-Stokes equations is selected, and the k-ε model is used to simulate complex turbulent flow fields. During the CFD simulation, a combination of unstructured meshes and a local encrypted mesh is used in the fluid domain to refine the flow field near the bridge surface to capture the details of the interaction with the bridge section. The mesh size and division strategy are adjusted according to the specific section shape and simulation precision requirements to ensure the calculation precision of important areas.

The flow field boundary conditions are set as free-stream boundary conditions to simulate the aerodynamic response of the bridge under the action of the wind field. The incoming wind speed in the simulation ranges from low speed to high speed. Simultaneously, the simulation is performed for conditions with different attack angles, and multiple sets of wind speed, attack angle, and aerodynamic data are obtained. Under each simulation condition, the calculation results output a series of time series data, mainly including lift and torque time series data. To accurately reflect the aerodynamic characteristics of the bridge section under different wind speeds and attack angles, the time step of the aerodynamic data is set to 0.01 seconds to ensure that the data sampling is fine enough to accurately capture the aerodynamic changes.

After obtaining aerodynamic data through CFD simulation, the modal decomposition method is used to extract flutter derivatives for each set of lift and torque time series data. First, the lift and torque time series data are analyzed in the frequency domain, and the time domain signal is converted into a frequency domain signal using a fast Fourier transform to identify the frequency components related to bridge flutter. The formula is:

$$X\left(f\right)={\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}x\left[n\right]e^{-j2\pi fn/N},$$(1)

x [n] is the time series data of the bridge lift and torque. X (f) is the converted frequency domain signal. f is the frequency. N is the number of time series data points.

In the frequency domain signal, the amplitude and phase information corresponding to the peak frequency are selected, combined with the natural frequency of the bridge. The mechanical derivatives related to the bridge flutter are obtained through the modal decoupling method. These derivative values are used as output labels in the dataset for the training of the deep learning model. To obtain the natural frequency of the bridge, the following modal equation needs to be solved:

$$M{\displaystyle \ddot{u}}\left(t\right)+C{\displaystyle \dot{u}}\left(t\right)+Ku\left(t\right)=F\left(t\right),$$(2)

M is the mass matrix of the bridge. C is the damping matrix. K is the stiffness matrix. u (t) is the displacement response of the bridge. F (t) is the effect of external aerodynamic force.

To facilitate subsequent model training, the time series data is standardized. Each time series data is first normalized, and the minimum-maximum normalization method is used to map all data to the interval [0,1] to eliminate the influence of different dimensions and amplitudes and reduce numerical instability during training. The normalization formula is:

$$x^{\prime }=\frac{x-\text{min}\left(x\right)}{\text{max}\left(x\right)-\text{min}\left(x\right)}.$$(3)

After normalization, the time window is segmented. In order to improve the adaptability of the model to different wind speeds and bridge forms, the time window length of each aerodynamic time series data is dynamically adjusted. According to the aerodynamic response characteristics of different wind speed conditions and bridge sections, the time window length changes adaptively to ensure better capture of key features in the data.

During the construction of the dataset, data enhancement was also performed. The determination of parameters is based on the statistical analysis of the original CFD simulation data and the understanding of the aerodynamic response characteristics of the bridge. The amplitude range of the random perturbation is set to 10%–20% of the standard deviation of the original data, the upper and lower limits of the time window length adjustment are 0.8 times and 1.2 times the original time step, respectively, and the random deletion and insertion ratio of data points is controlled at 5%–15%. The determination of these parameters is intended to enable the enhanced data to cover more aerodynamic response characteristics under actual working conditions, while avoiding data distortion caused by excessive enhancement. The enhanced dataset contains aerodynamic time series data of various wind speeds, angles of attack, and bridge cross-sectional shapes, covering various typical aerodynamic response conditions.

Fig. 1 Bridge cross-sections. (a) Prestressed concrete bridge with piers, (b) Steel box girder bridge, (c) Truss bridge, (d) Concrete box girder bridge.

3.2 Flutter derivative deep learning model

This paper constructs a hybrid architecture combining 1D-CNN and Bi-LSTM, in which 1D-CNN accurately extracts turbulence features through multiple convolutional layers and local convolution kernels to capture a wider frequency range. In order to further optimize feature extraction, the number of convolutional layers is increased, so that the model can handle more complex time series features and improve its ability to respond to high-frequency changes. At the same time, Bi-LSTM captures cross-period aerodynamic responses through bidirectional time series modeling, enhancing the model's expression of long-term dependencies. The architecture of the combined model is shown in Figure 2, and its structural parameters are shown in Table 2.

