Bridge damping ratio identification method based on PINNs under dimensionless governing equation

Abstract

The application discloses a bridge damping ratio identification method based on PINNs under dimensionless control equations, and relates to the technical field of bridge damping ratio identification.The bridge damping ratio identification method comprises the following steps: input variables are mapped by a network to obtain output displacement; each differential term is obtained by automatic differentiation of the output displacement; each parameter and the differential term are subjected to dimensionless processing, and a dimensionless free vibration differential equation containing unknown damping ratio is constructed; loss is calculated, and a total loss function is constructed; network parameters and trainable variables are updated, and a training end point is judged according to a network convergence mechanism.The bridge damping ratio identification method based on PINNs under dimensionless control equations fundamentally eliminates the order-of-magnitude difference between the coefficients of each differential term by implicitly containing bridge stiffness and linear density in dimensionless parameters, and realizes bridge damping ratio identification based on the PINNs method.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of bridge damping ratio identification, and particularly relates to a bridge damping ratio identification method based on PINNs under dimensionless control equations. BACKGROUND

[0002] Damping ratio is a key dynamic parameter reflecting the energy dissipation capacity of bridge structure, and is considered as an important indicator of structural health. Accurate identification of bridge damping ratio is of great significance to master the dynamic characteristics of bridge and evaluate the structural damage. The vibration response of bridge after excitation contains the damping characteristics related to the attenuation information. Therefore, the unknown damping ratio can be identified by analyzing the measured vibration response data.

[0003] Based on the sensor placement target, the damping ratio identification method can be divided into direct measurement method and indirect measurement method. In the direct measurement method, the sensor is directly installed on the bridge structure to obtain the vibration response. Li et al. analyzed the sensor data of the suspension bridge for a long time by using the Bayesian spectral density algorithm, and revealed the influence of different environmental conditions on modal frequency and damping ratio. Kim et al. determined the first modal damping ratio of the cable-stayed bridge by analyzing the vibration data obtained by a large number of sensors installed during the operation of the cable-stayed bridge. Magalhães et al. analyzed more than 2500 data sets during operation based on the covariance driven stochastic subspace identification method, and accurately estimated the multi-modal parameters such as damping of bridge structure. The above direct measurement method is widely used and has high identification accuracy. However, it usually depends on a large amount of vibration data, which is difficult to obtain in actual engineering due to environmental restrictions and cost restrictions. The indirect measurement method eliminates the requirement of directly installing sensors on the bridge structure. González et al. identified the bridge damping ratio by installing accelerometers on the axle based on the simplified half-car-bridge interaction model, and verified the effectiveness of the method under specific bridge span and vehicle speed. Yang et al. developed a bridge damping ratio identification method by using a double-axle mobile test vehicle equipped with accelerometers and laser sensors. The method extracts the attenuation characteristics of the scanning point response by Hilbert transform, and realizes the identification of simply supported beam damping ratio. Yang et al. analyzed the acceleration signal of the static detection vehicle based on the peak decay method, and extracted the first modal damping ratio of the bridge. The above indirect measurement method needs less sensors, but has lower accuracy compared with the direct method, and is easily disturbed by factors such as road roughness. Considering that the direct measurement method and the indirect measurement method have their own limitations and complementary advantages, it is very attractive to combine the advantages of the two methods, which not only retains the high accuracy of the direct measurement method, but also reduces the dependence of the direct measurement method on monitoring data.

[0004] Fusion of physical laws and data-driven methods is a feasible way to reduce the demand for monitoring data. Physics-informed neural networks (PINNs) is one of the representative methods of this kind. PINNs can set the parameters in the governing equation as unknown trainable variables, and then realize parameter identification according to partial monitoring data. PINNs can accurately identify unknown parameters with a small amount of monitoring data, which benefits from the constraint of physical laws on the solution space. Therefore, PINNs provide a new way to accurately identify the bridge damping ratio in the case of limited data and limited measurement points. As a physically constrained machine learning method, PINNs have achieved relatively mature research results in many fields such as fluid mechanics and solid mechanics. However, although its basic theory is relatively mature, its application in structural health monitoring (SHM) still needs to be continuously expanded. Yin et al. found that when directly applying PINNs to identify the bridge damping ratio, the information of the inertia term and the damping term was covered due to the large coefficient of the stiffness term in the vibration differential equation. Therefore, they proposed a function approximation guided physics-informed neural network (FA-PINN) to solve this problem from the network architecture level. However, the strict requirements of FA-PINN on data quality and fitting accuracy limit its application in actual engineering noise environment. Therefore, it is urgent to further explore and develop a PINNs-based bridge damping ratio identification method with strong adaptability. SUMMARY

