Method for predicting buffeting response of suspension bridge based on enhanced bayesian physically constrained extreme learning machine

Abstract

The application relates to a suspension bridge flutter response prediction method based on an enhanced Bayesian physical constraint extreme learning machine, and belongs to the technical field of suspension bridge wind resistance design. The method constructs a mixed base function form fusing a Fourier base function and a hyperbolic tangent base function for the established bridge flutter dynamic equation, and builds an enhanced extreme learning machine network. Then, a physical constraint condition dominated by a flutter dynamics control equation is introduced, a special form of a trial solution is constructed to perform hard constraint embedding processing on initial conditions, and a regularization ridge regression method is used to solve the output layer weight of the enhanced extreme learning machine network. Finally, based on a Bayesian inference framework, the posterior distribution of the output layer weight coefficient is estimated, and a bridge flutter response prediction result and an uncertainty interval are obtained. The application can realize high-precision prediction of bridge flutter time domain response, and can give reliable probability interval estimation. The application shows good robustness under the conditions of wind speed fluctuation and noise level change.

Description

Technical Field

[0001] This invention belongs to the field of wind-resistant design technology for suspension bridges, and relates to a method for predicting the flutter response of suspension bridges based on an enhanced Bayesian physical constraint limit learning machine. Background Technology

[0002] As the span of suspension bridges continues to increase, their natural frequencies and damping decrease significantly, greatly increasing their sensitivity to wind loads. While buffeting, a common form of wind-induced vibration, does not lead to catastrophic instability and failure like flutter, excessive buffeting responses still pose a serious threat to the safety of construction workers and the structural integrity during construction. During operation, the buffeting effect significantly shortens the fatigue life of structural components. Therefore, buffeting analysis of long-span suspension bridges has significant engineering implications.

[0003] In recent years, deep learning models have been widely used in alternative model construction and are increasingly being applied to predict bridge flutter response, thanks to their powerful feature extraction and data learning capabilities. For example, Castellon et al. used multilayer perceptrons and support vector regression models to study the flutter response of long-span cable-stayed bridges; Liu et al. used long short-term memory networks (LSTM) to predict future wind speed and direction and conducted bridge flutter analysis accordingly; Laima et al. constructed a frequency domain flutter response prediction framework composed of bidirectional LSTMs; Hu et al. proposed a flutter response prediction model for thin plates that integrates LSTM and transfer learning; Zhu et al. studied bridge tower flutter response based on a combination of convolutional neural networks (CNN) and BiLSTMs; and Nav et al. used autoencoders and LSTMs to predict the time-domain flutter response of suspension bridges. Although the above methods can effectively predict bridge flutter response to some extent, they are essentially "black box" models, lacking physical interpretability, and their prediction performance is highly dependent on the quality of the training data.

[0004] To overcome the aforementioned shortcomings, Physics-Informed Neural Networks (PINNs) have been increasingly introduced into the field of structural dynamics analysis in recent years. Wang et al. proposed a deep learning model incorporating physical knowledge to predict linear and nonlinear dynamic responses under wind loads; Zhang et al. constructed a physics-guided CNN and a physics-constrained multi-LSTM network for predicting nonlinear structural dynamic responses; Su et al. combined PINN with explicit temporal methods for predicting nonlinear structural responses; and Wang et al. proposed a physics-preserving graph learning model to achieve rapid computation of structural dynamic responses. Although PINN improves model accuracy and generalization ability by introducing physical information, its purely physics-driven training mode still suffers from low computational efficiency. PINN training requires numerous iterations, with gradients calculated via backpropagation in each iteration, resulting in significant computational time consumption.

[0005] To address this issue, Dwivedi et al. proposed a Physically Constrained Limit Learning Machine (PIELM) for solving time-varying linear partial differential equations. Its output layer weights can be directly solved using a linear least squares problem, resulting in significantly higher computational efficiency than PINN. Liu et al. constructed a Bayesian Physically Constrained Limit Learning Machine (BPIELM) based on Bayesian methods, achieving estimation of the posterior distribution of model hyperparameters. Li et al. proposed a PIELM model for solving biharmonic equations, using Fourier expansion as the activation function.

[0006] However, existing PIELM models still suffer from insufficient accuracy and efficiency when solving large-scale dynamic equations, mainly in the following three aspects: (1) Traditional PIELM models often use simple activation functions of a single form, which are difficult to characterize the complex dynamic characteristics of multi-degree-of-freedom long-time systems; (2) PIELM models usually embed boundary conditions into the loss function through penalty function terms, and their constraint errors are affected by both the equation residuals and boundary weight parameters, resulting in stability issues; (3) PIELM models are difficult to effectively assess the uncertainty of random response prediction. Although some studies on chattering response prediction have introduced Bayesian theory, most of them are still purely data-driven methods and lack physical interpretability. Summary of the Invention

[0007] In view of this, the purpose of this invention is to provide a method for predicting the flutter response of suspension bridges based on an enhanced Bayesian physical constraint extreme learning machine. This method deeply integrates the extreme learning machine (ELM) constructed based on mixed basis functions with the physical constraint flutter dynamics equation, and introduces a Bayesian inference framework to estimate the posterior distribution of the weights from the hidden layer to the output layer, thereby achieving probabilistic prediction of the flutter response.

