Vehicle-bridge system random response prediction method based on time convolution network and Gaussian process

Abstract

The invention relates to a vehicle-bridge system random response prediction method based on a time convolution network and a Gaussian process, and belongs to the field of VBI system random response prediction. The method comprises the following steps: establishing an axle interaction model; obtaining random dynamic response samples of the vehicle and the bridge; a Gaussian process and a time convolution network are fused, a GP-TCN network model is established, in the network model, weights and offsets between convolution layers are set as random parameters, and the positions of the random parameters are set in a first expansion convolution layer of each residual connection block; taking the track irregularity as the input of the GP-TCN model, taking the dynamic response of the vehicle and the bridge as the output of the GP-TCN model, and training the model by adopting sample data obtained by numerical simulation; and utilizing the trained GP-TCN model to predict the random dynamic response of the axle interaction system. According to the method, the dynamic response of the VBI system in the time domain and the frequency domain can be accurately evaluated.

Description

Technical Field

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

[0002] Due to the dynamic interaction between vehicles and bridges, the dynamic response of vehicles when traveling 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 increasing attention. At present, in most references, the coupled vibration analysis of the vehicle-bridge system is assumed to be a deterministic dynamic problem, ignoring the randomness of track irregularities as one of the main excitation sources. In fact, not only the track is irregular, but also the parameters of the vehicle and bridge structure inevitably have uncertain characteristics due to environmental influences, the inherent discreteness of concrete materials, and structural geometric deviations during construction. Therefore, in order to capture the random response characteristics of the vehicle-bridge system, it is necessary to perform random vibration analysis of the vehicle-bridge system while considering the influence of random track irregularities and random structural parameters.

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

[0004] Among them, MCM is a random analysis method based on a large number of discrete sample statistics, which can be widely used in complex random problems. However, in order 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 VBI systems affected by random track irregularities, PEM converts track irregularities into a series of deterministic harmonic excitations with phase lag. However, since PEM cannot consider random structural parameters and the nonlinear relationship between wheels and tracks, its application in random vibration analysis of VBI systems is limited. The hybrid pseudo-excitation-polynomial chaos expansion method (PEM-PCE) is developed for vehicle-track coupling systems with uncertain parameters, and the results obtained are highly consistent with the Monte Carlo method. Among them, the polynomial chaos (PC) expansion is a non-invasive method for evaluating the evolution and quantification of uncertainty in dynamic systems. Polynomial chaos (PC) expansion can be introduced into the model random process. The PC expansion is formed by Hermite polynomials and Gaussian random variables. The polynomial chaos expansion method can be used to analyze the dynamic response of the bridge under the action of moving vehicles considering the random parameters of the beam bridge model. It can also be used to predict the vehicle motion under uncertain vehicle parameters. The point estimation method is based on Gaussian integral theory and has the advantages of simple operation and simple principle. It can be used to study the dynamic behavior of the train-bridge system with random track irregularities in combination with Karhunen-Loéve expansion and point estimation method. 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 random vibration theories have been developed, hundreds of calculations are still required to ensure the accuracy of computer results, which will consume a lot 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 technology 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 technology has stronger data mining capabilities and nonlinear fitting capabilities using multi-layer neural networks. In recent years, a number of studies on random vibration analysis of vehicle-bridge interaction (VBI) systems based on deep learning networks have been carried out. For example, the random pseudo excitation method (SPEM) combined with a fully convolutional network and a gated recurrent unit can predict the random dynamic response of the vehicle-bridge interaction system under uncertain system parameters and random track irregularities; the long short-term memory (LSTM) model is combined with the ballasted track train interaction (TBTI) system to accurately predict the dynamic behavior of TBTI under a large number of random track geometry excitations; the Bayesian inference is combined with the LSTM network to analyze the random vibration response of the bridge under vehicle dynamic interaction.

[0006] However, most deep learning models used to predict the random dynamic response of VBI systems usually ignore the uncertainty of the model. In fact, estimating the uncertainty of deep learning models through statistical methods can greatly improve the nonlinear mapping ability of the proxy model. Therefore, this paper proposes a new model combining Gaussian process (GP) and temporal convolutional network (TCN) for predicting the random response of VBI systems. Summary of the invention

[0007] In view of this, the object of the present invention is to provide a vehicle-bridge system random response prediction method based on temporal convolutional network (TCN) and Gaussian process (GP), which uses TCN to extract the characteristics of track irregularity and predict the random dynamic response.