[0005] In view of the above technical problems, the present application provides a bridge damping ratio identification method based on PINNs under dimensionless governing equation, which implicitly includes bridge stiffness and linear density in dimensionless parameters, fundamentally eliminates the order of magnitude difference between the coefficients of each differential term, and realizes the identification of bridge damping ratio based on PINNs method.

[0006] To solve the above technical problems, the technical scheme provided by the present application is:

[0007] A bridge damping ratio identification method based on PINNs under dimensionless governing equation, comprising the following steps:

[0008] The input variables are mapped by the network to obtain the output displacement;

[0009] The output displacement is automatically differentiated to obtain each differential term;

[0010] The dimensionless treatment is performed on each parameter and differential term to construct a dimensionless free vibration differential equation containing unknown damping ratio;

[0011] The loss is calculated to construct a total loss function;

[0012] The network parameters and trainable variables are updated, and the training end point is determined according to the network convergence mechanism.

[0013] A further improvement to the above technical solution is as follows:

[0014] Preferably, the input variables are first seeded randomly before network mapping, and the training grid points for unsupervised learning are divided according to the bridge length and the testing time. The input variable corresponding to each grid point is ( x , t ), x For spatial variables, t The time variable is used as the input variable, and the network output is obtained through a non-linear mapping. w , w This represents the displacement perpendicular to the longitudinal direction of the bridge.

[0015] Preferably, the automatic differentiation generates parameter gradients through computational graphs:

[0016] ;

[0017] in, This is the matrix right multiplication operator. For loss function, k This represents the layer index in the computation graph. This represents the parameter vector of PINNs. v k For the first k Intermediate variables in the layer; This represents the output variable of the i-th layer. For the i-th layer function f right Jacobian matrix, Transpose it;

[0018] The gradient is formalized as a discrete integral computed over the path space of the graph:

[0019] ;

[0020] in, Indicates from parameter To loss function The set of all backpropagation paths, For path Jacobian matrix multiplication on top of each other, Indicates the first j The output variables of the layer, This represents the parent node index, identifying the input source layer of the current operator. This represents the child node index, identifying the output layer of the current operator.

[0021] Preferably, the dimensionless free vibration differential equation is:

[0022] ;

[0023] wherein, is the dimensionless output displacement, n denotes the modal order, denotes the dimensionless spatial variable, denotes the dimensionless time variable.

[0024] Preferably, the total loss function is expressed as:

[0025] ;

[0026] wherein, the subscripts c, d and b correspond to the physical domain, the data domain and the boundary domain, respectively, denotes the physical loss function, denotes the data loss function, denotes the boundary loss function, denotes the displacement output quantity of the physical domain, denotes the displacement output quantity of the data domain, denotes the displacement output quantity of the boundary domain, is the weight coefficient of the physical loss function, is the weight coefficient of the data loss function, is the weight coefficient of the boundary loss function, denotes the residual point of the boundary domain, denotes the residual point of the physical domain, denotes the residual point of the data domain, is the vibration acceleration signal collected by the sensor as the supervised data.

[0027] Preferably, the network convergence mechanism is to first set the minimum number of iteration steps to ensure that the network escapes from the local optimal solution, and then realize the termination judgment of the network through the parameter stability analysis.

[0028] Preferably, in the model training process, the damping ratio parameter inverted by the network is constantly updated to form a time series s wherein, is the total number of iteration steps; the mean and standard deviation of the damping ratio parameter are calculated by using a sliding time window: N

[0029] ;

[0030] wherein, denotes the result mean at the current iteration step, denotes the length of the sliding time window, denotes the damping ratio identification value at the current iteration step, ​It represents the standard deviation of the results at the current iteration step. When the minimum iteration step and the inversion parameter stability are met at the same time, the network is determined to be converged; the minimum iteration step is set, and when the minimum iteration step is reached, the standard deviation convergence condition is activated, the learning of the model to various types of information has been saturated, and when the convergence condition is met, the training is terminated, and the final As the network identification result.

