Axle coupling vibration response prediction method based on PIKAN

Abstract

The invention discloses an axle coupling vibration response prediction method based on PIKAN, and the method comprises the following steps: carrying out the discretization of the time-space domain of a bridge and a vehicle according to the structure of the bridge and the running speed of the vehicle on a bridge floor; constructing an axle coupling control equation according to a discrete result; selecting proper internal and external univariate functions, and respectively constructing a KAN1 module used for bridge vertical displacement response prediction and a KAN2 module used for vehicle vertical displacement response prediction; and constructing a residual loss function based on an axle coupling control equation, conditional constraints and real observation data to train and optimize the KAN1 module and the KAN2 module. The method has the beneficial effects that the method provided by the invention has a relatively high calculation rate, has relatively good calculation precision on the premise of considering the calculation efficiency, and can be used as a substitute model for axle coupling finite element numerical simulation. And the bridge vibration response of a heavy-load vehicle not getting on the bridge can be calculated in real time, and the safety risk of the bridge structure is rapidly evaluated.

Description

Technical Field

[0001] The present application relates to the field of vehicle-bridge coupling technology, and in particular to a vehicle-bridge coupling vibration response prediction method based on PIKAN. Background Art

[0002] Vehicle-bridge coupled vibration refers to the complex dynamic response phenomenon caused by the dynamic interaction between the vehicle and the bridge structure when a vehicle passes over a bridge. As modern transportation infrastructure develops toward larger spans, lighter weight, and higher speeds, the issue of vehicle-bridge coupled vibration is becoming increasingly important. On the one hand, high-speed vehicles can exert significant dynamic impact on bridge structures, directly leading to increased fatigue damage and shortened service life for bridge components. On the other hand, bridge vibration can affect the safety and stability of operating vehicles, particularly on curved beam bridges or under strong external excitation, potentially causing major accidents such as vehicle derailment. Therefore, accurately predicting this coupled response is crucial for bridge safety assessment, fatigue life prediction, structural health monitoring, and vehicle operation safety.

[0003] The current mainstream vehicle-bridge coupled vibration analysis methods are mainly based on physics-driven numerical simulation technology. Such methods establish the coupled dynamic equations of the vehicle-bridge system and use numerical integration methods to solve the response. Their computational complexity is exponentially related to the system's degrees of freedom, and solving the dynamic equations requires iterative calculations, which lacks real-time performance.

[0004] In recent years, machine learning methods have provided new insights into vehicle-bridge coupled vibration analysis. However, purely data-driven models suffer from "black box" nature and a lack of physical consistency. While these methods can improve efficiency, they lack physical constraints, exhibit weak extrapolation capabilities, and exhibit poor generalization. Furthermore, they require the generation of massive amounts of simulation data, while actual measured samples are scarce.

[0005] Physically-Informed Neural Networks (PINNs) attempt to embed physical mechanisms into data-driven models, forming a hybrid framework driven by physical information. For example, they couple the residuals of the governing equations to the loss function to enable unsupervised learning. However, existing PINNs still face bottlenecks. For example, the network model must be able to decouple multi-scale features from the high-frequency vibrations of vehicles and low-frequency vibrations of bridges, which is difficult to achieve with traditional MLP architectures. Furthermore, PINNs are sensitive to boundary conditions and residual functions, resulting in slow convergence and difficulty in training complex coupled systems. Summary of the Invention

[0006] One of the objectives of the present application is to provide a PIKAN-based vehicle-bridge coupled vibration response prediction method that can address at least one of the drawbacks of the above-mentioned background technology.

