A vehicle axle coupling system analysis method based on a physical information neural network

Abstract

The application discloses a vehicle-bridge coupling system analysis method based on a physical information neural network, relates to the technical field of vehicle-bridge coupling systems, and aims to solve the problem of low calculation precision of vehicle-bridge coupling system dynamics. The vehicle-bridge coupling system analysis method based on the physical information neural network comprises the following steps: establishing a vehicle model and a bridge model, and establishing a vehicle-bridge coupling system according to the vehicle model and the bridge model; taking a motion equation of the vehicle-bridge coupling system as a constraint condition, taking initial conditions and boundary conditions of the vehicle-bridge coupling system as a loss function, and constructing a physical information neural network; selecting a plurality of random points in the vehicle-bridge coupling system to perform iterative calculation on the physical information neural network, so as to obtain a trained neural network; and calculating the motion equation of the vehicle-bridge coupling system through the trained neural network, so as to obtain a vibration response of the vehicle-bridge coupling system.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of vehicle-bridge coupling system, and particularly relates to a vehicle-bridge coupling system analysis method based on physical information neural network. BACKGROUND

[0002] The vehicle-bridge coupling system dynamics problem is a research topic that never goes out of date, and the development of vehicle-bridge coupling system dynamics research has always been one of the research hotspots in the field of transportation. The existing calculation methods of vehicle-bridge coupling system, such as numerical simulation, finite element simulation, field test and the like, often face the complex and large amount of mathematical derivation and calculation brought by the nonlinearity of the vehicle-bridge time-varying system, or are limited by high-cost commercial software, large-scale sensor arrangement and long-period data acquisition. These methods usually have high requirements for the dynamics, mathematics and programming ability of researchers, and the analytical precision is difficult to accurately reflect the actual working conditions, such as the modal superposition method which inevitably brings calculation errors due to the modal truncation problem.

[0003] With the development of artificial intelligence, new opportunities have also been brought to the vehicle-bridge coupling dynamics research. By integrating physical information into the neural network, the complex mechanical and mathematical derivation process is avoided, and the large amount of data dependence of the conventional neural network method is reduced. At the same time, new research methods or ideas are provided for the deep integration of artificial intelligence and vehicle-bridge coupling system dynamics analysis, and intelligent bridge and vehicle parameter identification, damage identification and the like. SUMMARY

[0004] The present application aims to provide a vehicle-bridge coupling system analysis method based on physical information neural network, which is used to improve the calculation precision of vehicle-bridge coupling system dynamics problem with lower calculation period and cost.

[0005] In order to achieve the above-mentioned purpose, the present application provides the following technical scheme:

[0006] A vehicle-bridge coupling system analysis method based on physical information neural network, comprising:

[0007] establishing a vehicle model and a bridge model, and establishing a vehicle-bridge coupling system according to the vehicle model and the bridge model;

[0008] taking the motion equation of the vehicle-bridge coupling system as a constraint condition, and taking the initial condition and the boundary condition of the vehicle-bridge coupling system integration as a loss function, to construct a physical information neural network;

[0009] selecting a plurality of random points in the vehicle-bridge coupling system to perform iterative calculation on the physical information neural network, to obtain a trained neural network;

[0010] The trained neural network is used to calculate the motion equation of the vehicle-bridge coupling system to obtain a vibration response of the vehicle-bridge coupling system.

[0011] Further, the bridge model adopts an Euler-Bernoulli beam, and the motion equation of the vehicle-bridge coupling system is:

[0012]

[0013]

[0014] m * , which represents a linear element mass of the bridge model, , which represents a mass matrix of the vehicle model, , which represents a second-order derivative of the bridge model with respect to time, u '''' , which represents a fourth-order derivative of the bridge model with respect to a horizontal position of the bridge, , which represents a vertical acceleration vector of the vehicle model, u , which represents a vertical displacement of the bridge model, , which represents a vertical displacement vector of the vehicle system, E , which represents an elastic modulus of the bridge model, I , which represents a moment of inertia of the bridge model, , which represents a stiffness matrix of the vehicle model, , which represents a vertical displacement vector of the bridge at a driving position of the vehicle, , which represents a reaction force of the vehicle model on the bridge.

[0015] Further, the physical information neural network includes an input layer, a physical feature layer, a fully connected layer, a tensor processing layer, and an output layer.

