A vehicle axle coupling vibration response prediction method based on PIKAN

Abstract

The application discloses a PIKAN-based vehicle-bridge coupling vibration response prediction method, which comprises the following steps: discretizing the time-space domain of a bridge and a vehicle according to the structure of the bridge and the driving speed of the vehicle on the bridge surface; constructing a vehicle-bridge coupling control equation according to the discretization result; selecting appropriate internal and external single-variable functions, and respectively constructing a KAN1 module for bridge vertical displacement response prediction and a KAN2 module for vehicle vertical displacement response prediction; and constructing a residual loss function based on the vehicle-bridge coupling control equation, a condition constraint and real observation data to train and optimize the KAN1 module and the KAN2 module. The method has the beneficial effects that the method has a fast calculation rate, and has good calculation accuracy on the premise of taking into account the calculation efficiency, and can be used as a substitute model for vehicle-bridge coupling finite element numerical simulation. The bridge vibration response of a heavy-load vehicle that has not yet arrived on the bridge can also be calculated in real time, and the safety risk of the bridge structure can be quickly evaluated.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of axle coupling, in particular to a PIKAN-based axle coupling vibration response prediction method. BACKGROUND

[0002] The axle coupling vibration refers to a complex dynamic response phenomenon caused by the dynamic interaction between a vehicle and a bridge structure when the vehicle passes through the bridge. With the development of modern transportation infrastructure towards large span, lightweight and high speed, the problem of axle coupling vibration has become increasingly important. On the one hand, the high-speed vehicle can produce a significant dynamic impact effect on the bridge structure, directly leading to the aggravation of fatigue damage of the bridge components and the shortening of the service life. On the other hand, the vibration of the bridge can affect the safety and stability of the running vehicle, especially under the action of curve bridge or external strong excitation, which can cause vehicle derailment and other major accidents. Therefore, accurately predicting the coupling response is of great significance for bridge safety evaluation, fatigue life prediction, structure health monitoring and vehicle operation safety.

[0003] The current mainstream axle coupling vibration analysis method is mainly based on the numerical simulation technology driven by physics. Such method solves the response by using numerical integral method through establishing the coupling dynamics equation of the vehicle-bridge system, and the calculation complexity thereof is in exponential relationship with the system degrees of freedom, and the iterative calculation is required for solving the dynamics equation, which is not real-time.

[0004] In recent years, the machine learning method provides a new idea for the analysis of axle coupling vibration, but the pure data-driven model has the problems of "black box" and "lack of physical consistency". Although the efficiency can be improved, the physical constraints are lacked, the extrapolation ability is weak, the generalization is poor, a large amount of simulation data needs to be generated, and the measured samples are scarce.

[0005] The physical information neural network (PINN) attempts to embed the physical mechanism into the data-driven model to form a hybrid framework driven by physical information, such as coupling the residual error of the control equation in the loss function to realize unsupervised learning. However, the existing PINN still has bottlenecks, such as the need for the network model to have multi-scale feature decoupling capability for high-frequency vibration of the vehicle and low-frequency vibration of the bridge, the traditional MLP architecture is difficult to meet, the boundary conditions and residual function are sensitive, the convergence speed is slow, and the training is difficult in a complex coupling system. SUMMARY

[0006] One of the purposes of the present application is to provide a PIKAN-based axle coupling vibration response prediction method which can solve at least one of the defects in the background.

[0007] To achieve the above at least one purpose, the technical solution adopted by the present application is as follows: a PIKAN-based axle coupling vibration response prediction method, comprising the following steps:

[0008] S100: Discretize the space-time domain of the bridge and vehicle according to the structure of the bridge and the driving speed of the vehicle on the bridge surface; and construct a bridge-vehicle coupling control equation according to the discretization result;

[0009] S200: Select appropriate internal and external single-variable functions to construct a KAN1 module for predicting the vertical displacement response of the bridge and a KAN2 module for predicting the vertical displacement response of the vehicle, respectively;

[0010] S300: Train and optimize the KAN1 module and the KAN2 module based on a residual loss function constructed based on the bridge-vehicle coupling control equation, the conditional constraints and the real observation data.