[0007] To achieve at least one of the above objectives, the present application adopts a technical solution: a PIKAN-based vehicle-bridge coupled vibration response prediction method, comprising the following steps: S100: Discretize the time and space domains of the bridge and vehicle based on the bridge structure and the speed of the vehicle on the bridge deck; construct the vehicle-bridge coupling control equation based on the discretization results; S200: Select appropriate internal and external single-variable functions to construct the KAN1 module for predicting the vertical displacement response of the bridge and the KAN2 module for predicting the vertical displacement response of the vehicle, respectively; S300: Construct a residual loss function based on the vehicle-bridge coupling control equation, conditional constraints, and real observation data to train and optimize the KAN1 module and KAN2 module.

[0008] Preferably, in step S100, the vehicle-bridge coupling control equation is expressed as follows: ; ; ; Among them, F b (t) and F v (t) represents the contact coupling force of the bridge and the vehicle at the discrete nodes, u v Represents the vertical displacement matrix of the vehicle at the discrete node, u b Represents the vertical displacement matrix of the bridge at discrete nodes, M b and M v Denote the mass matrices of the bridge and vehicle respectively, C b and C v Denote the damping matrices of the bridge and vehicle respectively, K b and K v Denote the stiffness matrices of the bridge and vehicle respectively, k t Indicates the stiffness of the vehicle tire, c t represents the damping coefficient of the vehicle tire, x c represents the vehicle-bridge coupling position corresponding to discrete time t, and r represents the uneven sample data of the bridge surface.

[0009] Preferably, the discretization of the time domain in the space-time domain includes the following process: calculating the total analysis time t based on the bridge span and the vehicle speed; 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].

[0010] Preferably, the discretization of the spatial domain in the time-space domain includes the following process: dividing the bridge structure into unit grids to obtain a two-dimensional matrix S for discretizing the geometric dimensions of the bridge structure. B , S B=[(0, 0), (ΔL, 0), (2ΔL, 0), …, (L, 0)]; where L represents the span of the bridge and ΔL represents the unit grid size; the number of nodes used for the equivalent vehicle structure is set to A, and the coordinates of each node at the initial position (x 0,i ,y 0,i ), i∈{1, 2, ..., A}, discretize the spatial position of the vehicle at all times 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 )].

[0011] Preferably, in step S200, the internal function of the KAN1 module selects a combination of Fourier basis function and B-spline function, and the external function selects a linear weighted summation form; the internal function of the KAN2 module selects a combination of Chebyshev polynomial function and B-spline function, and the external function selects a linear weighted summation form; the input of the KAN1 module and the KAN2 module is the corresponding discrete time and space domain, and the output is the corresponding vertical displacement matrix; the outputs of the KAN1 module and the KAN2 module are spliced ​​to form a coupled prediction output.

[0012] Preferably, the internal function combination form of the KAN1 module and the KAN2 module adopts a dynamic weighted combination form based on machine learning. The specific combination process is as follows: Construct the internal function of the KAN1 module and the KAN2 module , ; Among them, α and β constitute the weights of the two basis functions X and Y of the internal function respectively; the weights α and β are assigned according to the local and global emphasis, the value of weight α gradually increases with the increase of global emphasis, and the value of weight β gradually increases with the increase of local emphasis, α + β = 1; construct a feature extraction network g to extract the features that affect the local and global emphasis of the KAN1 module and the KAN2 module, and adjust the weights α and β according to the extraction results.

[0013] Preferably, the system is divided into multiple stages according to the difference between local emphasis and global emphasis, and the values ​​of weights α and β corresponding to adjacent stages are different; a transition stage is set between adjacent stages, and when the changes in local emphasis and global emphasis are within the transition stage, the values ​​of weights α and β adopt the values ​​of the previous stage.

[0014] Preferably, step S300 includes the following process: weighting each residual loss function by setting the residual loss weight to obtain the physical information residual loss function; when training the KAN1 module and the KAN2 module, dynamically adjusting the learning rate and the loss function weight according to the descent of the loss curve of the physical information residual loss function until the loss curve converges.

[0015] Preferably, in step S300, the residual loss function L of the differential equation based on the vehicle-bridge coupling control equation is DEB and L DEV , and the residual loss function L of the real observation data DATA The expression is as follows: ; ; ; Where N represents the number of discrete nodes, Represents the actual measured value or finite element simulation value of the vertical displacement of the bridge.