[0016] The input layer is configured to input a plurality of random points in the vehicle-bridge coupling system, and the random points are used to represent real solutions of the vehicle-bridge coupling system at different time points.

[0017] The physical feature layer is configured to add inherent frequencies of the vehicle-bridge coupling system and time-varying motion features.

[0018] The fully connected layer is configured to receive the random points and perform iterative training in the neural network.

[0019] The tensor processing layer is configured to perform tensor clipping on the vehicle model, and only the output of the time dimension is reserved.

[0020] The output layer is configured to output vibration responses of the vehicle model and the bridge model with respect to time.

[0021] ​​​The further technical scheme is characterized in that the loss function comprises a partial differential equation loss, an initial constraint condition loss and a boundary constraint condition loss, and the loss function adopts a mean square error loss function.

[0022] The partial differential equation loss is used to measure the difference between the neural network prediction result and the partial differential equation describing the vehicle-bridge coupling system.

[0023] The initial constraint condition loss is used to ensure that the state of the vehicle-bridge coupling system at the initial time meets the given initial condition.

[0024] The boundary constraint condition loss is used to ensure that the state of the vehicle-bridge coupling system at the boundary meets the given boundary condition.

[0025] The further technical scheme is characterized in that the loss function equation is:

[0026]

[0027] In the formula, vehicle equation loss function weight, bridge equation loss function weight, vehicle initial condition loss function weight, bridge initial condition loss function weight, and bridge boundary condition loss function weight are respectively represented by , , , , vehicle equation loss function is represented by bridge equation loss function is represented by vehicle initial condition loss function is represented by bridge initial condition loss function is represented by bridge boundary condition loss function is represented by

[0028] Each mean square error loss function is specifically shown as follows:

[0029]

[0030]

[0031]

[0032]

[0033]

[0034]

[0035] In the formula, the number of internal points is represented by the number of initial points is represented by ​​​​​​​​representing the number of boundary points, representing the input value of the vehicle model at the initial time in the neural network, representing the true value of the vehicle model at the initial time, representing the input value of the bridge model at the initial time in the neural network, representing the true value of the bridge model at the initial time, representing the input value of the bridge model at the boundary in the neural network, representing the true value of the bridge model at the boundary.

[0036] Further, the technical solutions are as follows: the physical information neural network further comprises an optimizer, and the optimizer is used to adjust parameters of the network to minimize a loss function.

[0037] Further, the technical solutions are as follows: the optimizer comprises an Adam optimizer and an L-BFGS optimizer.

[0038] Compared with the prior art, the vehicle-bridge coupling system analysis method based on the physical information neural network has the following beneficial effects:

[0039] The fitting method for the motion state of the vehicle-bridge coupling system is based on a physical information neural network, constructs a vehicle-bridge coupling system motion equation as a physical constraint, constructs a loss function in combination with initial conditions and boundary conditions, and obtains a training result for fitting the motion state of the vehicle-bridge coupling model through training. The method introduces a dynamic differential equation into the loss function on the basis of a fully connected neural network, so that the model can still follow the physical law in the training process, and the accuracy and generalization ability of the model are improved. BRIEF DESCRIPTION OF DRAWINGS

[0040] The accompanying drawings, which are included to provide a further understanding of the application and are incorporated in and constitute a part of this application, illustrate embodiments of the application and together with the description serve to explain the application. In the drawings:

[0041] Figure 1 A method flowchart is provided for the embodiments of the application;

[0042] Figure 2 An internal structure diagram is provided for the embodiments of the application, taking a single-degree-of-freedom undamped vehicle-bridge coupling system as an example;

[0043] Figure 3 A specific principle diagram of the physical information neural network is provided for the embodiments of the application. DETAILED DESCRIPTION

[0044] ​In order to make the above features and effects of the present application more clear and easy to understand, the following embodiments are specifically described in conjunction with the accompanying drawings. The present specification discloses an embodiment comprising the features of the present application. The disclosed embodiments are only for illustration. The scope of protection of the present application is not limited to the disclosed embodiments, but is defined by the appended claims.

[0045] It should be noted that in the present application, "at least one" means one or more, and "multiple" means two or more. The relationship terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between the entities or operations.

[0046] Without more limitations, the elements defined by the phrase "comprising" do not exclude the presence of additional identical elements in the process, method, article, or device comprising the described elements.