[0008] To achieve the above objectives, the present invention provides the following technical solution: A method for predicting the flutter response of suspension bridges based on an enhanced Bayesian physics-constrained limit learning machine, characterized in that the method includes the following steps: S1. Considering the buffeting response of the bridge under wind load, establish the mechanical equation for bridge buffeting. S2. Construct a hybrid basis function form that integrates Fourier basis functions and hyperbolic tangent basis functions, and build an enhanced extreme learning machine network; S3. Introduce physical constraints dominated by the buffeting dynamics control equations, and construct a special form of trial solution to perform hard constraint embedding on the initial conditions. S4. The output layer weights of the augmented extreme learning machine network are solved using the regularized ridge regression method. S5. Based on the Bayesian inference framework, the posterior distribution of the weight coefficients of the output layer is estimated to obtain the prediction results of the bridge flutter response and its uncertainty interval.

[0009] Furthermore, in step S1, the mechanical equation for the bridge's buffeting response under wind load is first established. The bridge is simplified as a three-degree-of-freedom continuous system with linear damping. The displacements of the bridge deck in the lateral, vertical, and torsional directions are denoted as follows: In the wind axis coordinate system, aerodynamic drag, lift, and pitching moment are denoted as follows: An additional torsional aerodynamic damping term is introduced into the wind load model. Its velocity relative to the aerodynamic center of the cross section Since the distance between the shear center and the aerodynamic center is related to the ratio of the beam width, the mechanical equation for bridge flutter is expressed as:

[0010] In the formula: This is the quality matrix; Here is the damping matrix; Here is the stiffness matrix; and These are the static wind load matrix and the buffeting wind load matrix, respectively; This is the acceleration response vector; and These are the velocity and displacement response vectors, respectively. and These represent the aerodynamic damping matrix and the aerodynamic stiffness matrix, respectively. Average wind speed; and These represent the downwind and vertical pulsating wind speed components, respectively. Moving the aerodynamic damping and aerodynamic stiffness terms to the right-hand side of the equation, the bridge flutter force equation is rewritten as follows:

[0011] In the formula, The total wind load matrix can be expressed as a static wind term. Vibration force item pneumatic damping term and aerodynamic stiffness A linear combination of .

[0012] Furthermore, in step S2, an enhanced extreme learning machine (ESM) is used to construct a functional mapping relationship between the wind speed input and the bridge flutter response. The input layer of the single hidden layer of the ESM consists of a size of... The time series structure, with hidden layer output dimension as... ,in Indicates the structural degrees of freedom. Indicates the number of hidden layer nodes; then the th The output of each hidden layer node is represented as:

[0013] In the formula, Indicates the first The output of each hidden layer node; For activation functions; and These are the randomly generated input weights and biases, respectively. The activation function from the input layer to the hidden layer is designed as a hybrid basis function combining Fourier basis functions and hyperbolic tangent functions. The total output of the hidden layer is expressed as:

[0014] in, This indicates the hidden layer output using Fourier basis functions. This indicates the hidden layer output using the hyperbolic tangent basis function; The final output of the augmented extreme learning machine is then expressed as:

[0015] In the formula, This is the output matrix of the hidden layer; For hidden layers up to the 1st The output weight vector of each output node.

[0016] Furthermore, in step S2, the hidden layer output using Fourier basis functions... Its first and second derivatives are as follows:

[0017] In the formula, This represents the number of hidden layer nodes in the Fourier basis functions; Hidden layer output using hyperbolic tangent basis functions Its first and second derivatives are as follows:

[0018] In the formula, This represents the number of hidden layer nodes using the hyperbolic tangent basis function.

[0019] Furthermore, in step S3, a time-variable transformation function is introduced for the constructed augmented extreme learning machine model. The trial solution with embedded initial conditions is expressed as:

[0020] In the formula, and These represent the bridge's displacement and velocity at the initial moment, respectively. Perform time-related analysis on the constructed trial solution function. Taking the derivatives, we obtain the expressions for the velocity and acceleration of the chattering response:

[0021] In the formula, This represents the weights from the hidden layer to the output layer. This indicates the output of the hidden layer. This represents the first derivative of the hidden layer output. This represents the second derivative of the hidden layer output.

[0022] Furthermore, in step S4, the output layer weights are solved. The specific process includes: First, the physical loss function in the rewritten bridge flutter force equation is minimized, which is equivalent to minimizing the sum of squares of the output errors, i.e.:

[0023] Next, based on the constructed trial-and-error equations and the velocity and acceleration expressions for the chattering response, the physical loss function is equivalently transformed into solving the following system of linear equations:

[0024] In the formula, For dimension is The system matrix; For dimension is The weighted coefficient vector corresponds to all structural degrees of freedom; Let be the right-hand term vector of dimension n×1; System Matrix elements in Indicates the first The hidden layer node pairs with the first... The contribution of each output node. , , The expression is:

[0025] In the formula, , and These represent the mass, damping, and stiffness terms, respectively. , and These represent the coefficients of the mass, damping, and stiffness terms, respectively. Right-hand term vector The expression is:

[0026] Finally, the output layer weight coefficients can be solved with high accuracy using the regularized ridge regression method:

[0027] In the formula, It is the identity matrix. is the regularization coefficient.