[0011] Preferably, in step S100, the expression of the bridge-vehicle coupling control equation is as follows:

[0012]

[0013]

[0014]

[0015] wherein F b (t) and F v (t) represent the contact coupling forces of the bridge and the vehicle at the discretized nodes, respectively, u v represents the vertical displacement matrix of the vehicle at the discretized nodes, u b represents the vertical displacement matrix of the bridge at the discretized nodes, M b and M v represent the mass matrices of the bridge and the vehicle, respectively, C b and C v represent the damping matrices of the bridge and the vehicle, respectively, K b and K v represent the stiffness matrices of the bridge and the vehicle, respectively, k t represents the stiffness of the tire of the vehicle, c t represents the damping coefficient of the tire of the vehicle, x c represents the bridge-vehicle coupling position corresponding to the discretized time t, and r represents the uneven sample data of the bridge surface.

[0016] Preferably, the discretization of the time domain in the space-time domain includes the following processes: calculating the total analysis time t according to the span of the bridge and the driving speed of the vehicle; setting the bridge-vehicle coupling vibration analysis time step Δt based on the time history analysis of structural mechanics; and discretizing the total analysis time t into a one-dimensional vector T according to the obtained time step Δt, T = [Δt, 2Δt, …, t].

[0017] ​​​Preferably, the discretization of the spatial domain in the space-time domain includes the following process: the structure of the bridge is divided into a unit grid to obtain a two-dimensional matrix S which discretizes the geometric size of the bridge structure B B =[(0, 0), (ΔL, 0), (2ΔL, 0), …, (L, 0)]; wherein, L represents the span of the bridge, and ΔL represents the size of the unit grid; the number of nodes for the equivalent vehicle structure is set as A, and the spatial position of the vehicle at all times is discretized into a two-dimensional matrix S according to the coordinates (x 0,i , y 0,i ) of each node at the initial position, i∈{1, 2, …, A} v v =[(x t,1 , y t,1 ), (x t,2 , y t,2 ), …, (x t,A , y t,A )].

[0018] Preferably, in step S200, the internal function of the KAN1 module selects a combination form of Fourier basis functions and B-spline functions, and the external function selects a form of linear weighted summation; the internal function of the KAN2 module selects a combination form of Chebyshev polynomial functions and B-spline functions, and the external function selects a form of linear weighted summation; the inputs of the KAN1 module and the KAN2 module are the corresponding discretized space-time domains, and the outputs are corresponding vertical displacement matrices; the outputs of the KAN1 module and the KAN2 module are spliced to form a coupled prediction output.

[0019] Preferably, the combination form of the internal functions in the KAN1 module and the KAN2 module adopts a dynamic weighted combination form based on machine learning, and the specific combination process is as follows: the internal functions of the KAN1 module and the KAN2 module are constructed as , ; wherein, α and β respectively constitute the weights of two basis functions X and Y of the internal function; the weights α and β are valued according to the degree of emphasis on the local and global, the value of the weight α gradually increases with the increase of the global emphasis degree, and the value of the weight β gradually increases with the increase of the local emphasis degree, α+β=1; a feature extraction network g is constructed to extract the features affecting the local and global emphasis degrees of the KAN1 module and the KAN2 module, and the weights α and β are adjusted according to the extracted results.

[0020] ​​Preferably, according to the different local emphasis and global emphasis, the process is divided into multiple stages, the values of the weights α and β corresponding to adjacent stages are different, and a transition stage is arranged between adjacent stages, and when the change of the local emphasis and the global emphasis is located in the transition stage, the values of the weights α and β are the values of the last stage.

[0021] Preferably, the step S300 comprises the following process: the physical information residual loss function is obtained by weighting each residual loss function through setting a residual loss weight; when the training of the KAN1 module and the KAN2 module is performed, the learning rate and the loss function weight are dynamically adjusted according to the loss curve descending condition of the physical information residual loss function until the loss curve converges.

[0022] Preferably, in the step S300, the differential equation residual loss function L DEB and L DEV of the bridge coupling control equation, and the residual loss function L DATA of the real observation data are expressed as follows:

[0023] ;

[0024] ;

[0025] ;

[0026] wherein N represents the number of discrete nodes, represents the actual measured value or the finite element simulation value of the vertical displacement of the bridge.