The convolutional layer extracts local features from the aerodynamic time series data. At this stage, the convolution operation performs a sliding window process on the input data through a set of convolution kernels. The output of the convolutional layer is a two-dimensional feature matrix, where each column represents the feature representation of a time step, and the size of the matrix depends on the number of convolution kernels and the length of the input data. The pooling layer then uses the maximum pooling method to reduce the dimensionality of the convolutional layer output, reducing the data dimension and retaining important features. The output of each convolution kernel for the input sequence X is:

$$z^{\left(k\right)}\left(t\right)=\text{ReLU}\left({\displaystyle {\displaystyle \sum }_{i=1}^F}{\displaystyle {\displaystyle \sum }_{j=1}^h}{w}_{i,j}^{\left(k\right)}\cdot x_{i,t+j-1}+b^{\left(k\right)}\right),$$(4)

F = 2 is the number of input feature channels . h is the width of the convolution kernel. z^(k) (t) is the output of the k th convolution kernel at time t. (lift and torque)

After the convolutional layer extracts the local features of the aerodynamic time series data, Bi-LSTM is used to capture the long-term dependencies in the data. For input feature z_t at time t, the forward and backward hidden states of the Bi-LSTM are:

$${{\displaystyle \overrightarrow{h}}}_t=\text{LSTM}\left(z_t,{{\displaystyle \overrightarrow{h}}}_{t-1}\right)$$(5)

$${{\displaystyle \overleftarrow{h}}}_t=\text{LSTM}\left(z_t,{{\displaystyle \overleftarrow{\text{h}}}}_{t+1}\right).$$(6)

The final moment output is their concatenation:

$$h_t=\left[{{\displaystyle \overrightarrow{h}}}_t;{{\displaystyle \overleftarrow{h}}}_t\right]\in {ℝ}^{2d}.$$(7)

Bi-LSTM can consider both historical and future information on aerodynamic data. Its LSTM unit can remember important aerodynamic features in the early stage for a long time and pass them to subsequent moments for processing, thus improving the ability to identify medium-term and long-term trends in aerodynamic data. The ReLU activation function is used in the convolutional layer and the Bi-LSTM layer to avoid overfitting problems in the network.

After the Bi-LSTM layer, the output time series feature vector is fed into the fully connected layer. The main function of the fully connected layer is to map the time series features extracted by the Bi-LSTM to the target space, which is the flutter derivative of the bridge in this model. The fully connected layer performs a weighted summation on the output of the Bi-LSTM layer and finally generates an output that matches the target variable.

In order to improve the recognition ability of the model under complex wind speed change conditions, a multi-scale convolution structure is introduced. Multi-scale convolution extracts aerodynamic time series features at different scales, allowing the model to capture both short-term fluctuations and long-term trends. The multi-scale convolution layer can better process features in different frequency ranges, enhancing the performance of the model under complex aerodynamic conditions.

Since this task is a regression problem, the output layer does not use an activation function so that continuous numerical outputs can be generated. The output of the fully connected layer represents the flutter derivatives of the bridge under given wind speed conditions, which are used for subsequent calculations of critical flutter wind speeds. Table 3 shows examples of flutter derivative output, listing the model prediction results for some aerodynamic samples that cover rectangular, trapezoidal, and curved sections, circular fit sections, wind speed ranges, and attack angle settings. In each sample, the corresponding flutter derivatives H₁* and A₂* are identified by the deep learning model. Table 3 reflects the response characteristics of the structure to lift and torque aerodynamic forces, respectively. These example output results are used to intuitively demonstrate the predictive performance of the model under various working conditions, providing basic data support for subsequent critical wind speed assessments.

Fig. 2 1DCNN-BiLSTM combined model architecture.

3.3 Critical flutter wind speed calculation

After obtaining the flutter derivative output by the deep learning model, interpolation processing is performed between the wind speed and the flutter derivative. The flutter derivative data identified by the deep learning model is usually discrete, but the calculation of the critical flutter wind speed requires continuous wind speed as input, so these discrete data must be interpolated. To ensure the smoothness and precision of the data, the parabola fitting method is used to interpolate the flutter derivatives. Based on the given discrete data points, this method uses the least squares method to calculate a quadratic polynomial model to represent the flutter derivatives at different wind speeds. For each wind speed interval, the corresponding interpolation function is calculated according to the adjacent discrete data points during the interpolation process, thereby obtaining a continuous aerodynamic parameter curve. Figure 2 presents the flutter derivative parabolic interpolation curve and critical wind speed determination.