[0028] Furthermore, in step S5, a Bayesian framework is introduced to estimate the posterior distribution of the output layer weight coefficients. The process of probability quantization for the bridge flutter response prediction interval under different confidence levels is as follows: Let the linear regression model of the EELM model be: Error term Follows a Gaussian distribution with a mean of zero , Indicates noise hyperparameters; Assuming weight parameters The components are independent of each other and follow a Gaussian prior distribution with zero mean:

[0029] in, This is the a priori accuracy hyperparameter; Given weight parameters Under these conditions, observation data r The likelihood function is expressed as:

[0030] According to Bayes' theorem, the posterior distribution of the weight parameters is proportional to the product of the likelihood function and the prior distribution, that is:

[0031] Combined with weighted adoption number Gaussian prior distribution function and observed data r The likelihood function, by matching the quadratic and linear terms in the logarithmic form of the multivariate Gaussian distribution, yields the posterior distribution of the weight parameters as follows:

[0032] The corresponding posterior mean and covariance matrix are as follows:

[0033]

[0034] Among them, hyperparameters and The hyperparameters are updated iteratively using the evidence maximization method, with the convergence criterion set as follows: the change in hyperparameters between two consecutive iterations is less than a threshold. and The update formula is:

[0035] In the formula, and Represent matrices respectively The number of rows and columns; superscript k Indicates the number of iterations; Represents the posterior covariance matrix traces; Finally, the posterior distribution of the output layer weight parameters is obtained as follows:

[0036] For new input samples The corresponding chatter response prediction distribution is expressed as:

[0037] In the formula, This indicates the predicted output variable.

[0038] The beneficial effects of this invention are as follows: The Enhanced Bayesian Physically Constrained Limit Learning Machine (EPIELM) model proposed in this invention exhibits extremely high prediction accuracy for chattering displacement responses in different directions at multiple bridge locations. Based on Bayesian inference, this invention can provide a reliable probability prediction interval for node responses at any given confidence level.

[0039] The hybrid basis function form of this invention, composed of Fourier basis functions and tanh basis functions, significantly outperforms models that use only sine, cosine, or standard Fourier basis functions in terms of prediction performance. Its advantage mainly comes from the superior ability of this hybrid basis function to simultaneously characterize periodic oscillation characteristics and nonlinear dynamic behavior.

[0040] Although the EPIELM model of this invention is essentially a non-pure data-driven framework, it still exhibits significant robustness. Experiments show that although the prediction error increases with the increase of average wind speed, the EPIELM model of this invention can still maintain high-precision prediction performance even under strong wind conditions.

[0041] The EPIELM model of this invention exhibits good noise resistance and generalization performance. Under noise disturbance conditions, the RMSE of the lateral, vertical, and torsional buffeting responses at the mid-span location is also relatively low. Furthermore, under high-turbulence wind conditions, the model maintains stable and reliable predictive performance, fully validating its feasibility and practical value in real-world engineering applications.

[0042] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0043] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein: Figure 1 This is a schematic diagram of the overall architecture of the enhanced Bayesian physical constraint limit learning machine model according to an embodiment of the present invention; Figure 2 This is a schematic diagram of a suspension bridge according to an embodiment of the present invention; Figure 3 This is a schematic diagram of wind load acting on a bridge section according to an embodiment of the present invention; Figure 4 This is a schematic diagram of the Extreme Learning Machine structure according to an embodiment of the present invention; Figure 5 This is a schematic diagram of the first two vibration modes of the bridge in different directions according to an embodiment of the present invention; Figure 6 This is a schematic diagram comparing the time history curves and power spectra of simulated wind speeds in an embodiment of the present invention, wherein... Figure 6 (a) is a schematic diagram comparing the time history curves and power spectrum at downwind wind speeds. Figure 6 (b) is a schematic diagram comparing the time history curves and power spectrum under vertical wind speed; Figure 7 This is a schematic diagram comparing the evaluation indices of torsional chattering response under different input layer to hidden layer basis function forms in embodiments of the present invention; Figure 8 This diagram illustrates a comparison of evaluation metrics for torsional chattering response under different numbers of basis functions in an embodiment of the present invention. Figure 8 (a) is an indicator A comparison diagram with MSE. Figure 8 (b) is a diagram showing the comparison between the indicators RMSE and MAE; Figure 9 This diagram illustrates a comparison of torsional buffeting response prediction and evaluation indices under different random weight initialization intervals in an embodiment of the present invention. Figure 9 (a) is an indicator A comparison diagram with MSE. Figure 9 (b) is a diagram showing the comparison between the indicators RMSE and MAE; Figure 10 This is a schematic diagram showing the comparison between the predicted lateral buffeting response at different span positions and the target value, as well as the error distribution, in an embodiment of the present invention. Figure 10 (a) is a 1 / 4 span position. Figure 10 (b) is the mid-span position. Figure 10 (c) represents a 3 / 4 span position; Figure 11 This is a schematic diagram showing the comparison between the predicted vertical buffeting response and the target value at different span positions in an embodiment of the present invention, as well as the error distribution. Figure 11 (a) is a 1 / 4 span position. Figure 11 (b) is the mid-span position. Figure 11 (c) represents a 3 / 4 span position; Figure 12 This is a schematic diagram showing the comparison between the predicted torsional buffeting response at different span positions and the target value, as well as the error distribution, in an embodiment of the present invention. Figure 12 (a) is a 1 / 4 span position. Figure 12 (b) is the mid-span position. Figure 12 (c) represents a 3 / 4 span position; Figure 13 This is a schematic diagram of Bayesian interval prediction results for lateral buffeting response at different span positions in an embodiment of the present invention. Figure 13 (a) is a 1 / 4 span position. Figure 13 (b) is the mid-span position. Figure 13 (c) represents a 3 / 4 span position; Figure 14 This is a schematic diagram of the Bayesian interval prediction results for the vertical buffeting response at different span positions in an embodiment of the present invention. Figure 14 (a) is a 1 / 4 span position. Figure 14 (b) is the mid-span position. Figure 14 (c) represents a 3 / 4 span position; Figure 15 This is a schematic diagram of the Bayesian interval prediction results of torsional buffeting response at different span positions in an embodiment of the present invention. Figure 15 (a) is a 1 / 4 span position. Figure 15 (b) is the mid-span position. Figure 15 (c) is a 3 / 4 span position. Detailed Implementation