[0027] Preferably, in the step S300, the condition constraints comprise boundary condition constraints, initial condition constraints and coupling condition constraints, and the expressions of the corresponding boundary condition constraint residual loss function L BC , the initial condition constraint residual loss function L IC and the coupling condition constraint residual loss function L CC are as follows:

[0028] ;

[0029] ;

[0030] ;

[0031] wherein L represents the span of the bridge, and x represents the displacement of the vehicle.

[0032] Compared with the prior art, the application has the beneficial effects that:

[0033] The method has a faster calculation rate, and has better calculation accuracy on the premise of considering calculation efficiency, and can be used as a substitute model for vehicle-bridge coupling finite element numerical simulation. The bridge vibration response of the vehicle without the bridge under heavy load can also be calculated in real time, the safety risk of the bridge structure can be quickly evaluated, and intervention can be made in time. BRIEF DESCRIPTION OF DRAWINGS

[0034] Figure 1 The figure is a schematic diagram of the overall workflow of the present application. DETAILED DESCRIPTION

[0035] In the following, the present application will be further described in conjunction with specific embodiments. It should be noted that in the description of the present application, the description of the terms "one embodiment", "some embodiments", "an example", "a specific example", or "some examples" means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present application. In the present description, the illustrative description of the above terms should not be understood as necessarily referring to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art can combine and combine different embodiments or examples described in the present description.

[0036] In the description of the present application, it should be noted that for orientation words such as the terms "center", "transverse", "longitudinal", "length", "width", "thickness", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", "clockwise", "counterclockwise", etc. The orientation and positional relationship shown in the drawing is based on the orientation or positional relationship shown in the drawing, and is only for the convenience of describing the present application and simplifying the description, and does not indicate or imply that the device or element referred to must have a particular orientation, be constructed and operated in a particular orientation, and cannot be understood as limiting the specific protection scope of the present application.

[0037] It should be noted that the terms "first", "second", etc. in the specification and claims of the present application are used to distinguish similar objects, and do not necessarily describe a specific order or sequence.

[0038] In the present application, unless specifically defined and limited otherwise, the terms "mounting", "connected", "connection", "fixed", and the like, should be construed broadly and, for example, can be a connection, a detachable connection, or integral; can be a mechanical connection, or an electrical connection; can be a direct connection, or an indirect connection via an intermediate medium; can be a communication between two elements, or an interaction between two elements. The specific meaning of the above terms in the present application can be understood according to the specific circumstances by those skilled in the art.

[0039] In the present application, unless specifically defined and limited otherwise, the first feature "on" or "under" the second feature can include that the first and second features are in direct contact, or that the first and second features are not in direct contact but are in contact through another feature between them. Moreover, the first feature "on", "above" and "over" the second feature includes that the first feature is directly above and obliquely above the second feature, or only indicates that the horizontal height of the first feature is higher than that of the second feature. The first feature "under", "below" and "under" the second feature includes that the first feature is directly below and obliquely below the second feature, or only indicates that the horizontal height of the first feature is less than that of the second feature.

[0040] In the present application, the terms "comprising" and "having" and any variations thereof in the specification and claims are intended to cover non-exclusive inclusion, for example, a process, method, system, product or device comprising a series of steps or units does not have to be limited to only those steps or units clearly listed, but can include other steps or units not clearly listed or inherent to these processes, methods, products or devices.

[0041] One preferred embodiment of the present application, as shown in Figure 1 A PIKAN-based vehicle-bridge coupled vibration response prediction method, comprising the following steps:

[0042] S100: Discretize the space-time domain of the bridge and the vehicle according to the structure of the bridge and the driving speed of the vehicle on the bridge surface; and construct a vehicle-bridge coupled control equation according to the discretization result.

[0043] S200: Select appropriate internal and external single-variable functions to construct a KAN1 module for predicting the vertical displacement response of the bridge and a KAN2 module for predicting the vertical displacement response of the vehicle, respectively.

[0044] S300: Train and optimize the KAN1 module and the KAN2 module based on the residual loss function of the vehicle-bridge coupled control equation, the condition constraint and the real observation data.