[0016] Preferably, in step S300, the conditional constraints include boundary condition constraints, initial condition constraints and coupling condition constraints, and the corresponding boundary condition constraint residual loss function L BC , initial condition constrained residual loss function L IC And the coupling condition constrained residual loss function L CC The expression is as follows: ; ; ; Where L represents the span of the bridge and x represents the vehicle displacement.

[0017] Compared with the prior art, the present invention has the following advantages: The proposed method boasts a fast computational speed and high accuracy while maintaining efficiency, making it a viable alternative model for vehicle-bridge coupled finite element numerical simulations. It can also calculate the vibration response of a bridge in real time for a heavily loaded vehicle not on the bridge, enabling rapid assessment of structural safety risks and prompt intervention. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 This is a schematic diagram of the overall workflow of this application. DETAILED DESCRIPTION

[0019] Below, the present application is further described in conjunction with specific implementation methods. It should be noted that, in the description of this specification, the reference terms "one embodiment", "some embodiments", "example", "specific example", or "some examples" and the like are intended to mean 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 invention. In this specification, the schematic representation 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 may be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art may combine and combine the different embodiments or examples described in this specification.

[0020] In the description of this application, it should be noted that for directional words, such as the terms "center", "horizontal", "longitudinal", "length", "width", "thickness", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", "clockwise", "counterclockwise" and so on, indicating the orientation and position relationship are based on the orientation or position relationship shown in the accompanying drawings, which is only for the convenience of describing this application and simplifying the description, and does not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and cannot be understood as limiting the specific scope of protection of this application.

[0021] It should be noted that the terms "first", "second", etc. in the description and claims of this application are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence.

[0022] In this application, unless otherwise expressly specified or limited, terms such as "mounted," "connected," "connect," and "fixed" should be understood in a broad sense. For example, they may refer to connection, detachable connection, or integration; mechanical connection or electrical connection; direct connection or indirect connection through an intermediate medium; and internal communication between two components or interaction between two components. Those skilled in the art will understand the specific meanings of the above terms in this application based on specific circumstances.

[0023] In this application, unless otherwise expressly specified or limited, a first feature being "above" or "below" a second feature may include the first and second features being in direct contact, or may include the first and second features being in contact not directly but through another feature between them. Moreover, a first feature being "above," "above," and "above" a second feature may include the first feature being directly above or obliquely above the second feature, or may simply mean that the first feature is higher in level than the second feature. A first feature being "below," "below," and "below" a second feature may include the first feature being directly below or obliquely below the second feature, or may simply mean that the first feature is lower in level than the second feature.

[0024] The terms "comprises" and "having" and any variations thereof in the specification and claims of this application are intended to cover non-exclusive inclusions. For example, a process, method, system, product or apparatus that includes a series of steps or units is not necessarily limited to those steps or units expressly listed, but may include other steps or units not expressly listed or inherent to such process, method, product or apparatus.

[0025] One of the preferred embodiments of this application is as follows: Figure 1 As shown in FIG, a PIKAN-based vehicle-bridge coupled vibration response prediction method includes the following steps: S100: Discretize the time and space domains of the bridge and vehicle based on the bridge structure and the vehicle's speed on the bridge deck; construct the vehicle-bridge coupling control equation based on the discretization results.

[0026] S200: Select appropriate internal and external single-variable functions to construct the KAN1 module for predicting the vertical displacement response of the bridge and the KAN2 module for predicting the vertical displacement response of the vehicle, respectively.

[0027] S300: Construct a residual loss function based on the vehicle-bridge coupling control equation, conditional constraints, and real observation data to train and optimize the KAN1 module and KAN2 module.