[0044] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0045] The accompanying drawings are for illustrative purposes only and are schematic diagrams, not actual pictures. They should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some parts in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.

[0046] In the accompanying drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components. In the description of the present invention, it should be understood that if terms such as "upper," "lower," "left," "right," "front," and "rear" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe positional relationships in the drawings are only for illustrative purposes and should not be construed as limiting the present invention. For those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.

[0047] Please see Figures 1-15 This paper presents a method for predicting the flutter response of suspension bridges based on an enhanced Bayesian physical constraint limit learning machine.

[0048] Example 1 This embodiment details a method for predicting the buffeting response of suspension bridges based on an enhanced Bayesian physics-informed extreme learning machine (PIELM). The PIELM can be used to analyze bridge buffeting problems. PIELM is considered an efficient variant of Physically Constrained Neural Networks (PINNs), offering significant computational efficiency advantages in solving complex partial differential equations. However, its application in wind-induced bridge vibration problems remains relatively limited. This invention proposes an enhanced physics-informed extreme learning machine model incorporating Bayesian optimization, such as... Figure 1As shown, this method organically combines physical constraints, primarily based on the chattering dynamics equations, with probabilistic optimization methods. It includes at least the following specific processes: S1. Considering the buffeting response of the bridge under wind load, establish the mechanical equation for bridge buffeting. S2. Construct a hybrid basis function form that integrates Fourier basis functions and hyperbolic tangent basis functions, and build an enhanced extreme learning machine network; S3. Introduce physical constraints dominated by the buffeting dynamics control equations, and construct a special form of trial solution to perform hard constraint embedding on the initial conditions. S4. The output layer weights of the augmented extreme learning machine network are solved using the regularized ridge regression method. S5. Based on the Bayesian inference framework, the posterior distribution of the weight coefficients of the output layer is estimated to obtain the prediction results of the bridge flutter response and its uncertainty interval.

[0049] In step S1 of this embodiment, a long-span suspension bridge is taken as the research object. A schematic diagram of the suspension bridge is shown below. Figure 2 As shown, its main span is 446m, and the clearance height under the bridge is 50m. The main design parameters of the bridge are listed in Table 1: Table 1

[0050] The aerodynamic parameters of the bridge model are shown in Table 2, including the drag coefficient of the bridge section. Lift coefficient Pitch moment coefficient and its first derivative: Table 2

[0051] This study investigates the buffeting response of bridges under wind loads, neglecting the influence of self-excited aerodynamic effects. Quasi-steady linear theory is employed, and the bridge is simplified as a three-degree-of-freedom continuous system with linear damping. A schematic diagram of the beam cross-section and the wind load it experiences are shown below. Figure 3 As shown, the displacements of the bridge deck in the lateral, vertical, and torsional directions are denoted as follows: In the wind axis coordinate system, aerodynamic drag, lift, and pitching moment are denoted as follows: (Black markings), while the corresponding components in the system coordinate system are denoted as... (Highlighted in red). It should be noted that an additional torsional aerodynamic damping term has been introduced into the wind load model. This item is related to the velocity of the aerodynamic center of the cross section. The distance between the shear center and the aerodynamic center is related to the beam width. In this embodiment, this proportionality coefficient is taken as [value missing]. =0.25.

[0052] The equation for the vibration force of the bridge can be expressed as:

[0053] In the formula: This is the quality matrix; Here is the damping matrix; Here is the stiffness matrix; and These are the static wind load matrix and the buffeting wind load matrix, respectively; This is the acceleration response vector; and These are the velocity and displacement response vectors, respectively. and These represent the aerodynamic damping matrix and the aerodynamic stiffness matrix, respectively. Average wind speed; and These represent the downwind and vertical pulsating wind speed components, respectively.