[0045] It can be understood that the PIKAN is a machine learning model embedding physical laws in the KAN architecture, which approximates the solution or unknown parameters of a nonlinear PDE-ODE coupled system through a learnable combination of univariate functions, while ensuring that the obtained solution satisfies the given control equations, boundary and initial conditions, etc. through a physical residual loss.

[0046] In a popular way, the input parameters of the bridge coupling system are usually the span L of the bridge, the time t, the displacement x and the speed v of the vehicle, etc. When a full connection or CNN is used to construct the vehicle-bridge coupling model, the dimension will increase exponentially. In the technical solution of the present application, the vehicle and the bridge are modeled by KAN modules, and the parameters of the KAN module only increase linearly with the number of univariate functions, which makes the model size constructed by the KAN network significantly reduced, thereby effectively improving the calculation rate, while ensuring the calculation efficiency, and having good calculation accuracy, and thus can be used as a substitute model for vehicle-bridge coupling finite element numerical simulation. It can also calculate the bridge vibration response of the heavy vehicle not on the bridge in real time, quickly evaluate the bridge structure safety risk, and timely intervene.

[0047] In the embodiment, the time-space domain is divided into time domain and space domain. Since the bridge is fixed and different, the vehicle travels along the bridge deck, so the vehicle and the bridge are the total time t of the vehicle traveling on the bridge, that is, the total analysis time length when the time domain is discretized. Since the bridge has a span L, when the space domain is discretized, the discrete nodes are different position points of the bridge along the span direction, and the space domain discrete nodes of the vehicle are the displacement points of the vehicle traveling on the bridge deck. For the convenience of understanding, the following will take a two-dimensional simply supported beam and a 4-DOF vehicle model as an example to describe the specific discretization process of the time domain and the space domain in detail.

[0048] Specifically, the discretization of the time domain in the time-space domain includes the following processes: calculating the total analysis time t according to the bridge span L and the vehicle travel speed v, t=L / v; setting the vehicle-bridge coupling vibration analysis time step Δt based on the time history analysis of structural mechanics; and discretizing the total analysis time t into a one-dimensional vector T according to the obtained time step Δt, T=[Δt, 2Δt, …, t].

[0049] It should be known that the specific value of the analysis time step Δt is related to the driving speed of the vehicle, the number of discrete nodes and the span of the bridge, that is, the faster the driving speed of the vehicle, the shorter the time for the vehicle to pass through the bridge, and in order to have sufficient coupling data, the value of the analysis time step Δt is smaller; otherwise, the value of the analysis time step Δt can be appropriately increased. The number of discrete nodes of the bridge is related to the span of the bridge; generally, for a bridge with a span less than 30 m, the corresponding number of discrete nodes is about 10; for a bridge with a span greater than 50 m, the number of discrete nodes can be increased to more than 16. For example, the value of the analysis time step Δt can be 0.01 s and 0.05 s, etc., which can be selected by the person skilled in the art as needed.

[0050] Specifically, the discretization of the spatial domain in the space-time domain includes the following process: since the spatial position of the bridge is fixed, according to the span L of the bridge, the scope can be defined as [0, L]; then the structure of the bridge can be divided according to the set unit grid size ΔL, to obtain a two-dimensional matrix S B , S B =[(0, 0), (ΔL, 0), (2ΔL, 0), …, (L, 0)]. The number of nodes for the equivalent vehicle structure is set to A, and according to the coordinates (x 0,i , y 0,i ) of each node at the initial position, i∈{1, 2, …, A}, the spatial position of the vehicle at all times is discretized into a two-dimensional matrix S v , S v =[(x t,1 , y t,1 ), (x t,2 , y t,2 ), …, (x t,A , y t,A )].