[0028] It can be understood that PIKAN is a machine learning model that embeds physical laws in the KAN architecture. It approximates the solution or unknown parameters of a nonlinear PDE-ODE coupled system through a combination of learnable single-variable functions, while using physical residual loss to ensure that the obtained solution satisfies the given governing equations, boundaries, and initial conditions.

[0029] In layman's terms, the input parameters of the vehicle-bridge coupling system are usually the span L of the bridge, time t, vehicle displacement x and velocity v, etc. When the vehicle-bridge coupling model is constructed using the traditional fully connected or CNN method, its dimension will increase exponentially. In the technical solution of this application, the vehicle and bridge are modeled separately through the KAN module, and the parameters of the KAN module only increase linearly with the number of univariate functions. This significantly reduces the scale of the model constructed by the KAN network, thereby effectively improving the calculation rate. At the same time, while taking into account the calculation efficiency, it has good calculation accuracy, and can be used as an alternative model for vehicle-bridge coupling finite element numerical simulation. It can also calculate the bridge vibration response of heavy-loaded vehicles that are not on the bridge in real time, quickly assess the safety risks of the bridge structure, and make timely interventions.

[0030] In this embodiment, the spatiotemporal domain is divided into the time domain and the spatial domain. Since the bridge is fixed and the vehicle travels along the bridge deck, the vehicle and bridge are the total time t that the vehicle travels on the bridge, i.e., the total analysis duration when discretizing the time domain. Since the bridge has a span L, when discretizing the spatial domain, the discrete nodes are the different locations along the span of the bridge, and the spatial domain discrete nodes of the vehicle are the displacement points of the vehicle traveling on the bridge deck. For ease of understanding, the specific discretization process in the time and spatial domains will be described in detail below using a two-dimensional simply supported beam and a 4-DOF vehicle model as examples.

[0031] Specifically, the discretization of the time domain in the space-time domain includes the following process: calculating the total analysis time t based on the bridge span L and the vehicle speed v, t=L / v; setting the time step Δt for the vehicle-bridge coupled vibration analysis 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].

[0032] It should be noted that the specific value of the analysis time step Δt is related to vehicle speed, the number of discrete nodes, and the span of the bridge. Specifically, the faster the vehicle, the shorter the time it takes to cross the bridge. Therefore, to obtain sufficient coupling data, the analysis time step Δt should be smaller. Conversely, the analysis time step Δt can be appropriately increased. The number of discrete nodes on a bridge is related to its span. Generally speaking, for a bridge with a span less than 30m, the number of discrete nodes is around 10; for a bridge with a span greater than 50m, the number of discrete nodes can be increased to 16 or more. For example, the analysis time step Δt can be 0.01s or 0.05s, and the specific value can be selected based on the actual needs of those skilled in the art.

[0033] Specifically, the discretization of the spatial domain in the time-space domain includes the following process: since the spatial position of the bridge is fixed, according to the span L of the bridge, its scope can be defined as [0, L]; then the bridge structure can be divided according to the set unit grid size ΔL to obtain the two-dimensional matrix S that discretizes the geometric dimensions of the bridge structure B , S B =[(0, 0), (ΔL, 0), (2ΔL, 0), …, (L, 0)]. Assume that the number of nodes used for the equivalent vehicle structure is A, and the coordinates of each node at the initial position (x 0,i ,y 0,i ), i∈{1, 2, ..., A}, discretize the spatial position of the vehicle at all times 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 )].

[0034] It is understandable that the specific value of the unit grid size ΔL used for the grid division of the bridge structure can be selected according to the actual needs of those skilled in the art; the unit grid size ΔL can be equal to the distance between adjacent discrete nodes, or it can be less than the distance between adjacent discrete nodes. The value of the number of nodes A used for the equivalent vehicle structure is generally 4 to 12; that is, for a car, it can generally be equivalent through 4 nodes; and for large vehicles such as trucks, the corresponding number of equivalent nodes needs to be increased. In this embodiment, based on experience, the number of nodes A of the equivalent vehicle structure can be taken as 7. 7 nodes are just between general and complex, that is, they cover common working conditions without excessively increasing the amount of calculation. Then the two-dimensional coordinates of the vehicle at the initial position can be expressed 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 )].