Then, after obtaining the continuous curve of wind speed-flutter derivative, the flutter characteristic equation is established in combination with the improved Scanlan-Tomko flutter criterion. The Scanlan-Tomko criterion determines whether the wind speed reaches the critical value based on the relationship between aerodynamic damping and wind speed. In this paper, the flutter derivatives identified by the deep learning model are substituted into the criterion to calculate the aerodynamic damping. The improved Scanlan-Tomko criterion considers the dynamic effects of wind speed changes and constructs the flutter characteristic equation suitable for a specific bridge section based on the identified flutter derivatives and aerodynamic characteristics.

To more accurately fit the nonlinear relationship between wind speed and flutter derivatives, a higher-order interpolation method was introduced, building upon the existing parabolic fit. Specifically, the wind speed and flutter derivative data points were interpolated using cubic spline interpolation. After sorting the data points by wind speed, a smooth interpolation curve was constructed using a cubic spline function, ensuring that the relationship between wind speed and flutter derivatives was accurately captured within each wind speed range.

During the calculation process, an initial wind speed value is selected, and the corresponding aerodynamic characteristics are calculated based on the flutter derivatives identified by the deep learning model. Then, the wind speed is gradually increased in increments, and the corresponding aerodynamic damping is repeatedly calculated until the critical condition in the improved Scanlan-Tomko flutter criterion is met. Specifically, when the wind speed reaches a certain value, the aerodynamic damping approaches zero, indicating that the bridge is about to flutter. This wind speed is the critical flutter wind speed.

This paper adopts Newton's method to perform iterative calculations of wind speed to improve calculation precision and efficiency. The initial wind speed is set to U₀, and the corresponding aerodynamic damping coefficient C_a (U₀) is calculated using the identified flutter derivative. Then, the wind speed value is updated using Newton's method, and the gradient is calculated based on the aerodynamic damping at the current wind speed to quickly obtain a new wind speed estimate:

$$U_{n+1}=U_n-\frac{C_a\left(U_n\right)}{C_a^{\text{'}}\left(U_n\right)},$$(8)

$C_a^{\text{'}}\left(U_n\right)$ is the derivative of C_a (U_n) with respect to wind speed U. In each iteration, Newton's method adjusts the step size to ensure the convergence of the calculation process so as to find the critical wind speed that meets the flutter criterion in a small number of iterations. Compared with the traditional stepwise approximation method, Newton's method can significantly improve the calculation speed and reduce redundant calculations.

In each iteration, the aerodynamic parameters are updated based on the flutter derivatives output by the deep learning model. After each wind speed update, the flutter derivatives under the wind speed are recalculated to obtain new aerodynamic characteristics. In the actual calculation, to ensure the stability and accuracy of the calculation, appropriate convergence criteria are also utilized. Specifically, when the amplitude of the wind speed change is less than the set tolerance, the calculation is considered to have reached convergence, and the iteration is stopped. The current wind speed value is output as the critical flutter wind speed.

3.4 Experiment

The dataset is divided into a training set, validation set, and test set, with a ratio of 70%, 15%, and 15%. The training set is used for model training. The validation set is used to adjust the model's hyperparameters and avoid overfitting. The test set is used to assess the model's generalization ability. In the process of dataset division, a stratified sampling strategy is used to ensure that the data subsets are balanced in terms of bridge section type, wind speed range, and attack angle distribution. Assessment indicators such as MSE, MAE, and root mean squared error (RMSE) are used during the test. The input data is the aerodynamic time series data, and the output is the corresponding flutter derivative. All data are normalized before input into the model to ensure that the data between different wind speeds and bridge sections are processed on the same scale. To facilitate model learning, the aerodynamic time series data is divided into time windows, and each data sample can reflect the complete aerodynamic response process, thereby reducing the complexity of data processing. Table 4 lists the training parameter settings of the neural network.

During the training process, gradient clipping technology is used to prevent gradient explosion and maintain the numerical stability of the model during training. After each training epoch, the loss function of the training set and the validation set is calculated to evaluate the model's learning effect and generalization ability. The loss function of the validation set is used to adjust the hyperparameters during the training process to ensure that the model's generalization performance does not decrease. During the training process, the errors of the training set and the validation set are compared. If the error of the validation set continues to increase, it indicates overfitting, and the model structure or parameters should be adjusted in time. After the training is completed, the test set is used to verify the final precision of the model, and the final test error is recorded to ensure that the model's performance meets expectations.

This experiment is equipped with an Intel Core i7-10700K processor, 16 GB memory, and an NVIDIA GTX 1660 Ti graphics card. The operating system is Windows 10, and the deep learning framework is PyTorch 1.9.1. To verify the effect of the model in this paper, the model in this paper is used as the experimental group, and the combined model based on the deep residual network and LSTM is used as the control group.