[0051] It can be understood that the specific value of the unit grid size ΔL for the grid division of the bridge structure can be selected by the person skilled in the art as needed; the unit grid size ΔL can be equal to the distance between adjacent discrete nodes, or can be less than the distance between adjacent discrete nodes. The value of the number of nodes A for the equivalent vehicle structure is generally 4 to 12; that is, for a small car, it can be equivalent by 4 nodes; and for a large vehicle such as a truck, the number of equivalent nodes needs to be increased. In the present embodiment, the number of nodes A for the equivalent vehicle structure can be valued as 7 in combination with experience, which is between general and complex, that is, it covers common working conditions and does not excessively increase the calculation amount. The two-dimensional coordinates of the vehicle at the initial position can be represented as: [(x 0,1 , y 0,1), (x 0,2 , y 0,2 ), …, (x 0,7 , y 0,7 )], the corresponding discrete matrix S v = [(x t,1 , y t,1 ), (x t,2 , y t,2 ), …, (x t,7 , y t,7 )].

[0052] In this embodiment, according to the spatial domain discretization of the axle coupling system, the mass matrix M b and M v , the stiffness matrix K b and K v , and the damping matrix C b and C v of the bridge and the vehicle can be established. Then in step S100, the expression of the axle coupling control equation is as follows:

[0053] .

[0054] .

[0055] wherein F b (t) and F v (t) represent the contact coupling forces of the bridge and the vehicle at the discrete nodes respectively, u v represents the vertical displacement matrix of the vehicle at the discrete nodes, and u b represents the vertical displacement matrix of the bridge at the discrete nodes. It should be noted that for the damping matrix C b of the bridge, Rayleigh damping can be used for construction, i.e. , and all represent weight coefficients.

[0056] It can be understood that in the above control equation, the contact coupling forces F b (t) and F v (t) of the bridge and the vehicle at the discrete nodes can be calculated through the contact between the wheel nodes of the vehicle and the corresponding bridge nodes, and the specific calculation formula is as follows:

[0057] ;

[0058] wherein k t represents the stiffness of the vehicle tire, c t represents the damping coefficient of the vehicle tire, x cr represents uneven sample data of the bridge surface.

[0059] In the embodiment, in the step S200, the KAN1 module and the KAN2 module can form a network model for vehicle-bridge coupling vibration response prediction by splicing; wherein the bridge and the vehicle are respectively two sub-coefficients of the network model. For the KAN1 module, the input is the space-time domain discrete representation [T, S B ] of the bridge subsystem, [T, S B ]=[(Δt, 0, 0), (2Δt, ΔL, 0), …, (t, L, 0)]; and the output is the vertical displacement matrix u b of the bridge subsystem at the discrete nodes. For the KAN2 module, the input is the space-time domain discrete representation [T, S v ] of the vehicle subsystem, [T, S v ]=[(Δt, x t,1 , y t,1 ), (2Δt, x t,2 , y t,2 ), …, (t, x t,7 , y t,7 )]; and the output is the vertical displacement matrix u v of the vehicle subsystem at the discrete nodes. The outputs of the KAN1 module and the KAN2 module are spliced to form the coupling prediction output.

[0060] In the embodiment, the specific type selection of the internal function and the external function of the KAN1 module and the KAN2 module has multiple options. Generally, in order to realize the prediction of the local and the global, the internal function can select the combination of two unary functions respectively focusing on local solution and global solution. Specifically, the internal function of the KAN1 module selects the combination form of the Fourier basis function and the B-spline function, and the external function selects the form of linear weighted summation, that is, the external function is used for linear weighted summation of all channel results of the internal function. The internal function of the KAN2 module selects the combination form of the Chebyshev polynomial function and the B-spline function, and the external function selects the form of linear weighted summation.

[0061] It can be understood that when the vehicle-bridge coupling analysis is performed, two types of key factors, i.e., global scale and local scale, need to be considered. For the global factors, mainly include the overall modal of the beam bridge, the number and distribution of vehicles running on the bridge, and other environmental factors, etc. For the local factors, mainly include the contact force impact of the wheel and the bridge surface, the local structural performance of the bridge, and the local damage of the wheel or the bridge surface, etc. Therefore, when the internal functions are combined, the weight proportions of the two monomial functions respectively focusing on local solution and global solution are suitable for adaptive dynamic adjustment according to the degree of change from local to global. For the convenience of understanding, the internal function combination form in the KAN1 module and the KAN2 module will be described in detail below.

