Stochastic Response Prediction Method for Vehicle-Bridge Systems Based on Temporal Convolutional Networks and Gaussian Processes

Abstract

The present invention relates to a method for predicting the stochastic response of a vehicle-bridge system based on a temporal convolutional network and a Gaussian process, belonging to the field of stochastic response prediction of VBI systems. The method includes: establishing a vehicle-bridge interaction model; obtaining stochastic dynamic response samples of the vehicle and the bridge; fusing the Gaussian process and the temporal convolutional network to establish a GP-TCN network model, in which the weights and biases between convolutional layers are set as random parameters, and the positions of the random parameters are set in the first dilated convolutional layer of each residual connection block; using the track irregularity as the input of the GP-TCN model and the dynamic responses of the vehicle and the bridge as the output of the GP-TCN model, and training the model with the sample data obtained by numerical simulation; using the trained GP-TCN model to predict the stochastic dynamic response of the vehicle-bridge interaction system. The present invention can accurately evaluate the dynamic responses of the VBI system in the time domain and the frequency domain.

Description

Technical Field

[0001] The present invention belongs to the field of random response prediction of the VBI system, and relates to a method for predicting the random response of a vehicle-bridge system based on a temporal convolutional network and a Gaussian process. Background Art

[0002] Due to the dynamic interaction between vehicles and bridges, the dynamic response generated when vehicles travel on bridges has a significant impact on vehicle safety and passenger comfort. Therefore, with the increase in train speed and railway network density in recent years, the dynamic interaction between vehicles and bridges has received more and more attention. Currently, in most references, the coupled vibration analysis of the vehicle-bridge system is assumed to be a deterministic dynamics problem, ignoring the randomness of track irregularity as one of the main excitation sources. In fact, not only track irregularity, but also the parameters of vehicle and bridge structures inevitably have uncertain characteristics due to factors such as environmental influence, inherent discreteness of concrete materials, and structural geometric deviation during construction. Therefore, in order to capture the random response characteristics of the vehicle-bridge system, it is necessary to conduct random vibration analysis of the vehicle-bridge system considering the influence of random track irregularity and random structural parameters.

[0003] Currently, several methods have been proposed to study the random dynamic behavior of the VBI system (vehicle-bridge interaction system), such as the Monte Carlo method (MCM), the pseudo excitation method (PEM), the probability density evolution method (PDEM), polynomial chaos (PC), the point estimation method, etc.

[0004] Among them, MCM is a stochastic analysis method based on a large number of discrete samples and can be widely applied to complex stochastic problems. However, to ensure the calculation accuracy, a sufficiently large sample size is required, which limits the development of MCM in the field of vehicle-bridge coupling. PEM is a mature algorithm for analyzing deterministic vibrations under stationary or non-stationary random excitations. For the vibration analysis of the VBI system affected by random track irregularities, PEM converts the track irregularities into a series of deterministic harmonic excitations with phase lags. However, since PEM cannot consider the random structural parameters and the nonlinear relationship between the wheel and the track, its application in the random vibration analysis of the VBI system is limited. The hybrid pseudo-excitation-polynomial chaos expansion method (PEM-PCE) is developed for the vehicle-track coupling system with uncertain parameters, and its results are highly consistent with the Monte Carlo method. Among them, the polynomial chaos (PC) expansion is a non-intrusive method for evaluating the uncertainty evolution and quantification of dynamic systems. The polynomial chaos (PC) expansion can be introduced into the model stochastic process, and the PC expansion is constructed using Hermite polynomials and Gaussian random variables. The dynamic response of a bridge under the action of a moving vehicle considering the random parameters of the beam-bridge model can be analyzed using the polynomial chaos expansion method, and it can also be used to predict the vehicle motion with uncertain vehicle parameters. The point estimation method is based on the Gaussian integration theory and has the advantages of simple operation and principle. Combining the Karhunen-Loéve expansion and the point estimation method can be used to study the dynamic behavior of a train-bridge system with random track irregularities. PDEM can be used for random vibration analysis. Compared with PEM (pseudo-excitation method), the obvious improvement of PDEM is that it can consider the nonlinear characteristics of the dynamic system.

[0005] Although many efficient stochastic vibration theories have been developed, hundreds of calculations are still required to ensure the accuracy of computer results, which will consume a large amount of resources to analyze the random vibration of complex vehicle-bridge coupling systems. Fortunately, with the development of deep learning in recent years, combining advanced deep learning techniques with traditional vehicle-bridge coupling analysis methods has been proven to be a potential solution. This is mainly attributed to the fact that deep learning techniques have stronger data mining capabilities and nonlinear fitting capabilities using multi-layer neural networks. In recent years, many studies on the random vibration analysis of vehicle-bridge interaction (VBI) systems based on deep learning networks have been carried out. For example, by using the stochastic pseudo-excitation method (SPEM) combined with a fully convolutional network and a gated recurrent unit, the random dynamic response of the vehicle-bridge interaction system under uncertain system parameters and random track irregularities can be predicted; combining the long short-term memory (LSTM) model with the ballast track train interaction (TBTI) system can accurately predict the dynamic behavior of TBTI under a large number of random track geometry excitations; combining Bayesian inference with the LSTM network can analyze the random vibration response of a bridge under vehicle dynamic interaction.