4 Result analysis

4.1 Flutter derivative identification error analysis

To further verify the error control ability of the model in predicting derivative key features from the statistical distribution level, this paper conducts error analysis on eight aerodynamic flutter derivative feature items. Large sample simulations are performed on the absolute error values of the experimental group and the control group under each derivative item. Each group is sampled 100 times, and an error distribution histogram is constructed to explore the differences in stability and robustness of the model in multi-dimensional output scenarios, especially in terms of error distribution concentration and extreme value control ability.

Figure 3 shows the distribution of absolute errors of the prediction of the two models on the eight aerodynamic derivative parameters. From the distribution morphology, the error distribution of the experimental group is generally more concentrated, with a significant main peak and a small skewness, and most errors are concentrated within 0.005. In contrast, the errors of the control group show a wider distribution range and a higher long tail. In addition, the frequency of error peaks is also significantly different. The main peak of the experimental group is higher and more concentrated than that of the control group, showing a differential advantage in error control.

The above phenomena reflect the difference in the underlying modeling capabilities of the two types of models when dealing with unsteady aerodynamic response data. The experimental group model has stronger generalization ability and nonlinear mapping precision, and its error distribution is compact and stable, mainly due to a more reasonable structural design and sufficient feature extraction mechanism, especially in the micro-vibration response range with higher identification sensitivity. The control group may rely on traditional or parametric modeling methods, which have a tendency to respond bluntly or underfit when the derivative changes drastically or fluctuates nonlinearly, resulting in a significant increase in error.

In order to verify the generalization performance of the model under complex actual working conditions, this study examined the performance of the model in typical engineering scenarios such as turbulence intensity gradient changes, asymmetric wind attack angle disturbances, and extreme temperature-air pressure coupling conditions. Through simulation, a CFD aerodynamic data set containing different turbulence levels (5%–20%), asymmetric wind attack angles (±0° to ±10°), and extreme low temperatures (−20°C) was generated, focusing on evaluating the model's ability to maintain recognition accuracy in a strong nonlinear and strong interference environment. Table 5 shows the test parameters of each key working condition and the statistical results of flutter derivative identification errors.

The results show that under standard conditions, the MAE of the experimental group is 2.15%; under high turbulence intensity conditions, the MAE is 3.45%; and under asymmetric wind angle of attack conditions, it is 2.98%. Under extremely low temperature conditions, the MAE of the experimental group is 3.10%, and that of the control group is 4.50%. Under comprehensive complex conditions, the MAE of the experimental group is 3.25%, and that of the control group is 5.10%. The difference in the data in the table reflects the difference in the mechanisms of the two models when processing complex aerodynamic time series data. The experimental group model combines 1D-CNN and Bi-LSTM structures, which can simultaneously extract local features and long-term time dependencies. Under high turbulence intensity conditions, 1D-CNN captures high-frequency fluctuation characteristics in aerodynamic time series through local convolution kernels, while the bidirectional modeling mechanism of Bi-LSTM can integrate future and historical information and enhance the ability to capture long-term dependencies, thereby maintaining a low MAE under asymmetric wind angle of attack and extreme temperature conditions. In contrast, the control group model relies on unidirectional time series processing or a shallower network structure, and cannot effectively integrate multi-dimensional information when facing complex environmental disturbances, resulting in increased errors. In addition, the excellent performance of the experimental group under extremely low temperature conditions also shows that its feature extraction mechanism can adapt to the changes in air density caused by different temperatures, further verifying the adaptability and robustness of the model to actual engineering scenarios.

Fig. 3 Schematic diagram of flutter derivative parabolic interpolation curve and critical wind speed determination.

4.2 Critical flutter wind speed calculation error analysis

To further evaluate the error performance of the method in predicting critical wind speed under different bridge section types and wind speed conditions, this paper carries out a systematic comparative experiment on typical bridge aerodynamic sections. Four representative bridge section forms are selected, including rectangular closed box girder, circular streamlined box girder, trapezoidal slotted box girder, and curved box girder, and analyzed in combination with 5 fixed wind speed points to comprehensively characterize the model's generalization ability and stability in complex aerodynamic environments.

Figure 4 shows the relative error prediction results of the critical wind speed for four different professional cross-section box girders (rectangular closed, circular streamlined, trapezoidal slotted, and curved split) at wind speeds of 5 to 25 m/s. The horizontal axis represents the wind speed (in m/s), and the vertical axis represents the prediction error. Each sub-figure corresponds to a bridge section type. From the overall trend, the experimental group shows lower prediction errors in all section types and wind speed ranges, and the error increases significantly less with wind speed than the control group. For example, in the circular streamlined box girder, the error of the experimental group at a wind speed of 25 m/s is maintained at about 2.02%, while that of the control group is 4.94%; in the trapezoidal slotted box girder, the maximum error of the control group is 6.64%. In addition, the error curve of the experimental group is smoother overall, and the variance fluctuation is smaller, indicating that the model has better stability and adaptability under multiple sections and speed combinations.