[0062] Specifically, there are various specific ways for the dynamic weighted combination of the internal functions in the KAN1 module and the KAN2 module, and in the embodiment, the dynamic weighted combination form based on machine learning is preferred, and the specific combination process is as follows: the internal function combination form in the KAN1 module and the KAN2 module adopts the dynamic weighted combination form based on machine learning, and the specific combination process is as follows: constructing the internal functions of the KAN1 module and the KAN2 module , ; wherein, α and β respectively constitute the weights of two base functions X and Y of the internal function; the weights α and β are valued according to the degree of emphasis on local and global, the value of the weight α gradually increases with the increase of the degree of emphasis on global, and the value of the weight β gradually increases with the increase of the degree of emphasis on local, α+β=1; constructing a feature extraction network g to extract the features affecting the degree of emphasis on local and global of the KAN1 module and the KAN2 module, and adjusting the weights α and β according to the extracted results.

[0063] It should be known that in the KAN1 module, the Fourier base function corresponds to the base functions X and Y of the B-spline function, and in the KAN2 module, the Chebyshev polynomial function corresponds to the base functions X and Y of the B-spline function. For the feature extraction network g, a light gated network can be used, and the specific construction process is known to those skilled in the art, so it will not be described in detail here; the feature extraction network g can judge the emphasis on local factors and global factors corresponding to the different discrete node positions of the vehicle on the bridge surface according to the vehicle speed and structural parameters on the bridge surface, and the given environmental parameters and bridge structural parameters. If the global factors need to be emphasized, the value direction of the weight is in the direction from local to global; if the local factors need to be emphasized, the value direction of the weight is in the opposite direction from local to global.

[0064] It can be understood that, in order to simplify the calculation process of the KAN1 module and the KAN2 module, different stages can be divided according to the difference between the local emphasis and the global emphasis, so that different values can be assigned to the weights a and β corresponding to each stage; then the infinite dynamic adjustment of the weights a and β can be realized as dynamic adjustment of the corresponding stage. For example, 9 stages can be divided according to the difference between the local emphasis and the global emphasis, and the values [a, β] of the weights a and β of each stage are respectively: [0.1, 0.9], [0.2, 0.8], …, [0.8, 0.2], [0.9, 0.1].

[0065] It should be noted that, considering that the traffic flow of the bridge will be basically stable after a period of time after the completion of the construction, the values of the weights a and β can be relatively concentrated, such as mainly concentrated in [0.4, 0.6], [0.5, 0.5] and [0.6, 0.4]. This can cause the values of the weights a and β to be dynamically adjusted within the above three value ranges more frequently, so a hysteresis mechanism can be set for the values of the weights a and β to reduce the number of dynamic adjustments of the weights a and β.

[0066] Specifically, a transition stage is set between adjacent stages, and when the change of the local emphasis and the global emphasis is within the transition stage, the values of the weights a and β adopt the values of the last stage.

[0067] Taking the above values of the weights a and β concentrated in [0.4, 0.6], [0.5, 0.5] and [0.6, 0.4] as an example, the values of the weights a and β can be adjusted from [0.4, 0.6] to [0.5, 0.5], and the proportion threshold of the local emphasis and the global emphasis is a, and the values of the weights a and β can be adjusted from [0.5, 0.5] to [0.6, 0.4], and the proportion threshold of the local emphasis and the global emphasis is b. Then a hysteresis value Δd can be set to obtain the transition stages [a-Δd, a+Δd] and [b-Δd, b+Δd]. When the proportion of the local emphasis and the global emphasis is within the transition stage [a-Δd, a+Δd], if the values of the weights a and β at the previous moment are [0.4, 0.6], the values of the weights a and β at this moment are still [0.4, 0.6], and if the values of the weights a and β at the previous moment are [0.5, 0.5], the values of the weights a and β at this moment are still [0.5, 0.5]. Similarly, when the proportion of the local emphasis and the global emphasis is within the transition stage [b-Δd, b+Δd], if the values of the weights a and β at the previous moment are [0.5, 0.5], the values of the weights a and β at this moment are still [0.5, 0.5], and if the values of the weights a and β at the previous moment are [0.6, 0.4], the values of the weights a and β at this moment are still [0.6, 0.4].