[0031] The bridge damping ratio identification method based on PINNs under the dimensionless governing equation provided by the application has the following advantages compared with the prior art.

[0032] The bridge damping ratio identification method based on PINNs under the dimensionless governing equation provided by the application can accurately identify the bridge damping ratio by using a small amount of vibration data, thereby improving the applicability and efficiency of bridge health detection. The bridge damping ratio identification method provided by the application first solves the physical residual imbalance problem of the PINNs identification bridge damping ratio by dimensionless governing equation; secondly, based on numerical simulation, the influence of sensor configuration, bridge parameters and noise level on damping ratio identification accuracy is analyzed. The bridge damping ratio identification method provided by the application can accurately identify the first-order damping ratio of the bridge under the condition of data scarcity, and in addition, this method shows strong robustness to sensor configuration, bridge parameters and noise interference with SNR greater than or equal to 20 dB. Moreover, the engineering applicability of the method is verified by the bridge field test. BRIEF DESCRIPTION OF DRAWINGS

[0033] Figure 1 The flowchart of the bridge damping ratio identification method provided by the application.

[0034] Figure 2 The schematic diagram of the field test scheme provided by the application.

[0035] Figure 3 The original signal and the filtered signal diagram in the experimental verification of the application.

[0036] Figure 4 The test result diagram in the experimental verification of the application. DETAILED DESCRIPTION

[0037] The specific embodiments of the application will be described in detail below. It should be understood that the specific embodiments described herein are only used to illustrate and explain the application, and are not used to limit the application.

[0038] As Figure 1 shown, the bridge damping ratio identification method based on PINNs under the dimensionless governing equation provided by the application includes the following steps:

[0039] Step S1, the input variable is mapped to obtain the output variable by the network.

[0040] A random seed is first set, and a training grid point of unsupervised learning is divided according to the length of the bridge and the test time, and the input variables corresponding to each grid point are x , t ), x is a spatial variable, t is a time variable, and the input variables are mapped to the network output w , w represent the displacement perpendicular to the longitudinal direction of the bridge.

[0041] The nonlinear mapping is completed by a 4-layer fully connected feedforward neural network with 64-128-128-64 nodes, and the hyperbolic tangent activation function is used in each layer.

[0042] In step S2, the output displacement w is obtained by automatic differentiation (AD).

[0043] The displacement obtained by nonlinear mapping w needs to be differentiated under AD. Because the displacement solution of the beam free vibration equation in the inverse problem containing unknown parameters is an implicit solution, the differential terms required by the control equation cannot be obtained by analytical differentiation. In addition, numerical differentiation calculation has truncation and rounding errors, therefore, AD which is suitable for implicit problems and has high precision is the best choice for differential calculation of the method.

[0044] AD realizes accurate propagation of derivatives through the topological structure of the computation graph. Let the parameter vector of PINNs be , wherein d is the total dimension of the trainable parameters, is a real number set, and the objective function , wherein m is the dimension of the output physical quantity, such as the spatial coordinate, time or the joint dimension of a multi-element physical field. F can be decomposed into K a composite mapping of differentiable operators:

[0045] (1)

[0046] wherein, represents function combination, represents the K last differentiable operator, represents the first differentiable operator of the objective function F , processes the parameter vector and generates the initial intermediate output, represents the i differentiable operator, represents the parameter vector of PINNs. represents thei Local mapping of a layer operator, is the output dimension of the i th layer , denotes the output variable of the i th layer. The forward propagation is recursively generated by local mapping:

[0047] (2)

[0048] where is the parent node set of a node, i.e., the direct input variables required for computing v i ; E denotes the edge set in a computational graph, which is used to describe the data dependency relationship between nodes (variables or operators); denotes the output variable of the j th layer, denotes the parent node index, which identifies the input source layer of the current operator, denotes the child node index, which identifies the output layer of the current operator. In the backpropagation mechanism, the adjoint variable is defined, where is the loss function. The backpropagation follows the inverse accumulation rule of the chain rule:

[0049] (3)

[0050] where is the Jacobian matrix of the i th layer function f with respect to , and the transpose ensures the dimension compatibility of matrix multiplication ( ). is the child node set of a node v j , i.e., all the subsequent nodes with v j as the parent node.