The main reason behind this performance is the structural advantages and time series modeling capabilities of the method in this paper. The proposed model combines convolutional layers with a Bi-LSTM structure and has the ability to simultaneously extract local spatiotemporal features and global dynamic patterns, enabling it to more fully understand the aerodynamic response mechanisms of different sections at various wind speeds. In addition, by performing time window segmentation and standardization on the raw time series aerodynamic data, the adaptability of the model in input scale and structure is enhanced, and the influence of wind speed distribution differences on prediction precision is avoided. In contrast, the control group model lacks an effective temporal memory mechanism. Its anti-interference ability to wind speed changes is weak, and its error control ability is even worse in complex sections. Therefore, the experimental group shows better robustness and consistency under multiple sections and multiple wind speeds, reflecting its high ability to extract and generalize aerodynamic time series features.

In order to evaluate the adaptability and stability of the model under complex structural conditions, this study selected five special bridge deck cross-section forms that are relatively rare in actual engineering but have complex aerodynamic characteristics, constructed a targeted aerodynamic data set, and carried out systematic tests on recognition accuracy, critical wind speed prediction error, and computational efficiency on this basis. By quantitatively analyzing the performance of the model under these atypical bridge deck structures, its generalization ability and robustness when the structural geometry changes are verified. As shown in Table 6.

In terms of flutter derivative MAE, the error range of the experimental group was 3.06% to 3.58%, while that of the control group was as high as 5.12%. In terms of relative error of critical wind speed, the maximum value of the experimental group was 4.22%, and the maximum value of the control group was 6.91%. In terms of prediction time, the experimental group was between 18.5 seconds and 20.1 seconds, while the control group fluctuated between 24.4 seconds and 27.3 seconds. These data show that under complex geometric conditions, the experimental group model not only maintains a higher recognition accuracy, but also shows better computational efficiency and more stable wind speed prediction performance.

The above performance differences are mainly attributed to the deeper modeling ability of the 1D-CNN and Bi-LSTM combined structure adopted by the experimental group for aerodynamic time series characteristics. The 1D-CNN structure can effectively extract the instantaneous disturbance characteristics of the local irregular aerodynamic response under special bridge decks. For example, under the background of high-frequency fluctuations, the experimental group of the Wavy-Edged Open Box Girder still maintained a MAE of 3.15% and a wind speed error of 3.18%, which is much lower than the 4.66% and 5.79% of the control group. The Bi-LSTM integrates contextual information through a bidirectional temporal modeling mechanism. For structures such as the Tapered Section with Reinforced Ribs with a gradual change in cross-section along the length of the bridge, its long-term dependence characteristics are stronger, so it can better capture the aerodynamic evolution trend, thereby significantly reducing the error. The control group has a high error in the above-mentioned complex bridge type. The main reason is that its structure does not have the ability to fuse bidirectional information, and it does not capture local mutation characteristics in a non-stationary aerodynamic environment, resulting in higher recognition errors and time-consuming fluctuations under special structural conditions.

Fig. 4 Comparison of model prediction error distribution under various aerodynamic derivatives. (a) Vertical aerodynamic stiffness error distribution, (b) Vertical aerodynamic damping error distribution, (c) Vertical aerodynamic coupling error distribution, (d) Vertical added mass error distribution, (e) Torsional aerodynamic stiffness error distribution, (f) Torsional aerodynamic damping error distribution, (g) Torsional aerodynamic coupling error distribution, (h) Torsional added inertia error distribution.

4.3 Model robustness analysis

To assess the robustness differences of bridge flutter derivative identification models in different noise environments, this study adds gradient noise to the aerodynamic time series data generated by CFD and systematically compares the performance of the experimental group model based on 1D-CNN and Bi-LSTM and the control group model based on ResNet and LSTM.

In Figure 5, the horizontal axis represents the noise level (signal-to-noise ratio, SNR, in dB), and the vertical axis represents the mean square error, which shows the flutter derivative identification precision of the experimental group and the control group under different noise interferences. The MSE of the experimental group is 0.012 when SNR = 5 dB, lower than 0.014 of the control group. The standard deviations of the errors are 0.003 and 0.005, respectively, indicating that the experimental group has a stronger ability to extract aerodynamic time series features under strong noise. As the SNR increases to 30 dB, the errors of both groups converge to 0.0021, but the slope of the error decrease of the experimental group is more gentle. The error bars in the figure reflect that the hybrid model has a better ability to suppress noise interference.