[0008] In order to achieve the above object, the present invention provides the following technical solutions:

[0009] A vehicle-bridge system random response prediction method 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 simulations using the Monte Carlo method to obtain random dynamic response samples of vehicles and bridges;

[0012] 3) The Gaussian process and the temporal convolutional network are integrated to establish a GP-TCN network model, in which the weights and biases between convolutional layers are set as random parameters, and the position of the random parameters is set in the first dilated convolutional layer of each residual connection block;

[0013] 4) Using track irregularity as the input of the GP-TCN model and the dynamic response of vehicles and bridges as the output of the GP-TCN model, the model is trained using sample data obtained from numerical simulation;

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

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

[0016]

[0017] 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.

[0018] Furthermore, in the GP-TCN model, the output of the lth convolutional layer at time step t is It is expressed as:

[0019]

[0020] 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 is the output of the previous layer; Represents the output of the previous layer; Represents the noise term added to the convolution output.

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

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

[0023] Let 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;

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

[0025]

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

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

[0028]

[0029] 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;

[0030] Maximize the log-likelihood function:

[0031]

[0032] 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.

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

[0034] If the random parameters have the characteristics of independent Gaussian processes, then their prior distribution also follows the 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:

[0035]

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

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

[0038]

[0039] 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:

[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 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:

[0042]

[0043] 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.

[0044] 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 qΘ (θ R ), thereby obtaining random parameters.

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

[0046] S1, randomly initialized Gaussian process kernel hyperparameters Θ and deterministic parameters θ D ;

[0047] S2, select a batch of data from the sample data set;

[0048] 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 );

[0049] S4. From the predicted posterior distribution q Θ (θ R ) in the sample θ R ;

[0050] 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;

[0051] 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;

[0052] S7, use gradient descent and learning rate α to update the deterministic parameters and kernel hyperparameters of the Gaussian process;

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

[0054] The beneficial effect of the present invention is that the present invention proposes a deep learning model 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 TCN in capturing the fine details of a data set within a short time interval and predicting the response through a specific dilated convolution layer. At the same time, in order to map the uncertainty of the VBI system, the first convolution layer of each time block layer in the TCN is designated as a random layer, and a non-parametric estimation method based on a Gaussian process (GP) is used to evaluate the probability distribution of the weights and biases of the first layer of convolution, without assuming a specific probability distribution in advance.

[0055] The present invention can accurately evaluate the dynamic response of the VBI system in the time domain and 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, objectives and features of the present invention will be described in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the following examination and study, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below in conjunction with the accompanying drawings, wherein:

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

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

[0060] Figure 3 Schematic diagram of dilated causal convolution with dilation factors d=1, 2, 4 and filter size k=3;

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

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

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

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

[0065] Figure 8 This is a diagram showing how the training and test losses change over time. Figure 8 (a) is a schematic diagram of the training and testing of train dynamic response. Figure 8 (b) Schematic diagram of training and testing of bridge dynamic response;

[0066] Fig. 9 To compare the dynamic responses of trains based on MCM, SPEM and GP-TCN methods, Fig. 9 (a) is the average value of displacement, Fig. 9 (b) is the standard deviation of displacement, Fig. 9 (c) is the average value of acceleration, Fig. 9 (d) is the standard deviation of acceleration;

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

[0068] Fig.11 Comparison of the Fourier spectra of the bridge response obtained by GP-TCN and MCM methods. Fig.11 (a) is the displacement, Fig.11 (b) is acceleration;

[0069] Fig.12 This is a PDF comparison diagram of the maximum dynamic response based on the GP-TCN and MCM methods. Fig.12 (a) is the train displacement, Fig.12 (b) is the train acceleration, Fig.12 (c) is the bridge displacement, Fig.12 (d) is the bridge acceleration. DETAILED DESCRIPTION

[0070] The following describes the embodiments of the present invention by specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways 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 only illustrate the basic concept of the present invention in a schematic manner, and the following embodiments and features in the embodiments can be combined with each other without conflict.

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

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

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

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

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

[0076] based on Figure 1 According to the mechanical model and the principle of energy conservation shown in the figure, 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] This embodiment uses the GP-TCN deep learning model, which is based on a temporal convolutional network and a Gaussian process to predict the random response of the bridge system. The framework of the GP-TCN model is as follows: Figure 2 As shown, the model consists of four parts to realize the random vibration analysis of the VBI system.