[0047] The technical solutions in the embodiments of the present application will be described clearly and completely in conjunction with the accompanying drawings of the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the scope of protection of the present application.

[0048] The specific embodiments of the present application will be described in detail below in conjunction with the accompanying drawings.

[0049] Please refer to Figure 1 A vehicle-bridge coupling system analysis method based on a physical information neural network, comprising:

[0050] Step S1: establishing a vehicle model and a bridge model, and establishing a vehicle-bridge coupling system according to the vehicle model and the bridge model.

[0051] Taking a single-degree-of-freedom undamped vehicle-bridge coupling system as an example, the vehicle model is composed of an inertial element and an elastic element, the inertial element represents the lumped mass of the vehicle model, and the elastic element represents the stiffness of the vehicle model; the bridge model adopts an Euler-Bernoulli simply supported beam, and the model is as shown in Figure 2 The contact mode of the vehicle model and the bridge model is point contact, the mass matrix is obtained by collecting the mass of the inertial element, the stiffness matrix is obtained by collecting the stiffness coefficient of the elastic element, and the motion equation of the vehicle-bridge coupling system is constructed according to the mass matrix and the stiffness matrix, and the expression is as follows:

[0052] ;

[0053] ;

[0054] ;

[0055] In the formula: m * denotes the mass of the linear element of the bridge model, denotes the mass matrix of the vehicle model, denotes the second derivative of the bridge model with respect to time, u '''' denotes the fourth derivative of the bridge model with respect to the horizontal position of the bridge model, denotes the vertical acceleration vector of the vehicle model, u denotes the vertical displacement of the bridge model, denotes the vertical displacement vector of the vehicle model, E denotes the elastic modulus of the bridge model, I denotes the moment of inertia of the bridge model, denotes the stiffness matrix of the vehicle model, denotes the vertical displacement vector of the bridge model at the driving position of the vehicle model, denotes the contact force between the vehicle model and the bridge model.

[0056] In the motion equation of the vehicle-bridge coupling system, the Dirac function δ(x-xc) is usually used to reflect the coupling of the vehicle and bridge models, but due to the special mathematical properties of the transient change, it cannot be directly calculated in the fitting of the neural network. In the embodiment, the Gaussian function is used instead of the Dirac function, and the peak point of the Gaussian function is used instead of the transient change of the Dirac function, which ensures the normal iteration of the neural network, and at the same time, the Gaussian function is multiplied by a fixed coefficient to ensure the numerical accuracy. In the embodiment, the fixed coefficient is 0.06, and in order to avoid the long tail phenomenon brought by the Gaussian function, the Gaussian function is truncated in the actual network input, and only the peak range of is retained.

[0057] Step S2: constructing a physical information neural network with the motion equation of the vehicle-bridge coupling system as a constraint condition and the initial condition and boundary condition of the vehicle-bridge coupling system integrated as a loss function.

[0058] The initial condition of the vehicle-bridge coupling system is the motion state of the vehicle model at time (t) = 0, i.e. the initial displacement and initial velocity of the vehicle model; the boundary condition of the vehicle-bridge coupling system is the vibration state of the starting point and the ending point of the bridge model, i.e. the initial displacement of the starting point, the initial velocity of the starting point, the ending displacement and the ending velocity of the bridge model.

[0059] The physical information neural network includes an input layer, a physical feature layer, a full connection layer, a tensor processing layer and an output layer.

[0060] The input layer is used for inputting random points; the physical feature layer is used for adding inherent frequencies of the vehicle-bridge coupling system and time-varying motion characteristics; the fully connected layer is used for receiving random points obtained by the input layer and performing iterative training; the tensor processing layer is used for tensor clipping of the vehicle model, and only the output of the time dimension is reserved; and the output layer is used for outputting time-varying vibration responses of the vehicle model and the bridge model.

[0061] In the step S2, the loss function adopts a mean square error loss function, including a physical equation loss, an initial constraint condition loss and a boundary constraint condition loss.

[0062] The physical equation loss includes a motion equation of the vehicle model and a motion equation of the bridge model, wherein the motion equation of the bridge model is a partial differential equation, and the partial differential equation loss is used for measuring the difference between the neural network prediction result and the partial differential equation describing the vehicle-bridge coupling system. The initial constraint condition loss is used for ensuring that the state of the vehicle-bridge coupling system at the initial time meets the given initial condition. The boundary constraint condition loss is used for ensuring that the state of the vehicle-bridge coupling system at the boundary meets the given boundary condition.