[0006] However, most deep learning models for predicting the stochastic dynamic response of VBI systems usually ignore the uncertainty of the models. In fact, estimating the uncertainty of deep learning models through statistical methods can greatly improve the non-linear mapping ability of surrogate models. Therefore, the present invention proposes a new model combining Gaussian process (GP) and temporal convolutional network (TCN) for predicting the stochastic response of VBI systems. Summary of the Invention

[0007] In view of this, the purpose of the present invention is to provide a method for predicting the stochastic response of a vehicle-bridge system based on a temporal convolutional network (TCN) and a Gaussian process (GP), using TCN to extract the characteristics of track irregularities and predict the stochastic dynamic response.

[0008] To achieve the above object, the present invention provides the following technical solutions:

[0009] A method for predicting the stochastic response of a vehicle-bridge system based on a temporal convolutional network and a Gaussian process, the method comprising:

[0010] 1) Establish a vehicle-bridge interaction model;

[0011] 2) Perform numerical simulation through the Monte Carlo method to obtain stochastic dynamic response samples of the vehicle and the bridge;

[0012] 3) Integrate the Gaussian process and the temporal convolutional network to establish a GP-TCN network model. In this network model, the weights and biases between convolutional layers are set as random parameters, and the positions of the random parameters are set in the first dilated convolutional layer of each residual connection block;

[0013] 4) Use the track irregularity as the input of the GP-TCN model and the dynamic responses of the vehicle and the bridge as the output of the GP-TCN model, and train the model with the sample data obtained by numerical simulation;

[0014] 5) Use the trained GP-TCN model to predict the stochastic dynamic response of the vehicle-bridge interaction system.

[0015] Further, the vehicle-bridge interaction model is expressed as:

[0016]

[0017] In the formula, M, C, and K respectively represent the mass, damping, and stiffness matrices; X(t), and respectively represent the displacement, velocity, and acceleration vectors of the vehicle-bridge interaction system; F(t) represents the load vector.

[0018] Further, in the GP-TCN model, the output Expressed as:

[0019]

[0020] In the formula, and b (l) respectively represent the i-th weight and bias term of the l-th layer convolutional filter, which are deterministic parameters; and respectively represent the random parameters at the corresponding positions; represents the output of the previous layer at the dilated time step t - id of the l-th layer l ; represents the output of the previous layer; represents the noise term added to the convolutional output.

[0021] Furthermore, the deterministic parameters and random parameters are estimated by the maximum likelihood method and Gaussian process.

[0022] Among them, the process of estimating the deterministic parameters by the maximum likelihood method is as follows:

[0023] Let θ D represent the set of deterministic parameters, θ D = [w i b], and determine θ D by the maximum likelihood estimation method, D such that the joint probability of observing the data is maximized under θ

[0024] The likelihood function of θ D is expressed as:

[0025]

[0026] In the formula, x i , y i respectively represent the training input and output data, and N represents the amount of data;

[0027] Take the log-likelihood function of θ D :

[0028]

[0029] In the formula, f(x i , θ D ) represents the output of the network model under θ D and the input x i , represents the variance of the normal distribution caused by the noise;

[0030] Maximize the log-likelihood function:

[0031]

[0032] According to the above formula, maximizing the log-likelihood function is equivalent to minimizing the sum of squared residuals. Therefore, the parameters are determined by calculating the loss function between the true response and the predicted response of the VBI system. Obtain the set of deterministic parameters θ D , thus obtaining the deterministic parameters.

[0033] The process of estimating the random parameters by Gaussian process is as follows:

[0034] If the random parameters have the characteristics of an independent Gaussian process, then their prior distribution also follows a Gaussian distribution. Then, in the set of random parameters θ R The log-likelihood function of the observed data following a Gaussian distribution is expressed as:

[0035]

[0036] In the formula, X and Y represent the input and output of the model respectively, and I represents the identity matrix;

[0037] According to Bayes' theorem, the posterior distribution of the random parameters is expressed as:

[0038]

[0039] The posterior distribution is proportional to the likelihood function and the prior distribution; the posterior distribution solution of the random parameter θ R is transformed into a minimization problem of the log marginal likelihood with respect to the hyperparameters of the kernel function:

[0040] q Θ (θ R ) = arg max θ l(Θ) = arg max θ log p(Y|X; Θ)

[0041] In the formula, q Θ (θ R ) represents the posterior distribution of the random parameter θ R ; Θ represents the parameter vector containing the kernel hyperparameters and the noise variance; the analytical solution of the log marginal likelihood function is:

[0042]

[0043] In the formula, represents the variance of the normal distribution caused by noise, and K θ represents the covariance function of the prior distribution of the random parameters, and the covariance function with an automatic relevance determination in the form of a squared exponential is adopted;

[0044] According to the above formula, the log marginal likelihood function of the Gaussian function contains multiple local minima. Therefore, Θ is updated by the gradient descent method to infer qΘ (θ R ) to obtain random parameters.