The error advantage of the experimental group is due to the hierarchical processing mechanism of aerodynamic features in its model structure. 1D-CNN efficiently extracts instantaneous fluctuation features in lift and torque time series (such as high-frequency components caused by turbulence) through local convolution kernels. Bi-LSTM captures long-term dependencies of aerodynamic forces (such as periodic vibration modes) through bidirectional time series modeling. The two work together to enhance the robustness to noise. The ResNet in the control group relies on residual connections to transfer features, which may cause noise interference due to skip connections. The unidirectional LSTM can only utilize historical information and cannot simultaneously integrate future temporal contexts like the Bi-LSTM. In addition, the normalization and time window segmentation strategies in CFD data preprocessing further weaken the impact of noise on the experimental group. In a high-noise environment (SNR = 5 dB), the control group has an enlarged error due to insufficient local feature extraction and one-sided time series modeling, while the experimental group effectively suppresses the interference of noise on flutter derivative identification through multi-scale feature fusion.

Fig. 5 Comparison of the relative error of the critical wind speed with the wind speed for different bridge section types. (a) Critical wind speed error curve for a rectangular closed box girder, (b) Critical wind speed error curve for a circular streamlined box girder, (c) Critical wind speed error curve for a trapezoidal slotted box girder, (d) Critical wind speed error curve for a curve-separated box girder.

4.4 Calculation efficiency effect

To systematically evaluate the comprehensive performance of the deep learning model in the task of flutter derivative identification, this study conducts a statistical analysis of the calculation efficiency and error distribution through 50 independent tests, covering the time consumption and prediction error of a single flutter derivative identification. Figure 6 intuitively presents the differences in characteristics of the two models in computational resource usage and identification precision through scatter distribution. Figure 7a shows the computation time distribution of the two methods, and Figure 7b shows the computation error distribution of the two methods.

The calculation time of the experimental group ranges from about 15 to 23.1 seconds, with an average of about 17.68 seconds. The calculation time of the control group ranges from 19 to 34.28 seconds, with an average of nearly 23.43 seconds. The error of the experimental group falls in the range of 0.05–0.0111, with a mean of about 0.074. The error of the control group is concentrated in the range of 0.08–0.144, with a mean of about 0.1. In terms of the reduction in calculation time, the experimental group saves about 5.75 seconds on average, and the overall calculation time decreases by more than 24.54%, which reduces the engineering simulation cost and operating burden in CFD-assisted design scenarios with multiple bridge locations and sections. In edge devices or resource-constrained deployment environments, this lightweight structure can complete real-time predictions more quickly, improving the flexibility and timeliness of model applications. The error is reduced by about 26%, indicating that the model can more accurately capture the flutter derivative information implied in the changes in lift and torque.

Fig. 6 Comparison of model prediction errors with different signal-to-noise ratios.

Fig. 7 Comparison of calculation efficiency and error distribution. (a) Comparison of calculation time. (b) Comparison of calculation error.

4.5 Comparison of Reynolds number sensitivity of model performance

To further explore the adaptability and stability of the proposed deep learning model under different flow scales, this paper further conducts a Reynolds number sensitivity analysis and compares the identification performance of the model in the typical Reynolds number range of 1 × 10⁴ to 1 × 10⁶. For each Reynolds number level, the identification accuracy and error performance of the two groups of bridge flutter derivatives under different scale flow conditions are compared. The evaluation indicators include MAE, RMSE, and identification accuracy. In addition, the statistical significance of the performance difference between the two groups of models is verified by the significance test. Table 7 lists the results.

When the Reynolds number is 1 × 10⁴, the MAE of the experimental group model is 2.15%; the RMSE is 3.02%; the identification accuracy is 97.85%. As the Reynolds number increases to 1 × 10⁵, the MAE and RMSE of the experimental group are 2.08% and 2.95%, and the accuracy reaches 97.92%. Under high Reynolds number conditions, although the error of the experimental group increases slightly, with an MAE of 3.21% and an RMSE of 4.17%, it is still better than that of the control group. Moreover, the p-values are always less than 0.05, indicating that the performance difference between the two groups at each Reynolds number level is statistically significant.

When the Reynolds number is low, the flow near the bridge surface is mainly laminar. The separation point is relatively stable, the aerodynamic signal fluctuates less, and the feature frequency is concentrated, with a high signal-to-noise ratio. At this time, 1D-CNN can efficiently extract local features with stable frequencies through a fixed-size convolution kernel, and Bi-LSTM is more likely to capture the evolution law across time steps in relatively stable data, so the identification error is small. As the Reynolds number increases to 1 × 10⁵, although the boundary layer gradually turns into partial turbulence, since the model has been trained with a large number of samples at different scales, it can adapt to more complex local features through the enlarged convolution kernel receptive field. At the same time, the future information brought by the Bi-LSTM enables it to maintain low-error identification in an environment with increased volatility.