[0054] By moving the aerodynamic damping and aerodynamic stiffness terms to the right-hand side of the equation, the mechanical force equation for bridge flutter can be rewritten as follows:

[0055] in:

[0056] From the above equation, we can see that the total wind load matrix... This can be represented as a calm wind term. Vibration force item pneumatic damping term and aerodynamic stiffness The linear combinations of are expressed in the following specific expressions:

[0057]

[0058]

[0059]

[0060] In step S2 of this embodiment, an Enhanced Extreme Learning Machine (EELM) is used to construct a functional mapping relationship between wind speed input and bridge flutter response. The basic structure of a single hidden layer EELM is as follows: Figure 4 As shown. By Figure 4 It can be seen that the input layer consists of a size of The time series structure; the hidden layer output dimension is ,in Indicates the structural degrees of freedom. This indicates the number of hidden layer nodes. The output of each hidden layer node can be represented as:

[0061] In the formula, Indicates the first The output of each hidden layer node; For activation functions; and These are the randomly generated input weights and biases, respectively.

[0062] To better adapt to the nonlinear characteristics of the structural dynamics system, this embodiment designs the activation function from the input layer to the hidden layer as a hybrid basis function combining Fourier basis functions and hyperbolic tangent functions. Therefore, the total output of the hidden layer can be expressed as:

[0063] The hidden layer output using Fourier basis functions is denoted as... Its first and second derivatives are respectively:

[0064] In the formula, This represents the number of hidden layer nodes in the Fourier basis functions.

[0065] The hidden layer output using the hyperbolic tangent basis function is denoted as... Its first and second derivatives are respectively:

[0066] In the formula, This represents the number of hidden layer nodes using the hyperbolic tangent basis function.

[0067] The final output of the EELM model can be expressed as:

[0068] In the formula, This is the output matrix of the hidden layer; For hidden layers up to the 1st The output weight vector of each output node. Since the weights and biases between the input layer and the hidden layer are randomly generated and remain unchanged after initialization, the hidden layer output... This is a constant matrix. Therefore, solving for the model output only requires solving for the output layer weights. That should complete the task.

[0069] In step S3 of this embodiment, to effectively alleviate the ill-conditioned problem of the physical information constraint matrix, a special form of trial solution is constructed to perform hard embedding on the initial conditions. Using this method, the physical information constraint matrix contains only the constraint information generated by the chattering dynamics control equations, thereby significantly reducing the model weights. The computational complexity of the solution process is reduced, and numerical stability is effectively improved.

[0070] For the EELM model proposed in step S2, a time variable transformation function is introduced. The trial solution with embedded initial conditions can be expressed as:

[0071] In the formula, and These represent the bridge's displacement and velocity at the initial moment, respectively.

[0072] Perform time-related analysis on the constructed trial solution function. By differentiating , we can obtain the expressions for the velocity and acceleration of the chattering response:

[0073] In step S4 of this embodiment, the output layer weights are calculated. The specific process includes: First, the physical loss function in the rewritten bridge flutter force equation is minimized, which is equivalent to minimizing the sum of squares of the output errors, i.e.:

[0074] Next, based on the constructed trial-and-error equations and the expressions for the velocity and acceleration of the chattering response, the physical loss function can be further equivalently transformed into solving the following system of linear equations:

[0075] In the formula, For dimension is The system matrix; For dimension is The weighted coefficient vector (corresponding to all structural degrees of freedom); Let be the right-hand term vector of dimension n×1.

[0076] in, Indicates the first The hidden layer node pairs with the first... The contribution of each output node. , Its specific expression is:

[0077] In the formula, , and These represent the mass, damping, and stiffness terms, respectively. , and These represent the coefficients for the mass, damping, and stiffness terms, respectively; where each coefficient is defined as follows:

[0078] Right-hand term vector The expression is:

[0079] Finally, the output layer weight coefficients can be solved with high accuracy using the regularized ridge regression method:

[0080] In the formula, It is the identity matrix. is the regularization coefficient.

[0081] In step S5 of this embodiment, although the PIELM model, as an efficient variant of a physically constrained neural network, exhibits good computational efficiency without relying on purely data-driven input, the model weights in traditional EELM are usually uniquely determined using the least squares method; this deterministic solution method cannot reflect the inherent uncertainty in the flutter response prediction process. To address these shortcomings, this invention introduces a Bayesian framework to estimate the posterior distribution of the output layer weight coefficients, thereby achieving probability quantification of the bridge flutter response prediction interval under different confidence levels. The process is as follows: Let the linear regression model of the EELM model be: Error term Follows a Gaussian distribution with a mean of zero , Indicates noise hyperparameters; Further assume weight parameters The components are independent of each other and follow a Gaussian prior distribution with zero mean:

[0082] in, This is the prior accuracy hyperparameter.