[0045] Furthermore, the process of training the GP-TCN network model includes:

[0046] S1. Randomly initialize the kernel hyperparameters Θ and the deterministic parameter θ of the Gaussian process D ;

[0047] S2. Select a batch of data from the sample dataset;

[0048] S3. According to the current kernel hyperparameters Θ, use the Gaussian process to predict the posterior distribution q Θ (θ R );

[0049] S4. Sample θ from the predicted posterior distribution q Θ (θ R ); R ;

[0050] S5. Calculate the total loss L of the GP-TCN network model t = L mse + L Θ , where L mse is the mean squared error (MSE) loss function of the deterministic parameter θ D , and L Θ is the log marginal likelihood loss with respect to the kernel hyperparameters Θ;

[0051] S6. Use the backpropagation algorithm to calculate the gradient of L mse with respect to the deterministic parameter θ D and the gradient of L Θ with respect to the kernel hyperparameters Θ;

[0052] S7. Use the gradient descent method and the learning rate α to update the deterministic parameter and the kernel hyperparameters of the Gaussian process;

[0053] S8. Repeat steps S2 - S7 until the loss function converges to a predetermined stopping criterion.

[0054] The beneficial effects of the present invention are as follows: The present invention proposes a deep learning model by combining a temporal convolutional network (TCN) and a Gaussian process (GP) for predicting the random response of a VBI system. The model fully utilizes the advantages of the TCN in capturing fine details of the dataset within a short time interval and predicting the response through specific dilated convolutional layers. At the same time, in order to map the uncertainty of the VBI system, the first convolutional layer of each time block layer in the TCN is designated as a random layer, and a non-parametric estimation method based on the Gaussian process (GP) is used to evaluate the probability distribution of the weights and biases of the first layer of convolution without presupposing a specific probability distribution.

[0055] The present invention can accurately evaluate the dynamic responses of the VBI system in the time domain and the frequency domain. In addition, by introducing a Gaussian process-based method, the generalization ability and robustness of the TCN model are significantly improved.

[0056] Other advantages, objects, and features of the present invention will, to some extent, be set forth in the subsequent description, and to some extent, will be obvious to those skilled in the art based on an examination of the following text, or can be learned from the practice of the present invention. The objects and other advantages of the present invention can be achieved and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] In order to make the objects, technical solutions, and advantages of the present invention clearer, the present invention will be described in detail with reference to the accompanying drawings, where:

[0058] Figure 1 It is a schematic diagram of the vehicle-bridge interaction model;

[0059] Figure 2 It is a schematic diagram of the framework structure of the GP-TCN model;

[0060] Figure 3 It is a schematic diagram of dilated causal convolution with dilation factor d = 1, 2, 4 and filter size k = 3;

[0061] Figure 4 It is the improved TCN residual block;

[0062] Figure 5 It is a schematic diagram of the training process of the GP-TCN model;

[0063] Figure 6 It is a contour plot of the probability density function (PDF) of the bridge dynamic response, Figure 6 (a) is for displacement, Figure 6 (b) is for acceleration;

[0064] Figure 7 It is the time-varying power spectrum of the system dynamic response obtained by the spectral representation method (SPEM), Figure 7 (a) is for train displacement, Figure 7 (b) is for train acceleration, Figure 7 (c) is for bridge displacement, Figure 7 (d) is for bridge acceleration;

[0065] Figure 8 It is a schematic diagram of the variation of the training and test losses with the number of epochs, Figure 8 (a) is a schematic diagram of the training and test of the train dynamic response, Figure 8 (b) is a schematic diagram of the training and test of the bridge dynamic response;

[0066] Figure 9 For the comparison of train dynamic responses based on the MCM, SPEM, and GP-TCN methods, Figure 9 (a) is the average value of displacement, Figure 9 (b) is the standard deviation of displacement, Figure 9 (c) is the average value of acceleration, Figure 9 (d) is the standard deviation of acceleration;

[0067] Figure 10 For the comparison of bridge responses based on the MCM, SPEM, and GP-TCN methods, Figure 10 (a) is the average value of displacement, Figure 10 (b) is the standard deviation of displacement, Figure 10 (c) is the average value of acceleration, Figure 10 (d) is the standard deviation of acceleration;

[0068] Figure 11 For the comparison diagram of the Fourier spectra of bridge responses obtained by the GP-TCN and MCM methods, Figure 11 (a) is displacement, Figure 11 (b) is acceleration;

[0069] Figure 12 For the schematic diagram of the PDF comparison of the maximum dynamic responses based on the GP-TCN and MCM methods, Figure 12 (a) is train displacement, Figure 12 (b) is train acceleration, Figure 12 (c) is bridge displacement, Figure 12 (d) is bridge acceleration. Specific implementation manners

[0070] The following uses specific specific examples to illustrate the implementation manners 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 implementation manners. The details in this specification can also 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 diagrams provided in the following embodiments only illustrate the basic concept of the present invention in a schematic manner. Without conflict, the following embodiments and the features in the embodiments can be combined with each other.