[0051] AD generates parameter gradients through the computational graph:

[0052] (4)

[0053] where is the matrix right multiplication concatenation operator. Further, the gradient can be formalized as a discrete integral on the path space of the computational graph:

[0054] (5)

[0055] where denotes the path from a parameter to the loss function all the sets of backpropagation paths, is the Jacobian matrix multiplication on the path is the gradient from the parameters to the loss along the complete propagation route.

[0056] The present application adopts the backward mode, which can efficiently calculate the gradient of high-dimensional output (various losses) to low-dimensional parameters (damping ratio). In addition, in the adopted TensorFlow framework, the 'GradientTape' mechanism not only naturally adapts the backward mode AD, but also automatically tracks and optimizes all trainable parameters in the computation graph, simplifying the implementation process of parameter inversion under complex physical constraints.

[0057] Step S3, dimensionless processing of each parameter and differential term, constructing a dimensionless free vibration differential equation containing unknown damping ratio.

[0058] When PINNs identify unknown parameters, the control equation of the physical problem to be solved needs to be embedded for unsupervised learning. In the context of free vibration of a simply supported beam, the damping ratio of the bridge is identified, and the motion differential equation of the beam under each mode is expressed as:

[0059] (6)

[0060] In the formula, EI is the flexural rigidity of the beam; is the linear density of the beam; is the unit length viscous damping coefficient of the beam under the n mode, represents the transverse vibration displacement of the beam in the spatial variable x and the time variable t .

[0061] w The direction is perpendicular to the longitudinal direction of the bridge, that is, the transverse vibration displacement, and the properties are consistent, and the difference is that w is a general displacement of this attribute, while w ( x , t ) refers to the displacement in the spatial variable x and the time variable t coordinates when solving, that is, the transverse vibration displacement of the bridge longitudinal direction x at time t .

[0062] The partial differential equation PDE loss constructed by formula (6) is:

[0063] ​(7)

[0064] where, represents the residual point of the physical domain.

[0065] In actual bridges, the value of EI is usually higher than and five to six orders of magnitude, resulting in PDE loss being particularly sensitive to the calculation of the fourth-order partial derivative, thereby causing loss value explosion, network unable to correctly identify unknown parameters and other problems. Therefore, in order to solve the problem of too large difference of each differential term coefficient in formula (6), the present application carries out dimensionless processing on the free vibration equation of the beam.

[0066] The input x , t and output w ( x , t ) of the model are dimensionless:

[0067] (8)

[0068] In the formula: , and are dimensionless spatial variable, dimensionless time variable and dimensionless output displacement respectively; L is the spatial scale, the value is equal to the bridge span, and is normalized to the interval of 0 to 1; T is the time scale; D is the displacement scale, the value is the maximum value of the vibration response, the displacement is normalized to the interval of -1 to 1, and the displacement scale D is set to scale the dimensionless displacement used for network calculation. It should be noted that the value of the displacement scale has no strict limitation, and the core purpose is to enhance the weak signal characteristics by scaling the bridge vibration response and reduce the numerical calculation error. Therefore, in actual engineering, the displacement scale can be set flexibly to optimize the network identification effect.

[0069] Each partial differential term is transformed into:

[0070]

[0071] (9)

[0072] Substituting formula (9) into formula (6) and simplifying can obtain:

[0073] (10)

[0074] As can be seen from formula (10), the time scale T must contain to eliminate the influence of the coefficient of the fourth-order partial derivative, so , , is the natural angular frequency of the beam in the nth mode. and the damping ratio of each mode satisfies where, is the damping ratio of the bridge in the nth mode. Therefore, equation (10) is further simplified as: n

[0075] (11)

[0076] where, is the dimensionless output displacement, n denotes the mode order.

[0077] Equation (11) is the dimensionless free vibration differential equation of the beam. By embedding the equation as the physical principle guiding the network training into PINNs, the damping ratio of the bridge can be identified. The model proposed in the present application does not require the information of the stiffness and linear density of the bridge.

[0078] Step S4, calculate the loss.