[0083] Given weight parameters Under these conditions, observation data r The likelihood function can be expressed as:

[0084] According to Bayes' theorem, the posterior distribution of the weight parameters is proportional to the product of the likelihood function and the prior distribution, that is:

[0085] Combined with weighted adoption number Gaussian prior distribution function and observed data r The likelihood function, by matching the quadratic and linear terms in the logarithmic form of the multivariate Gaussian distribution, yields the posterior distribution of the weight parameters as follows:

[0086] The corresponding posterior mean and covariance matrix are as follows:

[0087]

[0088] Among them, hyperparameters and Iterative updates are performed using the evidence-maximizing method, with the convergence criterion set as the change in hyperparameters being less than a threshold between two consecutive iterations. The corresponding update formula is:

[0089] In the formula, and Represent matrices respectively The number of rows and columns; the superscript " k " indicates the number of iterations; Represents the posterior covariance matrix The traces.

[0090] Finally, the posterior distribution of the output layer weight parameters can be obtained as follows:

[0091] For new input samples The corresponding chatter response prediction distribution can be expressed as:

[0092] In the formula, This indicates the predicted output variable.

[0093] Example 2 To comprehensively evaluate the prediction accuracy of the proposed model, this embodiment selects four evaluation metrics: coefficient of determination (R²). 2 The mean square error (MSE), root mean square error (RMSE), and mean absolute error (MAE) are calculated using the following formulas:

[0094] In the formula, Indicates the true chatter response; This represents the mean of the actual chatter response; This represents the buffeting response predicted by the model. This embodiment systematically evaluates the performance of the proposed Bayesian optimization-based EPIELM model in predicting bridge buffeting response by performing time-domain buffeting analysis on a long-span bridge. Detailed design parameters of the bridge are given in Table 1 of Embodiment 1. This embodiment uses the modal superposition method to characterize the dynamic characteristics of the bridge and discretizes the bridge along the span direction into 30 equidistant elements.

[0095] Table 3 summarizes the first two natural frequencies of the bridge in the transverse, vertical, and torsional directions, showing the first two modal frequencies of the bridge in different directions: Table 3

[0096] It can be seen that the modal frequency results calculated in this embodiment are highly consistent with the results in the literature "Buffeting response of asuspension bridge in complex terrain. Engineering Structures", verifying the correctness and reliability of the established bridge model. The corresponding first two vibration modes in different directions are as follows: Figure 5 As shown, each vibration mode is identified using a coding system consisting of two letters and a number: the first letter (L, V, or T) indicates the direction of vibration (Lateral, Vertical, or Torsional), respectively; the second letter (S or A) indicates the mode's symmetry (Symmetric or Antisymmetric); and the number "1" indicates the mode order. For example, LS1 represents the first-order lateral symmetric mode.

[0097] In this embodiment, when applying a wind field to the main beam, only the along-wind and vertical fluctuating wind speed components at each node are considered, while the influence of the crosswind wind speed component is ignored. The average along-wind speed is denoted as U, and the average vertical wind speed is assumed to be zero. The Davenport coherence function is used to describe the correlation between fluctuating wind speeds at two spatial points. The wind speed samples along the span of the bridge deck are simulated using the Spectral Representation Method (SRM). The basic steps include: constructing a stochastic process in the frequency domain that satisfies the target power spectrum and the coherence function model, and transforming it to the time domain using the Inverse Fast Fourier Transform (IFFT) to generate a physically meaningful wind speed time history. The key parameters of the wind speed turbulence spectrum are listed in Table 4: Table 4

[0098] Therefore, the turbulent power spectrum model used can be expressed as: The sampling time interval for wind speed time history was set to 0.01 s. The simulation time histories of downwind and vertical wind speeds at the mid-span node of the bridge deck, along with their corresponding power spectra, are compared as follows: Figure 6 As shown, the average downwind speed is set to 10 m / s. Figure 6 As can be seen, the power spectrum of the generated wind speed sample is basically consistent with the target spectrum, indicating that the constructed wind field model has good reliability and rationality. It should be noted that the subsequent buffeting analysis only uses the wind speed time history data of the first 50 seconds.

[0099] This embodiment also systematically optimizes the hyperparameters in the EPIELM model, mainly including key parameters such as the type and number of basis functions and the random weight initialization interval. Using the prediction performance of torsional buffeting displacement as the evaluation benchmark, the prediction effects of the model under different hyperparameter combinations are compared and analyzed. It should be noted that although this embodiment does not explicitly consider the dynamic coupling effect between the lateral, vertical, and torsional components in the buffeting response prediction process, numerous numerical experimental results show that the optimal hyperparameters determined through torsional buffeting analysis are also applicable to the prediction problem of lateral and vertical buffeting responses, demonstrating good applicability.

[0100] Figure 7 Comparative results of torsional chattering response prediction performance under different basis function forms are presented. It can be seen that the model using a sine-cosine mixed Fourier basis function significantly outperforms models using only sine or only cosine functions in terms of prediction accuracy. Specifically, the MSE of the mixed Fourier model reaches [value missing]. Furthermore, in terms of computational efficiency, it is 88.96% and 90.39% higher than the pure sine model and the pure cosine model, respectively.