[0068] In this embodiment, step S300 includes the following process: the physical information residual loss function is obtained by weighting each residual loss function by setting a residual loss weight. When training the KAN1 module and the KAN2 module, the learning rate and the loss function weight are dynamically adjusted according to the loss curve descending condition of the physical information residual loss function until the loss curve converges.

[0069] Specifically, the conditional constraints include boundary condition constraints, initial condition constraints and coupling condition constraints; then in step S300, the residual loss function L of the entire network model is loss is the differential equation residual loss function based on the vehicle-bridge coupling control equation. DEB is the initial condition constraint residual loss function. DEV is the coupling condition constraint residual loss function. DATA is the residual loss function of the real observation data. BC is the boundary condition constraint residual loss function. IC is the initial condition constraint residual loss function. CC and the coupling condition constraint residual loss function L is the weighted sum of the above, and the specific expression is as follows:

[0070] .

[0071] .

[0072] .

[0073] .

[0074] .

[0075] .

[0076] .

[0077] wherein, represents the weight of the differential equation residual loss function L DEB . represents the weight of the differential equation residual loss function L DEV . represents the weight of the boundary condition constraint residual loss function L BC . represents the weight of the initial condition constraint residual loss function L IC . represents the weight of the coupling condition constraint residual loss function L CC . represents the weight of the residual loss function L DATA of the real observation data, and N represents the number of discrete nodes. The actual measured value or finite element simulation value representing the vertical displacement of the bridge, and x represents the vehicle displacement.

[0078] It can be understood that after the construction of the PIKAN-based vehicle-bridge coupled vibration response prediction network model is completed, it can be deployed in the front-end edge computing terminal. After the vehicle information is detected on the bridge, the vehicle space-time domain information is discretized, and is input into the network model together with the pre-set bridge space-time domain discrete information. The predicted value of the vertical dynamic displacement of the bridge is calculated, and is compared with the real-time monitored vertical displacement value of the bridge at the measuring point on the bridge. The error is analyzed, and the accuracy of the network model is judged according to the analysis result.

[0079] The above describes the basic principles, main features and advantages of the present application. Those skilled in the art should understand that the present application is not limited to the above-mentioned embodiments, and the above-mentioned embodiments and descriptions in the specification are only the principles of the present application. Without departing from the spirit and scope of the present application, various changes and improvements can be made to the present application, and these changes and improvements all fall within the scope of the claimed present application. The scope of protection claimed by the present application is defined by the appended claims and their equivalents.

Claims

1. A PIKAN-based method for predicting the response of a coupled vibration of an axle, characterized by, The method comprises the following steps: S100: Discretize the time-space domain of the bridge and the vehicle according to the structure of the bridge and the driving speed of the vehicle on the bridge surface; and construct a bridge-vehicle coupling control equation according to the discretization result; S200: Select appropriate internal and external single-variable functions, and respectively construct a KAN1 module for predicting the vertical displacement response of the bridge and a KAN2 module for predicting the vertical displacement response of the vehicle; S300: Train and optimize the KAN1 module and the KAN2 module based on the residual loss function of the bridge-vehicle coupling control equation, the condition constraint and the real observation data; The discretization of the time domain in the time-space domain comprises the following process: Calculate the total analysis time t according to the span of the bridge and the driving speed of the vehicle; Based on the time-history analysis of structural mechanics, the time step of the coupled vibration analysis of the bridge is set t; According to the time steps obtained t, the total analysis time t is discretized into a one-dimensional vector T, T=[ t,2 t, …, t]; The discretization of the space domain in the time-space domain comprises the following process: The bridge structure is divided into element meshes to obtain a two-dimensional matrix S that discretizes the geometric dimensions of the bridge structure. B S B =[(0, 0), ( L,0),(2 L, 0), ..., (L, 0)]; The number of nodes for the equivalent vehicle structure is set as A, the spatial position of the vehicle at all times is discretized into a two-dimensional matrix S according to the coordinates (x 0,i , y 0,i ) of each node at the initial position, i∈{1, 2, …, A}, S v , S v =[( x t,1 , y t,1 ), (x t,2 , y t,2 ), …, ( x t,A , y t,A )] wherein L represents the span of the bridge, L represents the size of the unit cell.