[0082] Specifically, firstly, a vertical vehicle-bridge interaction model is established considering the uncertain parameters of the VBI system and random track irregularities. Secondly, numerical simulations are performed using the MCM method to obtain random dynamic response samples of vehicles and bridges. Thirdly, in order to improve the robustness and uncertainty quantification ability of the surrogate model, an improved temporal convolutional network fused with Gaussian processes is established, in which the parameters of some convolutional layers are assumed to be random. Then, the GP-TCN model is trained and tested using the obtained MCM samples. Finally, the trained GP-TCN model is used to predict the random dynamic response of the VBI system under the updated uncertain system and random track irregularities, 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) Proxy model based on TCN network

[0084] Since TCN excels in capturing long-distance dependencies by using dilated convolutions, the 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 compromised. Therefore, this embodiment uses TCN to build a proxy model to predict dynamic responses.

[0085] 1) Dilated Convolution

[0086] In the surrogate model of the train-bridge coupled 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 response of the train and the bridge. Multiple stacked dilated causal convolutional layers are used to capture the complex features related to track irregularity.

[0087] For a one-dimensional sequence x∈R n The formula of the dilated convolution operation F for the filter f:{0,1,…,k-1} is expressed as:

[0088]

[0089] Among them, k is the filter size, d is the dilation factor, and “*” is the convolution operator. Figure 3 Examples of dilated causal convolutions with dilation factors d = 1, 2, 4 and filter size k = 3 are shown. It can be seen that the receptive field of TCN increases with the increase of network depth n, filter size k and dilation factor d. A larger receptive field indicates that the model is able to capture a wider range of temporal dependencies, thereby enhancing its ability to understand high-level features in the data.

[0090] 2) Residual Connection

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

[0092]

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

[0094] In order to enhance the stability of training and alleviate the gradient vanishing problem in deep networks, TCN uses a 1D fully convolutional network and multiple residual blocks. 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 the random behavior of the improved TCN, the location of the random parameters mentioned in Equation (3) is specified in the first dilated convolutional layer of each residual connection block. Specifically, the residual block of the improved TCN is as follows Figure 4 As shown in Figure 2. The residual block consists of two layers of dilated causal convolutions and rectified linear unit (ReLU) activation functions to extract nonlinear features from the data. Batch normalization is applied to the convolutional filters for normalization, and spatial dropout is added after each dilated convolution for regularization. In order to match the input and output widths of the residual network, an additional 1x1 convolution is used so that the element-wise addition receives tensors of the same shape.

[0097] (3) Gaussian process estimation of TCN parameters

[0098] 1) Deterministic parameters

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

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

[0101]

[0102] In order 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] Among them, f(x i ,θ D ) means that when input x i and parameter θ D The output of the network below; represents the variance of the normal distribution caused by noise. Formula (7) can be further simplified as:

[0107]

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

[0109]

[0110] Among them, 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 residual sum of squares. In other words, the parameter It can be determined by the loss function between the true series and the predicted series. However, maximizing the likelihood function is only suitable for estimating deterministic parameters mainly used in single point estimation models.

[0112] 2) Random parameters

[0113] Gaussian Process (GP) is a flexible nonparametric Bayesian inference method for modeling complex functional relationships and capturing the uncertainty therein. In the present invention, Gaussian Process is used to estimate the random parameter θ in the improved TCN model. R =[W rand ,b rand ], where W rand Because of w i,rand The matrix composed of.

[0114] Assume that the random parameter θ R are independent Gaussian processes (GPs) with zero mean and covariance function k θ , that is, θ R ~GP(0,k θ ). This assumption means that the prior distribution of the random parameters also follows a Gaussian distribution, which is expressed as:

[0115] p(θR |x)~N(0,K θ ) (10)

[0116] Among them, 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 parameter θ R The log-likelihood function of the observed data following a Gaussian distribution 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, we can get the posterior distribution of the predictions at the training samples as:

[0123]

[0124] Formula (14) shows that the posterior distribution can be completely determined by the conditional mean and conditional variance OK, they can be calculated as:

[0125]

[0126] in, Covariance function k in the prior θ Implicit statistical model assumptions such as isotropy and invariance. Therefore, the choice of 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 square exponential with automatic relevance determination (ARD) is used, and its expression is as follows:

[0127]

[0128] Among them, x a and x b is the input sample vector; variance and the length scale vector l 2represents the hyperparameter of the kernel function. Therefore, the random parameter θ R The posterior distribution of is transformed into the minimization problem of the log marginal likelihood with respect to the kernel function hyperparameters:

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

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

[0131]

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

[0133]

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

[0135] 3) Training of GP-TCN model

[0136] A hybrid training strategy combining Gaussian process and back-propagation algorithm is adopted to optimize the random parameters and deterministic parameters of the improved TCN model. Figure 5 The training process of the GP-TCN model is shown. The specific steps are as follows:

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

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

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

[0140] Step 4. From the predicted posterior distribution q Θ (θ R ) in the sample θ R .

[0141] Step 5. Calculate the total loss L of the improved TCN 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 (as shown in Equation (19)).

[0142] Step 6. Calculate L using the back-propagation algorithm mse Regarding the deterministic parameter θ D The gradient of ) and L Θ The gradient of the kernel hyperparameter Θ (i.e. ).

[0143] Step 7. Update the deterministic parameters and the kernel hyperparameters of the Gaussian process using gradient descent and a learning rate α (used to control the step size of parameter updates).

[0144]

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

[0146] It is worth noting that the training strategy will be repeatedly executed with different bridge irregularity inputs and dynamic response output batches until the loss function converges to a predetermined stopping criterion.

[0147] 3. Verification

[0148] (1) Establishing a random VBI system

[0149] In order to verify the accuracy and efficiency of the proxy model proposed in this paper, a random VBI system was established. In order to reduce the computational cost, the bridge model adopts a three-span simply supported beam widely used in railways. The length of the 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 8.89m 2 and 10.947m 4The damping form of the bridge structure is considered to be Rayleigh damping. The detailed properties of the beam, including Young's modulus, damping ratio and density, are considered as random variables, as shown in Table 1:

[0150] Table 1

[0151]

[0152] Meanwhile, this embodiment adopts an eight-carriage train model, which represents a model commonly used in China's high-speed railways. The attributes of vehicle parameters and their distribution patterns are listed in Table 2:

[0153] Table 2

[0154]

[0155] (2) Comparison between physical model and proxy model

[0156] The statistical results of 5000 samples are considered as the approximate solution of the vehicle-bridge interaction (VBI) system, and these results are used to verify the reliability of the proposed Gaussian process-temporal convolutional network (GP-TCN) model. To ensure that the train reaches a stable state before going on the bridge, the train travels 30 meters on the roadbed in advance. The track line length 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 200km / h. The Newmark-β method is used to solve the dynamic equations with a time step of 0.01 seconds.

[0157] Considering the 11 uncertain random parameters of the VBI system and the randomness of track irregularities, Figure 6 The probability density function (PDF) contour map of the bridge dynamic response calculated by Matlab program is shown. It can be seen that the high probability density area shows a clear trend of change and is concentrated in a specific area, which shows that despite considering many random parameters, the dynamic response of the bridge still has obvious statistical laws.

[0158] In addition, the spectral representation method (SPEM) is used as a comparison model to verify the reliability of the proposed surrogate model. The response surface methodology is combined with SPEM to predict the random response of the VBI system affected by multiple random factors. The response surface of the VBI system is constructed by a second-order polynomial with the cross terms of random parameters ignored. The response samples obtained by the Monte Carlo method (MCM) can be directly used to solve the coefficients of the second-order polynomial. Here, the spatial frequency range of track irregularity is set to 0.01×2π~1×2πrad / m, and the discrete frequency points are set to 100. Running the SPEM program once requires calling the VBI program 100 times. In order to construct the response surface of the VBI system, a total of 23 SPEM programs need to be run. Therefore, in order to predict the random dynamic response of the VBI system considering multiple random factors, a total of 2300 calls to the VBI program were issued.

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

[0160] The aforementioned GP-TCN model is used to predict the random response of the VBI system. First, the input and output data are normalized using MinMaxScaler to ensure that they are in a similar scale range. The training set is divided into 80% and the remaining samples are used as a test set to test the prediction accuracy of the GP-TCN model. PyTorch's dataloader is used for efficient batch processing of the dataset. The TCN architecture consists of multiple temporal block layers, each of which captures features from the input sequence through two convolutional layers, ReLU activation functions, dropout for regularization, and residual connections. Table 3 lists the hyperparameters of the model. The weights of the model are updated using the Adam optimizer, and the loss is calculated using the mean squared error (MSE) criterion.