[0063] Among them, the constraint condition is generally divided into soft constraint and hard constraint. The soft constraint is to include the initial and boundary constraint condition related items in the loss function, and to reduce the residual error through neural network iteration. The hard constraint is to transform in the output, and to force the value of the training point at the boundary to meet the boundary condition required by the physical equation through a specific function, for example, the output result is multiplied by sin(x), and when x is equal to 0, the output result is 0. In the embodiment, the hard constraint condition is adopted to ensure that the constraint condition of the vehicle-bridge coupling system meets the given condition.

[0064] Specifically, the total loss function equation of the physical information neural network is:

[0065]

[0066] In the formula, vehicle equation loss function weight, bridge equation loss function weight, vehicle initial condition loss function weight, bridge initial condition loss function weight, and bridge boundary condition loss function weight are respectively represented by , , , , vehicle equation loss function is represented by bridge equation loss function is represented by vehicle initial condition loss function is represented by bridge initial condition loss function is represented by bridge boundary condition loss function is represented by

[0067] The specific mean square error loss functions are as follows:​

[0068] ;

[0069] ;

[0070] ;

[0071] ;

[0072] ;

[0073] ;

[0074] In the formula, represents the number of internal points, represents the number of initial points, represents the number of boundary points, represents the input value of the vehicle model at the initial time in the neural network, represents the true value of the vehicle model at the initial time, represents the input value of the bridge model at the initial time in the neural network, represents the true value of the bridge model at the initial time, represents the input value of the bridge model at the boundary in the neural network, , represents the true value of the bridge model at the boundary.

[0075] The weight of the loss function directly affects the importance of different physical constraints and data fitting for the model, and further affects the performance and generalization ability of the model. When selecting, the physical nature of the vehicle-bridge coupling system and the corresponding model training stage should be considered, and appropriate loss function weight configuration should be selected to improve the rate and accuracy of iteration.

[0076] In this embodiment, the weights and biases of the physical information neural network are adjusted by introducing an optimizer. The optimizer determines how to adjust the network weights and biases through the gradient of the loss function, and satisfies the physical constraints. For the physical information neural network required by the vehicle-bridge coupling system, the Adam optimizer is usually selected. The Adam optimizer combines the ideas of Momentum and RMSProp, uses momentum to accelerate convergence, and uses an adaptive learning rate adjustment mechanism to automatically select different step sizes for different parameters, and is relatively insensitive to the selection of hyperparameters, which is more suitable for the complex situation of the vehicle-bridge coupling system.

[0077] In the physical information neural network, after optimization by the Adam optimizer, the L-BFGS optimizer is used for detailed optimization to minimize the loss function.

[0078] Step S3: Select multiple random points in the vehicle-bridge coupling system to perform iterative calculations on the physical information neural network to obtain a trained neural network.

[0079] like Figure 3 As shown, the specific training process is as follows: the input layer is used to input multiple random points of the vehicle in the current state of the vehicle-bridge coupled system, representing the true solution of the vehicle-bridge coupled system at different times. These random points include randomly generated initial points, internal points, and boundary points, representing the horizontal positions of the bridge model and the travel time of the vehicle model in the current state.

[0080] The physical information feature layer, located between the input layer and the fully connected layer, adds the natural frequencies of the bridge and vehicle models, as well as the time-varying motion characteristics of the vehicle-bridge coupling terms, to accelerate the iteration speed of the neural network and make the fitting results more accurate. The formula for calculating the natural frequencies of the Euler-Bernoulli beam is:

[0081] ;

[0082] The formula for calculating the natural frequency of a vehicle model is:

[0083] ;

[0084] In the formula , These represent the natural frequencies of the bridge model and the vehicle model, respectively. L Indicates the length of the bridge. E This represents the elastic modulus of the bridge. I This represents the moment of inertia of the bridge. m * This indicates the mass of the line unit of the bridge. This indicates the stiffness of the vehicle model. This indicates the mass of the vehicle model.

[0085] The physical information feature layer is used to concatenate the natural frequencies of the bridge and the vehicle using trigonometric functions, encoding the motion features of the bridge and vehicles into a multi-dimensional vector. This vector serves as input for subsequent machine learning tasks, and its concatenation method is as follows:

[0086] ;

[0087] .