[0101] The number of hidden layer nodes is another key factor affecting the prediction accuracy of the EPIELM model. Therefore, Figure 8 The evaluation metrics for mid-span torsional chattering response are compared under different numbers of basis functions. It should be noted that the horizontal axis represents the number of basis functions of a single type. The results show that the overall prediction accuracy of the model significantly improves with the increase of the number of basis functions; however, when the number of basis functions increases from 400 to 500, the improvement in prediction performance becomes relatively limited. Therefore, considering both computational efficiency and prediction accuracy, this invention uniformly selects 500 basis functions in subsequent analyses.

[0102] The weights and bias parameters between the input layer and the hidden layer are randomly sampled from a uniform distribution. To determine the optimal initialization interval, five parameter intervals were selected: [0.1, 10], [0.1, 20], [0.1, 30], [0.1, 40], and [0.1, 50], respectively, and the torsional chattering response prediction results were compared and analyzed. The evaluation metrics for torsional chattering response prediction under different random weight initialization intervals are as follows: Figure 9 As shown in the figure, the results indicate that the prediction error exhibits a non-monotonic trend of first decreasing and then increasing as the parameter initialization interval increases. The model's prediction performance reaches its optimal state when the parameter value interval is [0.1, 40]. Therefore, in all subsequent chattering response prediction analyses, this embodiment uniformly adopts [0.1, 40] as the initialization interval for random weights and bias parameters.

[0103] Based on hyperparameter optimization, the optimal parameter combination for the EPIELM model, which embeds the buffeting dynamics equations into the ELM framework, has been determined. This embodiment will focus on analyzing the model's predictive performance for bridge buffeting response.

[0104] Figure 10 –12 presents a comparison between the EPIELM model's predictions and finite element analysis (FEA) results for the lateral, vertical, and torsional buffeting responses (denoted as rx, rz, and rθ, respectively) at the 1 / 4 span, mid-span, and 3 / 4 span locations of a long-span suspension bridge, under an average wind speed of 10 m / s. Figure 10-12 As can be seen, the chattering time history curves at each location generally agree well with the FEA results, with only minor deviations at a few response peaks. The maximum prediction errors in the three directions are: lateral 0.054 mm, vertical 1.227 mm, and torsional 1.227 mm. rad. Furthermore, Table 5 shows the evaluation indicators for bridge displacement response prediction: Table 5

[0105] Table 5 quantitatively shows that the RMSE values ​​of the chattering response in the three directions at the mid-span location are only 0.026 mm, 0.032 mm, and 0.032 mm, respectively. The above results fully demonstrate that the EPIELM model possesses extremely high prediction accuracy and good stability for the multi-directional buffeting response of bridges at different locations.

[0106] Figure 13 –15 presents the interval prediction results of the lateral, vertical, and torsional buffeting displacement responses of long-span bridges using the Bayesian optimization-based EPEELM framework, under an average wind speed of 10 m / s. Figure 13 As shown in Figure 15, the predicted mean at each location is highly consistent with the target response, and the prediction intervals at all three locations can completely cover the actual response data. Furthermore, the differences in the width of the prediction intervals at different span locations indicate that the model can effectively characterize the inherent random fluctuations in the bridge's buffeting response.

[0107] Unlike traditional EPIELM, which only provides deterministic point predictions, this invention introduces a Bayesian framework to infer the posterior distribution of weights from the hidden layer to the output layer, thereby enabling reliable probability interval prediction of flutter responses at different locations. This method provides more comprehensive information support for the safety assessment and uncertainty analysis of bridge wind-induced vibrations.