[0035] In this embodiment, according to the spatial domain discretization of the vehicle-bridge coupling system, the mass matrix M of the bridge and vehicle can be constructed. b and Mv , the stiffness matrix K b and K v , and the damping matrix C b and C v Then in step S100, the vehicle-bridge coupling control equation is expressed as follows: .

[0036] .

[0037] Among them, F b (t) and F v (t) represents the contact coupling force of the bridge and the vehicle at the discrete nodes, u v Represents the vertical displacement matrix of the vehicle at the discrete node, u b Represents the vertical displacement matrix of the bridge at the discrete node. It should be noted that for the bridge damping matrix C b , can be constructed using Rayleigh damping, namely , and Both represent weight coefficients.

[0038] It can be understood that in the above control equations, the contact coupling force F between the bridge and the vehicle at the discrete nodes is b (t) and F v (t) can be calculated by the contact between the vehicle's wheel node and the corresponding bridge node. The specific calculation formula is as follows: ; Among them, k t Indicates the stiffness of the vehicle tire, c t represents the damping coefficient of the vehicle tire, x c represents the vehicle-bridge coupling position corresponding to discrete time t, and r represents the uneven sample data of the bridge surface.

[0039] In this embodiment, in step S200, the KAN1 module and the KAN2 module can be spliced ​​to form a network model for predicting the vehicle-bridge coupled vibration response; wherein the bridge and the vehicle are respectively used as two sub-coefficients of the network model. For the KAN1 module, its input is the time-space domain discrete representation of the bridge subsystem [T, S B ],[T,S B ]=[(Δt, 0, 0), (2Δt, ΔL, 0), …, (t, L, 0)]; its output is the vertical displacement matrix u of the bridge subsystem at the discrete node b For the KAN2 module, its input is the spatiotemporal discrete representation of the vehicle subsystem [T, S v ],[T,S v ]=[(Δt,xt,1 ,y t,1 ),(2Δt,x t,2 ,y t,2 ),……,(t,x t,7 ,y t,7 )]; its output is the vertical displacement matrix u of the vehicle subsystem at the discrete node v The outputs of the KAN1 module and the KAN2 module are concatenated to form the coupled prediction output.

[0040] In this embodiment, the specific types of internal and external functions for the KAN1 and KAN2 modules can be selected in a variety of ways. Generally speaking, to achieve both local and global predictions, the internal function can be a combination of two unary functions, focusing on local and global solutions, respectively. Specifically, the internal function of the KAN1 module is a combination of Fourier basis functions and B-spline functions, while the external function is a linear weighted summation. That is, the external function is used to perform a linear weighted summation of all channel results of the internal function. The internal function of the KAN2 module is a combination of Chebyshev polynomial functions and B-spline functions, while the external function is a linear weighted summation.

[0041] It is understandable that when conducting vehicle-bridge coupling analysis, two key factors need to be considered: global and local scales. Global factors mainly include the overall modal state of the beam bridge, the number and distribution of vehicles traveling on the bridge, and other environmental factors; local factors mainly include the contact force impact between the wheel and the bridge deck, the local structural performance of the bridge, and local damage to the wheel or bridge deck. Therefore, when combining internal functions, the weight ratio of the two unary functions that focus on local and global solutions, respectively, is suitable for adaptive dynamic adjustment based on the degree of change from local to global. For ease of understanding, the following will describe the internal function combination form in the KAN1 module and the KAN2 module in detail.