Under high Reynolds number conditions, the flow completely turns into a turbulent state. The separation point swings violently, and the vortex structure is complex and unstable. The aerodynamic signal shows stronger randomness and non-stationarity. This non-Gaussian disturbance characteristic makes the convolution kernel face the challenge of information aliasing when extracting features, especially when the window contains multiple frequency components simultaneously, resulting in a decrease in the expression ability of local features. Meanwhile, when Bi-LSTM faces non-periodic sequences with frequent signal mutations, the advantages of the long-term memory structure are relatively weakened, and the features of some key moments may be submerged or weakened in the back propagation, thereby affecting the final derivative identification precision. It is worth noting that even so, the experimental group is still significantly better than the control group, mainly due to the fact that the deep learning model fully fits the aerodynamic patterns of different Reynolds numbers during the training process, and its feature extraction and nonlinear mapping capabilities have surpassed the applicability limitations of traditional empirical methods in the high Reynolds number range. The main reason for the increase in error is that the non-stationarity of the aerodynamic response at high Reynolds numbers increases and the vortex structure becomes more complicated, which makes the local disturbance features in the time series more dense and increases the difficulty of the model to extract multi-scale features. Despite this, the synergistic mechanism of 1D-CNN and Bi-LSTM in the model structure still effectively captures the dominant aerodynamic mode in most cases, showing good robustness and stability. This result shows that the proposed method has a strong generalization ability for typical flow state transition processes and is practically feasible in engineering applications across Reynolds numbers.

5 Discussion

The model in this paper combines 1D-CNN and Bi-LSTM structures to extract local features and long-term dependencies in aerodynamic time series data, thereby improving the identification precision of flutter derivatives. This method shows stable generalization ability in different bridge section forms such as rectangle, trapezoid, and curved surface. The experimental results show that the model maintains a high identification accuracy (MAE≤3.21%) in the Reynolds number range of 1 × 10⁴ to 1 × 10⁶, verifying its adaptability to laminar and turbulent states. However, the slight increase in error at high Reynolds numbers (such as MAE of 3.21% at Reynolds number 1 × 10⁶) indicates that the model is still sensitive to the unsteady aerodynamic response caused by turbulence. Therefore, this method is more suitable for bridge section flutter analysis within the conventional Reynolds number range, and physical constraints need to be combined to enhance robustness under extremely turbulent conditions.

Model performance is highly dependent on the diversity and representativeness of CFD-generated data. This study ensures the comprehensiveness of the dataset by covering multiple section shapes, a wide range of wind speeds, and multiple attack angles. Data augmentation strategies (such as noise injection and time window segmentation) further improve the model's adaptability to practical scenarios. However, the CFD data is limited by the precision of the turbulence model (the k-ε model's ability to simulate separation flows) and the meshing strategy, resulting in some aerodynamic features not being fully captured, thus affecting the model's identification of subtle flutter phenomena. In the future, the quality of the dataset can be further optimized by integrating wind tunnel experimental data or applying high-fidelity simulations (such as large eddy simulations).

The critical wind speed calculation framework based on the improved Scanlan-Tomko criterion and Newton's method demonstrates high efficiency and reliability under various bridge sections and wind field conditions. For example, the relative error of the circular streamlined box girder at a wind speed of 25 m/s is only 2.02%, which verifies the method's ability to handle complex aerodynamic coupling. Its advantage is that it constructs a continuous wind speed-derivative curve through parabolic interpolation, avoiding the precision loss of traditional piecewise linear interpolation. However, the interpolation assumption of aerodynamic derivatives in this method cannot fully describe the strongly nonlinear aerodynamic behavior, and a higher-order interpolation or data-driven correction model is required in extreme wind attack angles or asymmetric sections. In addition, the current framework does not consider structural nonlinearity, which may limit its direct application in super-long span bridges.

Deep learning models have significant advantages in calculation efficiency. The identification of a single flutter derivative takes about 17.68 seconds, which is 24.54% less than the control group, and the error is reduced by 26%. Combined with GPU acceleration, this method can complete batch calculations of multiple sections and multiple working conditions in minutes. For GPU acceleration solutions, the CUDA platform and cuDNN library are used to optimize and accelerate deep learning models by allocating computing tasks to the parallel computing units of the GPU. In engineering design, this feature supports rapid iterative optimization, such as assessing the aerodynamic stability of different section shapes. However, the model training still requires high calculation costs, which may pose a challenge to resource-constrained scenarios. In the future, the deployment threshold can be lowered through knowledge distillation or lightweight network design.