[0108] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for predicting the flutter response of a suspension bridge based on an enhanced Bayesian physics-constrained limit learning machine, characterized in that: The method includes the following steps: S1. Considering the buffeting response of the bridge under wind load, establish the mechanical equation for bridge buffeting. In step S1, firstly, establish the mechanical equation for the buffeting response of the bridge under wind load, where the bridge is simplified as a three-degree-of-freedom continuous system with linear damping. The displacements of the bridge deck in the lateral, vertical, and torsional directions are denoted as follows: In the wind axis coordinate system, aerodynamic drag, lift, and pitching moment are denoted as follows: An additional torsional aerodynamic damping term is introduced into the wind load model. Its velocity relative to the aerodynamic center of the cross section Since the distance between the shear center and the aerodynamic center is related to the ratio of the beam width, the mechanical equation for bridge flutter is expressed as: In the formula: This is the quality matrix; Here is the damping matrix; Here is the stiffness matrix; and These are the static wind load matrix and the buffeting wind load matrix, respectively; This is the acceleration response vector; and These are the velocity and displacement response vectors, respectively. and These represent the aerodynamic damping matrix and the aerodynamic stiffness matrix, respectively. Average wind speed; and These represent the downwind and vertical pulsating wind speed components, respectively. Moving the aerodynamic damping and aerodynamic stiffness terms to the right-hand side of the equation, the bridge flutter force equation is rewritten as follows: In the formula, The total wind load matrix can be expressed as a static wind term. Vibration force item pneumatic damping term and aerodynamic stiffness A linear combination of; S2. Construct a hybrid basis function form that integrates Fourier basis functions and hyperbolic tangent basis functions, and build an enhanced extreme learning machine (ELM) network; in step S2, the ELM is used to construct a functional mapping relationship between wind speed input and bridge flutter response, wherein the input layer of the single hidden layer of the ELM consists of a size of The time series structure, with hidden layer output dimension as... ,in Indicates the structural degrees of freedom. Indicates the number of hidden layer nodes; then the th The output of each hidden layer node is represented as: In the formula, Indicates the first The output of each hidden layer node; For activation functions; and These are the randomly generated input weights and biases, respectively. The activation function from the input layer to the hidden layer is designed as a hybrid basis function combining Fourier basis functions and hyperbolic tangent functions. The total output of the hidden layer is expressed as: in, This indicates the hidden layer output using Fourier basis functions. This indicates the hidden layer output using the hyperbolic tangent basis function; The final output of the augmented extreme learning machine is then expressed as: In the formula, This is the output matrix of the hidden layer; For hidden layers up to the 1st The output weight vector of each output node; S3. Introduce physical constraints dominated by the chattering dynamics control equations, and construct a special form of trial solution to perform hard constraint embedding on the initial conditions; in step S3, for the constructed enhanced extreme learning machine model, introduce a time variable transformation function. The trial solution with embedded initial conditions is expressed as: In the formula, and These represent the displacement and velocity at the initial moment, respectively. Perform time-related analysis on the constructed trial solution function. Taking the derivatives, we obtain the expressions for the velocity and acceleration of the chattering response: In the formula, This represents the weights from the hidden layer to the output layer. This indicates the output of the hidden layer. This represents the first derivative of the hidden layer output. This represents the second derivative of the hidden layer output; S4. The output layer weights of the augmented extreme learning machine network are solved using the regularized ridge regression method. S5. Based on the Bayesian inference framework, the posterior distribution of the weight coefficients of the output layer is estimated to obtain the prediction results of the bridge flutter response and its uncertainty interval.

2. The method for predicting the flutter response of a suspension bridge based on an enhanced Bayesian physics-constrained limit learning machine according to claim 1, characterized in that: The hidden layer output using Fourier basis functions in step S2 Its first and second derivatives are as follows: In the formula, This represents the number of hidden layer nodes in the Fourier basis functions; Hidden layer output using hyperbolic tangent basis functions Its first and second derivatives are as follows: In the formula, This represents the number of hidden layer nodes using the hyperbolic tangent basis function.

3. The method for predicting the flutter response of a suspension bridge based on an enhanced Bayesian physics-constrained limit learning machine according to claim 1, characterized in that: In step S4, the output layer weights are calculated. The specific process includes: First, the physical loss function in the rewritten bridge flutter force equation is minimized, which is equivalent to minimizing the sum of squares of the output errors, i.e.: Next, based on the constructed trial-and-error equations and the velocity and acceleration expressions for the chattering response, the physical loss function is equivalently transformed into solving the following system of linear equations: In the formula, For dimension is The system matrix; For dimension is The weighted coefficient vector corresponds to all structural degrees of freedom; Let be the right-hand term vector of dimension n×1; System Matrix elements in Indicates the first The hidden layer node pairs with the first... The contribution of each output node. , , The expression is: In the formula, , and These represent the mass, damping, and stiffness terms, respectively. , and These represent the coefficients for mass, damping, and stiffness, respectively. Right-hand term vector The expression is: Finally, the output layer weight coefficients can be solved with high accuracy using the regularized ridge regression method: In the formula, It is the identity matrix. is the regularization coefficient.

4. The method for predicting the flutter response of a suspension bridge based on an enhanced Bayesian physics-constrained limit learning machine according to claim 3, characterized in that: In step S5, a Bayesian framework is introduced to estimate the posterior distribution of the output layer weight coefficients. The process of probability quantization for the predicted interval of bridge flutter response at different confidence levels is as follows: Let the linear regression model of the EELM model be: Error term Follows a Gaussian distribution with a mean of zero , Indicates noise hyperparameters; Assuming weight parameters The components are independent of each other and follow a Gaussian prior distribution with zero mean: in, This is the a priori accuracy hyperparameter; Given weight parameters Under these conditions, observation data r The likelihood function is expressed as: According to Bayes' theorem, the posterior distribution of the weight parameters is proportional to the product of the likelihood function and the prior distribution, that is: Combined with weighted adoption number Gaussian prior distribution function and observed data r The likelihood function, by matching the quadratic and linear terms in the logarithmic form of the multivariate Gaussian distribution, yields the posterior distribution of the weight parameters as follows: The corresponding posterior mean and covariance matrix are as follows: Among them, hyperparameters and The hyperparameters are updated iteratively using the evidence maximization method, with the convergence criterion set as follows: the change in hyperparameters between two consecutive iterations is less than a threshold. and The update formula is: In the formula, and Represent matrices respectively The number of rows and columns; superscript k Indicates the number of iterations; Represents the posterior covariance matrix traces; Finally, the posterior distribution of the output layer weight parameters is obtained as follows: For new input samples The corresponding chatter response prediction distribution is expressed as: In the formula, This indicates the predicted output variable.

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