2. The PIKAN-based coupled vibration response prediction method for vehicle axles as claimed in claim 1, wherein, In step S100, the expression of the bridge-vehicle coupling control equation is as follows: ； ； ； where F b (t) and F v (t) represent the contact force between the bridge and the vehicle at the discrete nodes, u v represents the vertical displacement matrix of the vehicle at the discrete nodes, u b represents the vertical displacement matrix of the bridge at the discrete nodes, M b and M v represent the mass matrix of the bridge and the vehicle, respectively, C b and C v represent the damping matrix of the bridge and the vehicle, respectively, K b and K v represent the stiffness matrix of the bridge and the vehicle, respectively, k t represents the stiffness of the vehicle tire, c t represents the damping coefficient of the vehicle tire, x c represents the corresponding vehicle-bridge coupling position at the discrete time t, and r represents the unevenness sample data of the bridge surface.

3. The PIKAN-based coupled vibration response prediction method for vehicle axles as claimed in claim 2, wherein, In step S200, the internal function of the KAN1 module is in the form of a combination of Fourier basis functions and B-spline functions, and the external function is in the form of linear weighted summation; The internal function of the KAN2 module is in the form of a combination of Chebyshev polynomial functions and B-spline functions, and the external function is in the form of linear weighted summation; The inputs of the KAN1 module and the KAN2 module are the corresponding discretized time-space domains, and the outputs are the corresponding vertical displacement matrices; The outputs of the KAN1 module and the KAN2 module are spliced to form a coupling prediction output.

4. The PIKAN-based coupled vibration response prediction method for vehicle axles as claimed in claim 3, wherein, The internal function combination form in the KAN1 module and the KAN2 module adopts a dynamic weighted combination form based on machine learning, and the specific combination process is as follows: Internal functions of constructing KAN1 module and KAN2 module , = αX + βY; wherein, α and β respectively constitute weights of two base functions X and Y of the internal function; The weights α and β are valued according to the local and global emphasis degrees, the value of the weight α gradually increases with the increase of the global emphasis degree, and the value of the weight β gradually increases with the increase of the local emphasis degree, and α+β=1; A feature extraction network g is constructed to extract the features of the local and global emphasis degrees affecting the KAN1 module and the KAN2 module, and the weights α and β are adjusted according to the extraction results.

5. The PIKAN-based coupled vibration response prediction method for vehicle axles as claimed in claim 4, wherein, According to the difference between the local emphasis and the global emphasis, multiple stages are divided, and the values of the weights α and β corresponding to adjacent stages are different; A transition stage is set between adjacent stages, and when the change of the local emphasis and the global emphasis is within the transition stage, the values of the weights α and β adopt the values of the previous stage.

6. The PIKAN-based coupled vibration response prediction method for vehicle axles according to any one of claims 2-5, wherein, Step S300 comprises the following process: The physical information residual loss function is obtained by weighting each residual loss function by setting a residual loss weight; When training the KAN1 module and the KAN2 module, the learning rate and the loss function weight are dynamically adjusted according to the loss curve descent of the physical information residual loss function until the loss curve converges.

7. The PIKAN-based coupled vibration response prediction method for vehicle axles as claimed in claim 6, wherein, In step S300, the differential equation residual loss function L of the vehicle axle coupling control equation DEB and L DEV , and the residual loss function L of the real observation data is expressed as follows: DATA ​ ； ； ； wherein N represents the number of discrete nodes, denotes the actual measured or finite element simulated value of the vertical displacement matrix of the bridge, and x denotes the vehicle displacement.

8. The PIKAN-based coupled vibration response prediction method for vehicle axles as claimed in claim 6, wherein, In step S300, the conditional constraints include boundary condition constraints, initial condition constraints and coupling condition constraints, and the expressions of the corresponding boundary condition constraint residual loss function L BC , the initial condition constraint residual loss function L IC and the coupling condition constraint residual loss function L CC are as follows: ； ； ； Wherein, L represents the span of the bridge, x represents the displacement of the vehicle, and N represents the number of discrete nodes.

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