[0071] Due to the unique stacked dilated convolutional structure of the TCN, it can not only capture fine details within short time intervals but also capture trends and dependencies over longer time periods. In addition, the TCN has the advantages of fewer parameters and good model stability. The dynamic response of the VBI (vehicle-bridge interaction) system exhibits significant periodic and time-correlated characteristics over a relatively long time period. Therefore, the TCN model has significant advantages in predicting the stochastic vibration response of time-varying vehicle-bridge systems. To improve the uncertainty quantification ability of the surrogate model when mapping the stochastic VBI system, Gaussian process (GP) can be used to evaluate the probability distribution of some convolutional layers of the TCN. GP is a non-parametric method that does not require prior assumptions about the form of the data distribution, making it more flexible in capturing complex and non-linear relationships. Due to the excellent ability of GP in quantifying uncertainty, it has been widely used in the calculation of confidence intervals in wind speed prediction tasks.

[0072] Based on this, an embodiment of the present invention proposes a vehicle-bridge stochastic response prediction method based on a temporal convolutional network and Gaussian process, and the specific content of this method is as follows:

[0073] 1. Physical modeling of stochastic vibration analysis of the VBI system

[0074] To balance model complexity and solution accuracy, a two-dimensional vehicle-bridge interaction model is adopted in this embodiment. Among them, the bridge is modeled as a simple Euler-Bernoulli beam. The train is assumed to be a multi-rigid body system, including the car body, two bogies, and four wheelsets. Two suspension subsystems are used to connect the rigid bodies. The primary suspension represents the connection between the bogie and the wheelset, while the secondary suspension represents the connection between the bogie and the car body. The suspension subsystems are simulated as linear spring-damping elements. The train model contains six independent degrees of freedom, including the vertical and rotational motions of the car body (denoted by y c , θ c respectively), and the vertical and rotational motions of the front bogie (denoted by y b1 , θ b1 respectively) and the rear bogie (denoted by y b2 , θ b2 respectively). The wheelset is assumed to be in continuous and rigid contact with the track.

[0075] The vehicle-bridge interaction model is as Figure 1 shown. Wherein, m and J represent mass and moment of inertia respectively; the subscripts "c", "b", and "w" represent the car body, bogie frame, and wheelset respectively; k and c represent spring stiffness and damping coefficient respectively; the subscripts "p" and "s" represent the primary suspension system and the secondary suspension system respectively; l c represents the half longitudinal distance between the centers of gravity of the front bogie and the rear bogie; l t represents the half distance of the front bogie or the rear bogie.

[0076] Based on Figure 1 the mechanical model and the principle of conservation of energy shown above, the time-varying motion equation of the VBI system can be obtained as follows:

[0077]

[0078] where M, C, and K represent the mass, damping, and stiffness matrices respectively; X(t), and are the displacement, velocity, and acceleration vectors of the VBI system respectively; F(t) represents the load vector.

[0079] 2. Surrogate model for random vibration analysis of VBI system

[0080] (1) Model framework

[0081] In this embodiment, a GP-TCN deep learning model is adopted. This model is based on a temporal convolutional network and a Gaussian process to predict the random response of the vehicle-bridge system. The framework of the GP-TCN model is as Figure 2 shown. This model includes four parts to achieve the random vibration analysis of the VBI system.

[0082] Specifically, first, considering the uncertain parameters of the VBI system and the random track irregularity, a vertical vehicle-bridge interaction model is established. Second, numerical simulations are carried out by the MCM method to obtain samples of the random dynamic responses of the vehicle and the bridge. Third, in order to improve the robustness and uncertainty quantification ability of the surrogate model, an improved temporal convolutional network integrating a Gaussian process is established, where the parameters of some convolutional layers are assumed to be random. Then, the obtained MCM samples are used to train and test the GP-TCN model. Finally, under the updated uncertain system and random track irregularity conditions, the trained GP-TCN model is used to predict the random dynamic response of the VBI system, and the accuracy and efficiency of the proposed GP-TCN model are verified by comparing the prediction results with those obtained by MCM and SPEM.

[0083] (2) Surrogate model based on TCN network

[0084] Since TCN performs excellently in capturing long-range dependencies by using dilated convolutions, this network has a large receptive field without significantly increasing the number of parameters. In addition, causal convolutions are used in TCN, which ensures that information from the future to the past is not damaged. Therefore, in this embodiment, TCN is adopted to construct a surrogate model to predict the dynamic response.

[0085] 1) Dilated convolution

[0086] In the surrogate model of the vehicle-bridge coupling system, the input sequence is the track irregularity, which is regarded as the main excitation of the VBI system. The output sequence is the dynamic responses of the train and the bridge. The multi-layer stacked dilated causal convolutional layers are used to capture the complex features related to the track irregularity.

[0087] For a one-dimensional sequence x ∈ R n with elements s, the dilated convolution operation F for a filter f: {0, 1, …, k - 1} is formulated as:

[0088]

[0089] where k is the filter size, d is the dilation factor, and "*" is the convolution operator. Figure 3 Shows examples of dilated causal convolutions with dilation factors d = 1, 2, 4 and filter size k = 3. It can be seen that the receptive field of the TCN increases with the increase of the network depth n, filter size k, and dilation factor d. The larger the receptive field, the more extensive the temporal dependencies the model can capture, thus enhancing its ability to understand high-level features in the data.