[0042] Specifically, there are many specific ways to dynamically weight the internal functions in the KAN1 module and the KAN2 module. In this embodiment, a dynamic weighted combination based on machine learning is preferably used. The specific combination process is as follows: The internal functions in the KAN1 module and the KAN2 module are dynamically weighted. The specific combination process is as follows: Construct the internal functions of the KAN1 module and the KAN2 module. , ; Among them, α and β constitute the weights of the two basis functions X and Y of the internal function respectively; the weights α and β are assigned according to the local and global emphasis, the value of weight α gradually increases with the increase of global emphasis, and the value of weight β gradually increases with the increase of local emphasis, α + β = 1; construct a feature extraction network g to extract the features that affect the local and global emphasis of the KAN1 module and the KAN2 module, and adjust the weights α and β according to the extraction results.

[0043] It should be noted that in the KAN1 module, the Fourier basis function and the B-spline function correspond to the basis functions X and Y; in the KAN2 module, the Chebyshev polynomial function and the B-spline function correspond to the basis functions X and Y. For the feature extraction network g, a lightweight gated network can be used. The specific construction process is common knowledge to those skilled in the art and will not be described in detail here. The feature extraction network g can judge the emphasis on local and global factors corresponding to vehicles traveling at different discrete node positions on the bridge deck based on the speed and structural parameters of the vehicles traveling on the bridge deck, as well as the given environmental parameters and bridge structural parameters. If it is necessary to emphasize global factors, the weight value direction is from local to global; if it is necessary to emphasize local factors, the weight value direction is in the opposite direction from local to global.

[0044] It is understood that to simplify the calculation process of the KAN1 and KAN2 modules, the calculation process can be divided into multiple stages based on the difference between local and global emphasis. This allows different values ​​to be assigned to the weights α and β corresponding to each stage. This allows the infinite dynamic adjustment of the weights α and β to be transformed into dynamic adjustment of the corresponding stage. For example, the calculation process can be divided into nine stages based on the difference between local and global emphasis, and the values ​​of the weights α and β in each stage [α, β] are: [0.1, 0.9], [0.2, 0.8], ..., [0.8, 0.2], [0.9, 0.1] respectively.

[0045] It's important to note that, given that traffic flow stabilizes for a period of time after a bridge is completed, the values ​​of weights α and β may be relatively concentrated, such as [0.4, 0.6], [0.5, 0.5], and [0.6, 0.4]. This may result in frequent dynamic adjustments to the values ​​of weights α and β within the three ranges mentioned above. Therefore, a hysteresis mechanism can be implemented to reduce the frequency of dynamic adjustments to the values ​​of weights α and β.

[0046] Specifically, a transition stage is set between adjacent stages. When the change between the local emphasis and the global emphasis is within the transition stage, the values ​​of the weights α and β are taken from the previous stage.

[0047] Taking the above-mentioned weights α and β as examples, where the values ​​are concentrated in [0.4, 0.6], [0.5, 0.5], and [0.6, 0.4], we can set the threshold of the ratio of local emphasis to global emphasis when the values ​​of weights α and β are adjusted from [0.4, 0.6] to [0.5, 0.5] to be a, and the threshold of the ratio of local emphasis to global emphasis when the values ​​of weights α and β are adjusted from [0.5, 0.5] to [0.6, 0.4] to be b. Then, we can set a hysteresis value Δd to obtain the transition stages [a-Δd, a+Δd] and [b-Δd, b+Δd]. When the ratio of local emphasis to global emphasis is in the transition stage [a-Δd, a+Δd], if the values ​​of weights α and β at the previous moment were [0.4, 0.6], then the values ​​of weights α and β at this moment are still [0.4, 0.6]; if the values ​​of weights α and β at the previous moment were [0.5, 0.5], then the values ​​of weights α and β at this moment are still [0.5, 0.5]. Similarly, when the ratio of local emphasis to global emphasis is in the transition stage [b-Δd, b+Δd], if the values ​​of weights α and β at the previous moment were [0.5, 0.5], then the values ​​of weights α and β at this moment are still [0.5, 0.5]; if the values ​​of weights α and β at the previous moment were [0.6, 0.4], then the values ​​of weights α and β at this moment are still [0.6, 0.4].