[0079] (1) Calculate the physical residual based on equation (11) obtained in step S3, and further obtain the physical loss.

[0080] (2) According to the simply supported boundary condition, the displacement and bending moment at both ends of the beam are 0. Combined with the dimensionless parameters and differential terms in step S3, the boundary residual is calculated, and further the boundary loss is obtained.

[0081] (3) Arrange acceleration sensors on the beam to collect the free vibration response signal of the beam. The collected acceleration response is used as the data constraint of the network for supervised learning.

[0082] In order to calculate the data residual, the output displacement w obtained in step S1 needs to be twice differentiated with respect to time: and then compared with the collected acceleration signal, and further the data loss is obtained.

[0083] In the present embodiment, the lowest order can identify the first order damping ratio with high precision under 2 acceleration sensors. The arrangement mode of the sensor can be selected: the midpoint of the bridge span + another arbitrary point, then based on the spatial symmetry characteristics of the acceleration vibration response in the first mode, the response data is mirror mapped at the symmetric position of the non-midpoint collection point, that is, 3 measurement points are obtained by 2 sensors. The supervised data of the network training.

[0084] Step S5, construct the total loss function.

[0085] Based on the various sub-losses obtained in step S4, the total loss function is constructed. The total loss function of the model in the present application is represented as:​

[0086] (12)

[0087] where the subscripts c, d and b correspond to the physical domain, data domain and boundary domain respectively, represents the physical loss function, represents the data loss function, represents the boundary loss function, represents the displacement output of the physical domain, represents the displacement output of the data domain, represents the displacement output of the boundary domain, is the weight coefficient of the physical loss function, is the weight coefficient of the data loss function, is the weight coefficient of the boundary loss function, represents the residual point of the boundary domain, represents the residual point of the physical domain, represents the residual point of the data domain, is the vibration acceleration signal collected by the sensor as the supervised data. In order to align the network output with the sensor data, the second-order time derivative of the network output needs to be taken.

[0088] , and are the weight coefficients corresponding to the three loss functions respectively. Suitable loss function weights can ensure that the network learns the features of each domain sufficiently. However, the manual tuning process of the weight coefficient is time-consuming and lacks generalization, and it is difficult to adapt to different scene requirements. Therefore, the loss function weight coefficients of the model of the present application are set to adaptive weight coefficients. The trainable logarithmic weight parameters , and are defined, which are transformed by the exponential function to ensure the non-negativity of the weight and prevent gradient explosion. The loss function weight coefficients can be expressed as:

[0089] (13)

[0090] , and are all set to 0, corresponding to the initial weight .

[0091] In the network training stage, the logarithmic weight coefficients are optimized through back propagation:

[0092] (14)

[0093] The model can realize the magnitude balance between the loss terms by adaptively adjusting the loss function weight coefficients. Specifically, if the gradient of a certain loss term is dominant, the weight coefficient of the loss term will automatically decrease to suppress the dominant effect of the loss. Conversely, the weight coefficient of the loss term will automatically increase to suppress the dominant effect of the loss. The learning rate hyperparameter used in the model is .

[0094] In step S6, the network parameters and trainable variables are updated, and the training endpoint is determined according to the network convergence mechanism.

[0095] The network parameters and the damping ratio to be identified are updated in the direction of reducing the total loss. Then, whether the training is terminated is determined according to the network convergence mechanism.

[0096] A reasonable training termination condition can control the computing resources while ensuring that the model converges sufficiently. The present application designs a hybrid convergence judgment mechanism that combines qualitative and quantitative characteristics. The mechanism first sets a minimum number of iterations to ensure that the network escapes from a local optimal solution, and then realizes the termination judgment of the network through parameter stability analysis.

[0097] In the model training process, the damping ratio parameter is updated with the iteration step s to form a time series , wherein N is the total number of iterations. The mean and standard deviation of the damping ratio parameter are calculated using a sliding time window (length W =1000):

[0098] (15)

[0099] wherein represents the result mean at the current iteration step, represents the sliding time window length, represents the damping ratio identification value at the current iteration step, represents the result standard deviation at the current iteration step. When the minimum number of iterations and the inversion parameter stability are satisfied at the same time, the network is determined to be converged. In the present embodiment, the minimum number of iterations satisfies , and the minimum number of iterations is artificially selected through a small amount of simple pre-training. The simple pre-training can be a certain number of training tests under a set of parameters to determine reasonable and stable related indicators.