[0088] The full connection layer is used to receive random points for iterative training in the physical information neural network, and the input data is extracted and fused with the corresponding physical information through each layer of neurons with different weights. Each neuron in the same layer is not linked, but is linked to the previous layer. At the same time, an activation function and an initialization function are introduced in the full connection layer. The activation function applies a nonlinear mapping to the output of each neuron, which can theoretically enable the neural network to approximate any complex function, and at the same time maps the input to a specific range to prevent gradient explosion. The initialization function adjusts the distribution of the initial weights, which directly affects the iterative results of the neural network. The two functions work together to select the corresponding initialization function according to the selected activation function. In this example, the sin activation function is used to adapt to the analytical solution form in the vehicle-bridge coupling model, and the sin activation function can more easily learn the vibration characteristics of the vehicle-bridge coupling model. The initialization function uses the Orthogonal function to prevent the physical feature neural network from falling into gradient vanishing during iteration, thereby reducing the learning error.

[0089] The tensor processing layer is located between the full connection layer and the output layer, and mainly performs tensor cropping on the vehicle model. The vehicle model is implicitly related to the horizontal position variable of the bridge model, i.e., only related to the current position of the vehicle. In the motion equation of the vehicle model, the vertical displacement of the vehicle model is only related to the time independent variable, i.e., the motion equation of the vehicle model is a ordinary differential equation, and the motion equation of the bridge model is a partial differential equation. In training, the tensor is defined as a two-dimensional tensor by default, which does not conform to the physical reality. To meet this requirement, the output transformation is performed on the training results of the vehicle model, and the tensor representing the vehicle model is cropped to only retain the time dimension output. This satisfies the implicit relationship between the vehicle model and the bridge model, and the output result of the vehicle model is also a two-dimensional result, i.e., a function of time.

[0090] The constraint conditions such as initial conditions and boundary conditions are added in the tensor processing layer, i.e., the outputs of the bridge model and the vehicle model are forced to be fixed values at time zero and the boundary. For the Euler-Bernoulli simply supported beam, the initial value and the boundary value are both zero.

[0091] The output layer is connected to the tensor processing layer, which is used to output the vibration response of the vehicle model and the bridge model with time. Specifically, the vibration response includes the vertical displacement of the vehicle model and the vertical displacement of the bridge model, which represent the instantaneous motion state of each point of the vehicle model and the bridge model at different times, respectively.

[0092] Based on the physical information neural network fitting the model of the vehicle-road-bridge vehicle-bridge coupling motion system, the neural network model can learn the global motion state change of the vehicle-road-bridge vehicle-bridge coupling system. Compared with the traditional data-driven method, the physical information is added in the data iteration process, which makes the iterative result more accurate.

[0093] After the iteration ends, the error is gradually reduced and brought closer to the true value through the loss function during the iteration process. That is, the loss function is used to determine whether the iteration result meets the required accuracy requirement. If the accuracy requirement is met, the result is output. If the accuracy requirement is not met, the iteration continues until the required accuracy requirement is met.

[0094] The loss function includes the initial conditions and boundary conditions (initial speed conditions of the vehicle, initial speed conditions of the bridge, etc.) of the vehicle and bridge models, along with their corresponding physical information. The loss function primarily measures the difference between the model's predictions and the actual situation, guiding the model's parameter updates. The mean squared error (MSE) function quantifies the gap between the predicted data and the known physical information. By calculating this difference, the model can understand its own prediction performance, providing positive feedback for reducing prediction errors.

[0095] Step S4: Calculate the motion equation of the vehicle-bridge coupling system using the trained neural network to obtain the vibration response of the vehicle-bridge coupling system.

[0096] Compared with traditional numerical methods, this invention reduces the model simplification error caused by the modal superposition method. The physical information neural network embeds the system's motion equations as physical information into the neural network, enabling the model to follow physical laws during training and improving the accuracy and generalization ability of the solution.

[0097] Although the invention has been described herein in conjunction with various embodiments, those skilled in the art will understand and implement other variations of the disclosed embodiments by reviewing the accompanying drawings, the disclosure, and the appended claims in carrying out the claimed invention. In the claims, the word "comprising" does not exclude other components or steps, and "a" or "an" does not exclude a plurality. A single processor or other unit can implement several functions listed in the claims. While different dependent claims may recite certain measures, this does not mean that these measures cannot be combined to produce good results.