[0048] In this embodiment, step S300 includes the following process: The residual loss functions are weighted by setting residual loss weights to obtain a physical information residual loss function. During the training of the KAN1 module and the KAN2 module, the learning rate and loss function weights are dynamically adjusted according to the decrease in the loss curve of the physical information residual loss function until the loss curve converges.

[0049] 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 residual loss function L of the differential equation based on the vehicle-bridge coupling control equation DEB and L DEV , the residual loss function L of the real observation data DATA , boundary condition constrains the residual loss function L BC , the initial condition constrains the residual loss function L IC And the coupling condition constrained residual loss function L CC The specific expression is as follows: .

[0050] .

[0051] .

[0052] .

[0053] .

[0054] .

[0055] .

[0056] in, Represents the residual loss function L of the differential equation DEB The weight of Represents the residual loss function L of the differential equation DEV The weight of Represents the boundary condition constraint residual loss function L BC The weight of Represents the initial condition constraint residual loss function L IC The weight of Represents the coupling condition constraint residual loss function L CC The weight of Residual loss function L representing the real observation data DATA The weight of N represents the number of discrete nodes. represents the actual measured value or finite element simulation value of the vertical displacement of the bridge, and x represents the vehicle displacement.

[0057] It is understandable that after completing the construction of the PIKAN-based vehicle-bridge coupled vibration response prediction network model, it can be deployed in the front-end edge computing terminal. After detecting vehicle information on the bridge, the vehicle's spatiotemporal information is discretized and input into the network model together with the preset bridge spatiotemporal discrete information. The predicted value of the bridge's vertical dynamic displacement is calculated and compared with the actual value of the bridge's vertical displacement monitored in real time at the measurement points on the bridge, and error analysis is performed. The accuracy of the network model is judged based on the analysis results.

[0058] 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-described embodiments. The above-described embodiments and the specification merely illustrate the principles of the present application. Various changes and improvements may be made to the present application without departing from the spirit and scope of the present application. These changes and improvements fall within the scope of the present application for which protection is sought. 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 vehicle-bridge coupled vibration response, characterized in that: The steps include: S100: Discretize the time and space domains of the bridge and vehicle based on the bridge structure and the speed of the vehicle on the bridge deck; construct the vehicle-bridge coupling control equation based on the discretization results; S200: Select appropriate internal and external single-variable functions to construct the KAN1 module for predicting the vertical displacement response of the bridge and the KAN2 module for predicting the vertical displacement response of the vehicle, respectively; S300: Construct a residual loss function based on the vehicle-bridge coupling control equation, conditional constraints, and real observation data to train and optimize the KAN1 module and KAN2 module.

2. The vehicle-bridge coupled vibration response prediction method based on PIKAN according to claim 1, characterized in that: In step S100, the vehicle-bridge coupling control equation is expressed as follows: ； ； ； Among them, F b (t) and F v (t) represents the contact coupling force of the bridge and the vehicle at the discrete nodes, u v Represents the vertical displacement matrix of the vehicle at the discrete node, u b Represents the vertical displacement matrix of the bridge at discrete nodes, M b and M v Denote the mass matrices of the bridge and vehicle respectively, C b and C v Denote the damping matrices of the bridge and vehicle respectively, K b and K v Denote the stiffness matrices of the bridge and vehicle respectively, k t Indicates the stiffness of the vehicle tire, c t represents the damping coefficient of the vehicle tire, x c represents the vehicle-bridge coupling position corresponding to discrete time t, and r represents the uneven sample data of the bridge surface.

3. The vehicle-bridge coupled vibration response prediction method based on PIKAN according to claim 2, characterized in that: The discretization of the time domain in the space-time domain includes the following processes: Calculate the total analysis time t based on the bridge span and vehicle speed; Based on the time history analysis of structural mechanics, the time step Δt of the vehicle-bridge coupled vibration analysis is set; According to the obtained time step Δt, the total analysis time t is discretized into a one-dimensional vector T, T=[Δt, 2Δt,…, t].