[0090] 2) Residual connection

[0091] In traditional TCNs, all weights and biases are obtained through supervised learning, which trains the network on a loss function between the estimated sequence and the output sequence. Due to the randomness of the VBI system, this embodiment proposes an improved TCN to predict the random responses of the train and the bridge, where the weights and biases between layers are regarded as random parameters. The output of the improved TCN at the l-th layer at time step t can be expressed as:

[0092]

[0093] In the formula, and b (l) represent the i-th weight and bias term of the l-th layer convolutional filter respectively, which are deterministic parameters; and represent the random parameters at the corresponding positions; represents the output of the previous layer at the dilated time step t - id l ; represents the output of the previous layer; represents the noise term added to the convolutional output, which further improves its robustness and generalization ability.

[0094] To enhance the stability of training and alleviate the vanishing gradient problem in deep networks, a 1D fully convolutional network and multiple residual blocks are adopted in the TCN. Each residual block contains a branch that is used to learn the transformation function F(x). The output of F(x) is added to the input x of the block:

[0095] o = activation(x + F(x))(4)

[0096] Considering improving the stochastic behavior of the TCN, the positions of the stochastic parameters mentioned in Equation (3) are specified in the first dilated convolutional layer of each residual connection block. Specifically, the residual block of the improved TCN is as Figure 4 shown. The residual block consists of two layers of dilated causal convolution and a rectified linear unit (ReLU) activation function, which is used to extract non-linear features from the data. Batch normalization is applied for normalization in the convolutional filter, and spatial dropout is added after each dilated convolution for regularization. To match the input and output widths of the residual network, an additional 1x1 convolution is used so that element-wise addition receives tensors of the same shape.

[0097] (3) Gaussian process to estimate TCN parameters

[0098] 1) Deterministic parameters

[0099] Maximum Likelihood Estimation (MLE) is a commonly used method for estimating parameters in deep learning. Suppose there is a training dataset with input x and output y, denoted as and the set of deterministic parameters θ D = [w i b] in the proposed model. The goal of MLE is to determine the parameters θ D such that the joint probability of observing the data under these parameters is maximized.

[0100] The Likelihood Function represents the joint probability of observing the data given the parameters θ D . Assuming that the observed data is independent and identically distributed, the likelihood function can be expressed as:

[0101]

[0102] To simplify the calculation, the logarithm of the likelihood function is usually taken, which is called the Log-Likelihood Function. Assuming that the network output follows a normal distribution, then:

[0103]

[0104] The log-likelihood function can be expressed as:

[0105]

[0106] where f(x i , θ D ) represents the output of the network for input x i and parameter θ D ; represents the variance of the normal distribution caused by noise. Equation (7) can be further simplified to:

[0107]

[0108] The parameter is determined by maximizing the log-likelihood function log L(θ D ), which can be expressed as:

[0109]

[0110] where θ D represents the set of deterministic parameters in the TCN model.

[0111] It can be seen that maximizing the log-likelihood function is equivalent to minimizing the sum of squared residuals. In other words, the parameter can be determined by the loss function between the true sequence and the predicted sequence. However, maximizing the likelihood function is only applicable to estimating the deterministic parameters mainly used in single-point estimation models.

[0112] 2) Random parameters

[0113] The Gaussian Process (GP) is a flexible non-parametric Bayesian inference method used to model complex functional relationships and capture the uncertainty therein. In the present invention, the Gaussian Process is used to estimate the random parameter θ R = [W rand , b rand , where W rand is a matrix composed of w i,rand .

[0114] Assume that the random parameter θ R is an independent Gaussian Process (GP) with zero mean and covariance function k θ , i.e., θ R ~GP(0, k θ ). This assumption means that the prior distribution of the random parameter also follows a Gaussian distribution, and its expression is:

[0115] p(θR |x) ∼ N(0, K θ ) (10)

[0116] where K θ = k θ (x, x) represents the covariance matrix encoding the prior distribution of the random parameter to be inferred. Based on the above statistical assumptions, the log-likelihood function of the observed data following a Gaussian distribution under the parameter θ R can be written as:

[0117]

[0118] where I represents the identity matrix. According to Bayes' theorem, the posterior distribution of the random parameter can be expressed as:

[0119]

[0120] It can be seen that the posterior distribution is proportional to the likelihood function and the prior distribution. According to the conditional distribution of the multivariate Gaussian distribution, the analytical solution of the Gaussian likelihood function can be expressed as:

[0121]

[0122] Using the same Gaussian process analogy, the posterior distribution predicted at the training samples can be obtained as:

[0123]

[0124] Equation (14) shows that the posterior distribution can be completely determined by the conditional mean and the conditional variance which can be calculated as:

[0125]

[0126] where the covariance function k in the prior θ implies statistical model assumptions such as isotropy and invariance. Therefore, the choice of the covariance function directly affects the smoothness and complexity of the function generated by the Gaussian process. Therefore, a covariance function in the form of a squared exponential with Automatic Relevance Determination (ARD) is adopted, and its expression is as follows:

[0127]

[0128] where x a and x b are input sample vectors; the variance and the length-scale vector l 2denotes the hyperparameters of the kernel function. Thus, the random parameter θ R The posterior distribution of is transformed into a minimization problem of the log marginal likelihood with respect to the hyperparameters of the kernel function:

[0129] q Θ (θ R ) = arg max θ l(Θ) = arg max θ log p(Y|X; Θ) (18)

[0130] where q Θ (θ R ) represents the posterior distribution of the random parameter θ R ; denotes the parameter vector containing the kernel hyperparameters and the noise variance. The Gaussian log marginal likelihood can be expressed as:

[0131]

[0132] The log marginal likelihood function of the Gaussian function contains multiple local minima. Although gradient descent optimization can only produce local minima, unlike global stochastic optimization which can produce global minima, gradient descent optimization is still used to update the hyperparameters. This is because gradient descent optimization not only saves a large amount of computational cost but also can produce the analytical derivative of the marginal likelihood with respect to the hyperparameters. The analytical partial derivative of Equation (19) with respect to Θ is expressed as:

[0133]

[0134] where tr(·) represents the trace of a matrix. In addition, by generating different initial hyperparameters and repeating the optimization process, the optimized hyperparameters can be considered "good enough". Generally, Θ can be updated by the gradient descent algorithm to infer q Θ (θ R ), which is an algorithm for conventional neural networks with deterministic parameters.

[0135] 3) Training of the GP-TCN model

[0136] Adopt a hybrid training strategy that combines the Gaussian process and the backpropagation algorithm to optimize the random parameters and deterministic parameters of the improved TCN model. Figure 5 shows the training process of the GP-TCN model, and the specific steps are as follows:

[0137] Step 1. Randomly initialize the kernel hyperparameters Θ and the deterministic parameter θ of the Gaussian process D .

[0138] Step 2. Select a batch of data from the dataset.

[0139] Step 3. According to the current kernel hyperparameter Θ, use Gaussian process to predict the posterior distribution q Θ (θ R ).

[0140] Step 4. Sample θ Θ (θ R ) from the predicted posterior distribution q R .

[0141] Step 5. Calculate the total loss L t = L mse + L Θ , where L mse is the mean squared error (MSE) loss function of the deterministic parameter θ D , and L Θ is the log marginal likelihood loss with respect to the kernel hyperparameter Θ (as shown in Equation (19)).

[0142] Step 6. Use the backpropagation algorithm to calculate the gradient of L mse with respect to the deterministic parameter θ D (i.e., ) and the gradient of L Θ with respect to the kernel hyperparameter Θ (i.e., ).

[0143] Step 7. Use the gradient descent method and the learning rate α (used to control the step size of parameter update) to update the deterministic parameter and the kernel hyperparameter of the Gaussian process.

[0144]

[0145] Step 8. Repeat Steps 2 to 7 until the loss function converges to a predetermined stopping criterion.

[0146] It should be noted that the training strategy will be repeatedly executed with different batches of bridge irregularity inputs and dynamic response outputs until the loss function converges to a predetermined stopping criterion.

[0147] 3. Verification

[0148] (1) Establish a stochastic VBI system

[0149] To verify the accuracy and efficiency of the surrogate model proposed in the present invention, a stochastic VBI system was established. To reduce computational consumption, a three-span simply supported beam widely used in railways was adopted for the bridge model. The length of a single-span beam is 32 meters, and the cross-sectional shape is box-shaped. The cross-sectional area and mass moment of inertia of the beam are 8.89 m 2 and 10.947 m 4。The damping form of the bridge structure is considered as Rayleigh damping. The detailed properties of the beam, including Young's modulus, damping ratio, and density, are regarded as random variables, as shown in Table 1:

[0150] Table 1

[0151]

[0152] Meanwhile, in this embodiment, a train model with eight carriages is adopted, which represents a type commonly used in Chinese high-speed railways. The properties and their distribution patterns of the vehicle parameters are listed in Table 2:

[0153] Table 2

[0154]

[0155] (2) Compare the physical model with the surrogate model

[0156] Considering the statistical results of 5000 samples as the approximate solution of the vehicle-bridge interaction (VBI) system, these results are used to test the reliability of the proposed Gaussian process-time convolutional network (GP-TCN) model. To ensure that the train reaches a steady state before crossing the bridge, the train travels 30 meters in advance on the subgrade. The length of the track line is 156 meters. Using the trigonometric series method, a total of 5000 track irregularity samples with a total length of 300 meters are generated according to the German low-interference track spectrum. Under this assumption, the train runs at a constant speed of 200 km / h. The Newmark-β method is used to solve the dynamic equation, and the time step is 0.01 seconds.

[0157] Considering the 11 uncertain random parameters of the VBI system and the randomness of the track irregularity, Figure 6 The contour plot of the probability density function (PDF) of the bridge dynamic response calculated by the Matlab program is shown. It can be seen that the high probability density regions show an obvious changing trend and are concentrated in specific areas, indicating that although many random parameters are considered, the dynamic response of the bridge still has obvious statistical laws.