[0161] Table 3

[0162]

[0163]

[0164] In order to incorporate the uncertainty of the vehicle-bridge system, a Gaussian process is used in the first convolutional layer of each time block layer, where the weights and biases are treated as random variables. This architecture combines the advantages of TCN in modeling sequence data while incorporating the flexibility and uncertainty estimation capabilities of the Gaussian process, thereby improving the robustness and predictive performance of the model.

[0165] Figure 8 It shows how the training and test losses change as the number of training epochs increases. Figure 8 It is shown in Figure 2 that in the first 20 rounds, the training loss and test loss values ​​of the TCN proxy model used to predict the dynamic response of trains and bridges decrease significantly with the increase in rounds. After 20 rounds, the loss value remains almost unchanged. It can be seen that when the error converges, the loss value is almost close to zero, which shows that the modified TCN model has excellent training performance. The adjusted model is used to predict the new dynamic response of the VBI system under the influence of unknown multiple random parameters and track irregularities.

[0166] Fig. 9 and Fig.10 The mean and standard deviation of the dynamic responses of the train and bridge obtained by the Monte Carlo method (MCM), the spectral representation method (SPEM), and the Gaussian process-temporal convolutional network (GP-TCN) based method are shown respectively. It can be seen that the mean values ​​of the dynamic responses of the VBI system calculated based on 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 response obtained by MCM, especially where there are sharp local fluctuations.

[0167] In order to better demonstrate the prediction performance of the proposed GP-TCN method compared with other methods, three evaluation indicators on the standard deviation of dynamic response are listed in Table 4, where the bold part represents the best evaluation result of the two models.

[0168] Table 4

[0169]

[0170] It can be seen that the GP-TCN model outperforms SPEM in the three evaluation indicators of predicting the standard deviation of dynamic response. 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, the prediction accuracy of the GP-TCN model in predicting the standard deviation of train displacement is improved by 94.2% compared with SPEM. The results show that the proposed GP-TCN model performs well in predicting the random dynamic response 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 NVIDIAGeForce RTX 4070GPU, it takes 8443.4 seconds to obtain 5000 samples from MCM, 3893.6 seconds for SPEM, and less than 245 seconds for the training time of GP-TCN and the total time to predict the new 5000 responses. The results show that the efficiency of GP-TCN has been greatly improved over traditional random vibration analysis methods.

[0171] Fig.11 The comparison of the Fourier spectrum of the bridge response obtained by the proposed GP-TCN model and the MCM method is shown. It is worth noting that in order to reduce the workload, only the approximate solution results of the bridge response obtained by MCM are compared with the GP-TCN model. It can be observed that both the Fourier spectrum curves of the bridge displacement and the 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 about 4.72Hz. These results show that the GP-TCN model can accurately evaluate the frequency information of the VBI dynamic response while considering the uncertainty of system parameters and the influence of track irregularities.

[0172] Fig.12 The comparison of the probability density functions (PDFs) of the maximum dynamic responses of the train and bridge obtained by the GP-TCN model and the MCM, respectively, is shown. Overall, the PDFs of the dynamic responses of the VBI system obtained based on the GP-TCN model are consistent with the results calculated by the MCM. However, slight differences can be observed in the peak position of the PDF regarding the maximum acceleration response of the bridge.

[0173] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution, which should be included in 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 temporal convolutional network are integrated to establish the GP-TCN network model. In this network model, the weights and biases between the convolutional layers are set as random parameters, and the position of the random parameters is set in the first dilated convolutional layer of each residual connection block. 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: In the GP-TCN model, the output of the lth convolutional layer at time step t is It is expressed as: In the formula, and 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 is the output of the previous layer; Represents the output of the previous layer; Represents the noise term added to the convolution output.

4. The method according to claim 3, characterized in that: In the GP-TCN model, the deterministic parameters and random parameters are estimated by maximum likelihood method and Gaussian process.

5. The method according to claim 4, 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 the 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.

6. The method according to claim 4, 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: q Θ (i R )=argmax θ l(Θ)=argmax θ logp(Y|X;Θ) 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.

7. 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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