4. The vehicle-bridge coupled vibration response prediction method based on PIKAN according to claim 3, characterized in that: The discretization of the spatial domain in the space-time domain includes the following processes: The bridge structure is divided into unit grids to obtain a two-dimensional matrix S that discretizes the geometric dimensions of the bridge structure. B , S B =[(0,0),(ΔL,0),(2ΔL,0),…,(L,0)]; Assume that the number of nodes used for the equivalent vehicle structure is A, and the coordinates of each node at the initial position (x 0,i ,y 0,i ), i∈{1, 2, ..., A}, discretize the spatial position of the vehicle at all times 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 )]; Where L represents the span of the bridge and ΔL represents the unit grid size.

5. The vehicle-bridge coupled vibration response prediction method based on PIKAN according to claim 2, characterized in that: In step S200, the internal function of the KAN1 module selects the combination of Fourier basis function and B-spline function, and the external function selects the form of linear weighted summation; The internal function of the KAN2 module uses a combination of Chebyshev polynomial function and B-spline function, and the external function uses linear weighted summation. The input of KAN1 module and KAN2 module is the corresponding discrete time and space domain, and the output is the corresponding vertical displacement matrix; The outputs of the KAN1 module and the KAN2 module are concatenated to form a coupled prediction output.

6. The vehicle-bridge coupled vibration response prediction method based on PIKAN according to claim 5, characterized in that: The internal function combination form in the KAN1 module and the KAN2 module adopts a dynamic weighted combination form based on machine learning. The specific combination process is as follows: Construct internal functions of KAN1 module and KAN2 module , ; Among them, α and β constitute the weights of the two basis functions X and Y of the internal function respectively; Assign weights α and β according to the degree of emphasis on local and global. The value of weight α gradually increases with the increase of global emphasis, and the value of weight β gradually increases with the increase of local emphasis. α + β = 1. A feature extraction network g is constructed to extract features that affect the local and global emphasis of the KAN1 module and the KAN2 module, and the weights α and β are adjusted according to the extraction results.

7. The vehicle-bridge coupled vibration response prediction method based on PIKAN according to claim 6, characterized in that: It is divided into multiple stages according to the difference between local emphasis and global emphasis, and the values ​​of weights α and β corresponding to adjacent stages are different; A transition stage is set between adjacent stages. When the change between local emphasis and global emphasis is within the transition stage, the values ​​of the weights α and β are taken from the previous stage.

8. The vehicle-bridge coupled vibration response prediction method based on PIKAN according to any one of claims 2 to 7, characterized in that: Step S300 includes the following process: By setting the residual loss weight, each residual loss function is weighted to obtain the physical information residual loss function; When training the KAN1 module and the KAN2 module, the learning rate and loss function weight are dynamically adjusted according to the decrease of the loss curve of the physical information residual loss function until the loss curve converges.

9. The vehicle-bridge coupled vibration response prediction method based on PIKAN according to claim 8, characterized in that: In step S300, the residual loss function L of the differential equation based on the vehicle-bridge coupling control equation is DEB and L DEV , and the residual loss function L of the real observation data DATA The expression is as follows: ； ； ； Where N represents the number of discrete nodes, Represents the actual measurement value or finite element simulation value of the bridge vertical displacement matrix.

10. The vehicle-bridge coupled vibration response prediction method based on PIKAN according to claim 8, characterized in that: In step S300, the conditional constraints include boundary condition constraints, initial condition constraints and coupling condition constraints. The corresponding boundary condition constraint residual loss function L BC , initial condition constrained residual loss function L IC And the coupling condition constrained residual loss function L CC The expression is as follows: ； ； ； Where L represents the span of the bridge and x represents the vehicle displacement.

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