[0158] In addition, the spectral representation method (SPEM) was also adopted as a comparison model to verify the reliability of the proposed surrogate model. The response surface method was combined with SPEM to predict the stochastic response of the VBI system affected by multiple random factors. The response surface of the VBI system was composed of a second-order polynomial that ignored the cross terms of random parameters. The response samples obtained by the Monte Carlo method (MCM) could be directly used to solve the coefficients of the second-order polynomial. Here, the spatial frequency range of track irregularity was set to 0.01×2π~1×2π rad / m, and the discrete frequency points were set to 100. Running the SPEM program once required calling the VBI program 100 times. To construct the response surface of the VBI system, a total of 23 runs of the SPEM program were required. Therefore, to predict the stochastic dynamic response of the VBI system considering multiple random factors, a total of 2300 calls were made to the VBI program.

[0159] Figure 7 The mean time-varying power spectral density (PSD) of the train and bridge dynamic responses obtained by the spectral representation method is shown. It can be seen that there is significant energy concentration around a specific frequency band (mainly between 0 Hz and 4 Hz) in the time-varying PSD of the vehicle dynamic response, which is consistent with the main frequency of vehicle motion. The energy of the bridge PSD is mainly concentrated around 10 Hz and 30 Hz.

[0160] The aforementioned GP-TCN model was used to predict the stochastic response of the VBI system. First, MinMaxScaler was used to normalize the input and output data to ensure that they were within a similar scale range. The training set was divided into 80%, and the remaining samples were used as the test set to test the prediction accuracy of the GP-TCN model. The dataloader of PyTorch was used for efficient batch processing of the dataset. The TCN architecture consisted of multiple temporal block layers, and each layer captured features from the input sequence through two convolutional layers, a ReLU activation function, dropout for regularization, and residual connections. Table 3 lists the hyperparameters of the model. The Adam optimizer was used to update the weights of the model, and the mean squared error (MSE) criterion was used to calculate the loss.

[0161] Table 3

[0162]

[0163]

[0164] To incorporate the uncertainties of the vehicle-bridge system, Gaussian processes were used in the first convolutional layer of each temporal block layer, where the weights and biases were regarded as random variables. This architecture combined the advantages of TCN in sequence data modeling while incorporating the flexibility and uncertainty estimation ability of Gaussian processes, thus improving the robustness and prediction performance of the model.

[0165] Figure 8 It shows the changes in training and test losses as the number of training epochs increases. Figure 8 As shown in [reference], in the first 20 epochs, the training loss and test loss values of the TCN surrogate model for predicting the dynamic responses of trains and bridges decreased significantly as the number of epochs increased. After 20 epochs, the loss values almost remained unchanged. It can be seen that when the error converges, the loss value is almost close to zero, indicating that the modified TCN model has excellent training performance. The adjusted model was used to predict the new dynamic responses of the VBI system under the influence of multiple unknown random parameters and track irregularities.

[0166] Figure 9 and Figure 10 respectively show the mean and standard deviation of the dynamic responses of trains and bridges obtained by the Monte Carlo method (MCM), spectral representation method (SPEM), and the method based on Gaussian process - temporal convolutional network (GP - TCN). It can be seen that the mean values of the dynamic responses of the VBI system calculated by the GP - TCN method and SPEM are in good agreement with the results of MCM. However, there are some minor differences in the standard deviation of the VBI system responses obtained by MCM, especially in places with sharp local fluctuations.

[0167] To better demonstrate the prediction performance of the proposed GP - TCN method and other methods, Table 4 lists three evaluation metrics regarding the standard deviation of the dynamic response, where the bold part represents the best evaluation results among the two models.

[0168] Table 4

[0169]

[0170] It can be seen that the GP-TCN model outperforms SPEM in the three evaluation metrics for predicting the standard deviation of dynamic responses. For example, for train displacement, the mean absolute percentage error (MAPE) values of SPEM and GP-TCN are 3.79715% and 0.22136% respectively. Correspondingly, compared with SPEM, the prediction accuracy of the GP-TCN model in predicting the standard deviation of train displacement is improved by 94.2%. The results show that the proposed GP-TCN model performs excellently in predicting the stochastic dynamic responses of the VBI system. Regarding the prediction efficiency of the proposed GP-TCN model, based on a computer equipped with a 13th generation Intel Core i9-13900K CPU and an NVIDIA GeForce RTX 4070 GPU, it takes 8443.4 seconds to obtain 5000 samples from MCM, 3893.6 seconds for SPEM, while the total training time of GP-TCN and the time for predicting 5000 new responses are less than 245 seconds. The results show that the efficiency of GP-TCN has been greatly improved compared with traditional stochastic vibration analysis methods.

[0171] Figure 11 Shows the comparison of the Fourier spectra of the bridge responses obtained by the proposed GP-TCN model and the MCM method. It should be noted that in order to reduce the workload, only the approximate solution results of the bridge responses obtained by MCM are compared with the GP-TCN model. It can be observed that the Fourier spectrum curves of both the bridge displacement and bridge acceleration predicted by GP-TCN are in good agreement with the results calculated by MCM. At the same time, it can be clearly observed that the first-order main frequency of the bridge is approximately 4.72 Hz. These results indicate that the GP-TCN model can accurately evaluate the frequency information of the VBI dynamic responses considering the system parameter uncertainties and the influence of track irregularities.