To more clearly position this research within the existing state of the art, Table 5 quantitatively compares the performance of our method with other data-driven approaches. As shown in Table 8, the proposed 1D-CNN-BiLSTM model not only achieves superior recognition accuracy but also the fastest single-sample inference speed.

Model optimization can be carried out from two aspects: structural improvement and physical constraint integration. In terms of structure, the ability to capture key time series features can be enhanced through the attention mechanism. In terms of physical constraints, the Navier-Stokes equation is embedded in the loss function as a regularization term to improve the physical consistency of the model for unsteady flow fields. In addition, a multi-task learning framework is constructed to simultaneously predict flutter derivatives and critical wind speed, or transfer learning is used to adapt real wind tunnel data to further improve engineering practicality. At the data level, combining generative adversarial networks to synthesize extreme working condition data or developing digital twin systems to achieve real-time data-model interaction is expected to break through the applicable boundaries of current methods.

Although the proposed model shows good recognition accuracy and generalization ability under various bridge sections and typical wind field conditions, there is still room for further optimization in the selection of model hyperparameters. The current training mainly optimizes basic parameters such as learning rate and batch size, while the specific impact of key structural parameters such as the number of LSTM units, convolution kernel scale, and time window length on model accuracy and stability has not been systematically analyzed. In the future, grid search or Bayesian optimization methods can be combined to further explore the sensitivity of each hyperparameter in the high-dimensional parameter space to the prediction performance, so as to achieve refined control and performance improvement at the structural level.

On the other hand, although this paper has quantitatively compared the proposed model with the control group in terms of error, calculation time and other dimensions, the in-depth comparison with the existing mainstream technologies in terms of robustness, scalability and adaptability to complex working conditions is still insufficient. Subsequent research can introduce more representative algorithms to conduct a more comprehensive horizontal evaluation from the dimensions of error distribution, time complexity, convergence stability, etc., so as to clarify the relative advantages of this method in engineering practicality.

6 Conclusion

The proposed model robustly identifies flutter derivatives under diverse conditions, achieving an average absolute error of ≤3.21%. In the calculation of critical wind speeds for a 25 m/s circular streamlined box girder, computational efficiency is improved by 24.54% and error is reduced by 26%. These results highlight the practical significance of this approach, providing a reliable alternative to wind tunnel testing and CFD simulations, thereby facilitating rapid aerodynamic assessments during bridge design. Future research will focus on embedding physical laws to enhance the model's robustness, adapting the model to more complex structural nonlinearities, and validating its performance using real-world monitoring data to further promote its engineering application.

Nomenclature

Funding

This work was supported by Guangxi Natural Science Foundation (2023GXNSFAA026418), National Natural Science Foundation of China (52178468; 52268023), and Basic scientific research business expenses in Heilongjiang Province (YWF10236230212).

Conflicts of interest

The authors declare no conflict of interest

Data availability statement

This article has no associated data generated and/or analyzed / Data associated with this article cannot be disclosed due to legal/ethical/other reasons.

Author contribution statement

Fangjin Sun, Jun Peng designed the research study. Daming Zhang and Yufei Li analyzed the data. Fangjin Sun wrote the manuscript. All authors contributed to editorial changes in the manuscript. All authors read and approved the final manuscript.

References

All Tables

All Figures

	Fig. 1 Bridge cross-sections. (a) Prestressed concrete bridge with piers, (b) Steel box girder bridge, (c) Truss bridge, (d) Concrete box girder bridge.
In the text

	Fig. 2 1DCNN-BiLSTM combined model architecture.
In the text

	Fig. 3 Schematic diagram of flutter derivative parabolic interpolation curve and critical wind speed determination.
In the text

Fig. 4 Comparison of model prediction error distribution under various aerodynamic derivatives. (a) Vertical aerodynamic stiffness error distribution, (b) Vertical aerodynamic damping error distribution, (c) Vertical aerodynamic coupling error distribution, (d) Vertical added mass error distribution, (e) Torsional aerodynamic stiffness error distribution, (f) Torsional aerodynamic damping error distribution, (g) Torsional aerodynamic coupling error distribution, (h) Torsional added inertia error distribution.

In the text

Fig. 5 Comparison of the relative error of the critical wind speed with the wind speed for different bridge section types. (a) Critical wind speed error curve for a rectangular closed box girder, (b) Critical wind speed error curve for a circular streamlined box girder, (c) Critical wind speed error curve for a trapezoidal slotted box girder, (d) Critical wind speed error curve for a curve-separated box girder.

In the text

	Fig. 6 Comparison of model prediction errors with different signal-to-noise ratios.
In the text

	Fig. 7 Comparison of calculation efficiency and error distribution. (a) Comparison of calculation time. (b) Comparison of calculation error.
In the text