[0098] Although the invention has been described in conjunction with specific features and embodiments, it is obvious that various modifications and combinations can be made therein without departing from the spirit and scope of the invention. Accordingly, this specification and drawings are merely exemplary descriptions of the invention as defined by the appended claims, and are considered to cover any and all modifications, variations, combinations, or equivalents within the scope of the invention. Clearly, those skilled in the art can make various alterations and modifications to the invention without departing from its spirit and scope. Thus, if such modifications and modifications of the invention fall within the scope of the claims and their equivalents, the invention is also intended to include such modifications and modifications.

Claims

1. A physical information neural network-based axle coupling system analysis method, characterized by, The application relates to a method for establishing a vehicle-bridge coupling system. The bridge model adopts an Euler-Bernoulli beam, and the motion equation of the vehicle-bridge coupling system is as follows: The physical information neural network is constructed by taking the motion equation of the vehicle-bridge coupling system as a constraint condition and taking the initial condition and the boundary condition of the vehicle-bridge coupling system as a loss function. ； ； wherein: m * a linear element mass of the bridge model, a mass matrix of the vehicle model, a second order derivative of the bridge model with respect to time, u'''' a fourth order derivative of the bridge model with respect to the horizontal position of the bridge, a vertical acceleration vector of the vehicle model, u a vertical displacement of the bridge model, a vertical displacement vector of the vehicle system, E a modulus of elasticity of the bridge model, I a moment of inertia of the bridge model, a stiffness matrix of the vehicle model, a vertical displacement vector of the bridge at the position of the vehicle, a reaction force of the vehicle model on the bridge; The input layer is used for inputting a plurality of random points in the vehicle-bridge coupling system, the random points are used for representing real solutions of the vehicle-bridge coupling system at different time points, the physical characteristic layer is used for adding inherent frequencies and time-varying motion characteristics of the vehicle-bridge coupling system, the full connection layer is used for receiving the random points for iterative training in the neural network, the tensor processing layer is used for tensor clipping of the vehicle model, and only the time dimension output is reserved, and the output layer is used for outputting time-varying vibration responses of the vehicle model and the bridge model. The plurality of random points in the vehicle-bridge coupling system are selected to perform iterative calculation on the physical information neural network, so that a trained neural network is obtained. The trained neural network is used to calculate the motion equation of the vehicle-bridge coupling system, so that the vibration response of the vehicle-bridge coupling system is obtained. The loss function includes a partial differential equation loss, an initial constraint condition loss and a boundary constraint condition loss, and the loss function adopts a mean square error loss function.

2. The method of claim 1, wherein, The partial differential equation loss is used for measuring the difference between the neural network prediction result and the partial differential equation describing the vehicle-bridge coupling system. The initial constraint condition loss is used for ensuring that the state of the vehicle-bridge coupling system at the initial time meets the given initial condition. The boundary constraint condition loss is used for ensuring that the state of the vehicle-bridge coupling system at the boundary meets the given boundary condition. The loss function equation is as follows:

3. The method of claim 2, wherein, Each mean square error loss function is as follows: ； wherein: , , , , respectively represent the weight of the vehicle equation loss function, the weight of the bridge equation loss function, the weight of the vehicle initial condition loss function, the weight of the bridge initial condition loss function, the weight of the bridge boundary condition loss function; represents the vehicle equation loss function, represents the bridge equation loss function, represents the vehicle initial condition loss function, represents the bridge initial condition loss function, represents the bridge boundary condition loss function; The physical information neural network further comprises an optimizer, and the optimizer is used for adjusting the parameters of the network to minimize the loss function. ； ； ； ； ； ； In the formula, represents the number of internal points, represents the number of initial points, represents the number of boundary points, represents the input value of the vehicle model at the initial time in the neural network, represents the true value of the vehicle model at the initial time, represents the input value of the bridge model at the initial time in the neural network, represents the true value of the bridge model at the initial time, represents the input value of the bridge model at the boundary in the neural network, , represents the true value of the bridge model at the boundary.

4. The method of claim 1, wherein, The optimizer comprises an Adam optimizer and an L-BFGS optimizer.

5. The method of claim 4, wherein, ​

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