[0172] Figure 12 Shows the comparison of the probability density functions (PDFs) of the maximum dynamic responses of the train and the bridge obtained by the GP-TCN model and MCM respectively. Generally speaking, the PDFs of the dynamic responses of the VBI system obtained based on the GP-TCN model are in agreement with the results calculated by MCM. However, regarding the peak position of the PDF of the maximum acceleration response of the bridge, subtle differences can be observed.

[0173] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not restrictive. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the purpose and scope of the present technical solution, and they should all be covered within the scope of the claims of the present invention.

Claims

1. A vehicle-bridge system random response prediction method based on temporal convolutional network and Gaussian process, characterized by: Establish vehicle-bridge interaction model; Numerical simulation is performed using the Monte Carlo method to obtain random dynamic response samples of vehicles and bridges; The Gaussian process and the time convolution network are integrated to establish the GP-TCN network model. In this network model, the output of the lth convolution layer at time step t It is expressed as: In the formula, and b (l) They represent the i-th weight and bias term of the l-th convolution filter, respectively, which are deterministic parameters; and They represent the random parameters at the corresponding positions respectively; Indicates the expansion time step t-id at layer l l The output of the previous layer; Represents the output of the previous layer; represents a noise term added to the convolution output; wherein the position of the random parameter is set in the first dilated convolution layer of each residual connection block; in the GP-TCN model, the deterministic parameter is estimated by the maximum likelihood method, and the random parameter is estimated by the Gaussian process; The track irregularity is used as the input of the GP-TCN model, and the dynamic response of the vehicle and bridge is used as the output of the GP-TCN model. The model is trained using sample data obtained from numerical simulation. The trained GP-TCN model is used to predict the random dynamic response of the vehicle-bridge interaction system.

2. The method according to claim 1, characterized in that: The vehicle-bridge interaction model is expressed as: Where M, C and K represent the mass, damping and stiffness matrices respectively; X(t), and They represent the displacement, velocity and acceleration vectors of the vehicle-bridge interaction system respectively; F(t) represents the load vector.

3. The method according to claim 1, characterized in that: Estimating the deterministic parameters by the maximum likelihood method includes: setting θ D represents the set of deterministic parameters, θ D =[w i b], and determine θ by maximum likelihood estimation D , so that in θ D The joint probability of observing the data is maximized; θ D The likelihood function is expressed as: In the formula, x i ,y i Respectively represent the input and output data of training, and N represents the amount of data; Take θ D The log-likelihood function of : In the formula, f(x i ,θ D ) indicates that in θ D and input x i The output of the network model is as follows: represents the variance of the normal distribution caused by noise; Maximize the log-likelihood function: According to the above formula, maximizing the log-likelihood function is equivalent to minimizing the residual sum of squares. Therefore, the parameters are determined by calculating the loss function between the true response and the predicted response of the VBI system. Get the deterministic parameter set θ D , thus obtaining the deterministic parameters.

4. The method according to claim 1, characterized in that: Estimating the random parameters by Gaussian process includes: if the random parameters have independent Gaussian process characteristics, then their prior distribution also follows Gaussian distribution, then in the random parameter set θ R The log-likelihood function of the observed data following a Gaussian distribution is expressed as: In the formula, X and Y represent the input and output of the model respectively, and I represents the unit matrix; According to Bayes' theorem, the posterior distribution of the random parameter is expressed as: The posterior distribution is proportional to the likelihood function and the prior distribution; the random parameter θ R The posterior distribution solution of is transformed into the minimization problem of the log-marginal likelihood with respect to the kernel function hyperparameters: In the formula, q Θ (θ R ) represents the random parameter θ R The posterior distribution of ; Θ represents the parameter vector containing the kernel hyperparameters and the noise variance; the analytical solution of the logarithmic marginal likelihood function is: In the formula, represents the variance of the normal distribution caused by noise, K θ The covariance function representing the prior distribution of the random parameter is a squared exponential covariance function determined by autocorrelation. According to the above formula, the logarithmic marginal likelihood function of the Gaussian function contains multiple local minima, so the gradient descent method is used to update Θ to infer q Θ (θ R ), thereby obtaining random parameters.

5. The method according to claim 1, characterized in that: The process of training the GP-TCN network model includes: S1, randomly initialized Gaussian process kernel hyperparameters Θ and deterministic parameters θ D ; S2, select a batch of data from the sample data set; S3. Based on the current kernel hyperparameter Θ, use the Gaussian process to predict the posterior distribution q of the random parameter Θ (θ R ); S4. From the predicted posterior distribution q Θ (θ R ) in the sample θ R ; S5. Calculate the total loss L of the GP-TCN network model t =L mse +L Θ , where L mse is the deterministic parameter θ D The mean square error (MSE) loss function, L Θ is the log-marginal likelihood loss with respect to the kernel hyperparameter Θ; S6. Calculate L using the back propagation algorithm mse Regarding the deterministic parameter θ D The gradient and L Θ The gradient with respect to the kernel hyperparameter Θ; S7, use gradient descent and learning rate α to update the deterministic parameters and kernel hyperparameters of the Gaussian process; S8. Repeat steps S2 to S7 until the loss function converges to a predetermined stopping criterion.

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