[0100] The inversion parameter stability is judged by the standard deviation of the final 1000-step inversion result after the minimum number of iterations, that is, it needs to satisfy . When the minimum number of iterations is reached, the standard deviation convergence condition is activated to ensure that the model has saturated learning of various types of information. When the convergence condition is satisfied, the training is terminated, and the final As the network identification result.

[0101] Experimental verification

[0102] A certain bridge is selected as the field test bridge, and the target bridge is a thirteen-span, double-lane simply supported concrete beam bridge with a total length of 331.46 meters. Each span is 25.0 m long and 8.5 m wide. In the case that there is no traffic load on the bridge deck and no regular vibration source near the bridge site, the DH5907N dynamic signal analysis system is used to measure the small vibration response of the bridge span structure caused by random load excitation such as wind load, ground microseismicity and water flow. The natural frequency and damping ratio of the structure are measured. The effective measurement frequency range of the analysis system is 0-39.6Hz, which can cover the test requirements of the structure base frequency. The modal parameters of the structure are extracted by using the cross power spectrum function between each response measuring point and the reference point, and the first-order modal natural frequency of the bridge is 6.25Hz and the damping ratio is 3.21%.

[0103] The method of the present application is based on the free vibration response of a simply supported beam. As shown in Figure 2 , a double-vehicle side-by-side driving mode is used to excite the bridge vibration, and the process is divided into three stages, Figure 2 The field test scheme diagram, in which (a) is the loading scheme, (b) is the photo of the loading vehicle, and (c) is the schematic diagram of the loading vehicle. The first stage is the adjustment stage, and the vehicle accelerates to 30-40km / h before entering the test beam section. The second stage is the excitation stage, and the driving vehicle passes through the measured beam section to excite the bridge vibration. The third stage is the test stage, and after the vehicle drives away from the measured beam section, the free vibration response of the beam body is collected by three acceleration sensors arranged at 1 / 4, 1 / 2 and 3 / 4 span.

[0104] Based on the first-order natural frequency of the bridge, a 3Hz bandwidth zero-phase Butterworth band-pass filter is used to extract the first-order modal acceleration response, and the original acceleration response and the extracted response are as shown in Figure 3 . Since the amplitude of the bridge vibration response generated by the vehicle excitation is small, a small displacement scale can be manually selected to avoid numerical instability problems in the training process. In terms of data acquisition, only 1s long free vibration decay section effective signal is needed to complete the modeling. In this test, the vibration response in the interval of 0.5s-1.5s is selected as the data constraint of the model for network training.

[0105] Table 1 shows the results of training with different measurement point data. When the network is trained with vibration data of 3 measurement points, the error of the identification result and the target bridge test data is about 1%. It is worth noting that in No. 2, the data at 0.75L is taken from 0.25L. In No. 2, the model only uses the data of 2 measurement points (the data at 0.25L is also used at 0.75L position). This is because the acceleration response of the simply supported beam is symmetrically distributed along the longitudinal direction under the first order mode. Based on this characteristic, the vibration data of the symmetric position measurement points can be complemented by mirror mapping, thereby enhancing the data constraint effect. In actual engineering, due to the influence of noise and filtering and other factors, there may be some differences in the vibration response of the symmetric position. Using this mirror mapping method not only can reduce the number of sensor layout, but also is helpful to learn the symmetry characteristics of the vibration system.

[0106] Table 1 test results

[0107]

[0108] The experimental results show that the method of the present application can realize effective identification of the first order damping ratio of the simply supported beam only with two sensors.

[0109] The above implementation cases are only preferred embodiments of the present application, and do not limit the present application in any form. Although the present application has been disclosed as above with preferred embodiments, it is not intended to limit the present application. Therefore, any simple modification, equivalent change and modification made to the above embodiments without departing from the technical solution of the present application, according to the technical essence of the present application, shall fall within the protection scope of the technical solution of the present application.

Claims

1. A method for identifying the bridge damping ratio based on PINNs under dimensionless governing equations, characterized in that, Includes the following steps: The input variables are mapped through a network to obtain the output displacement; The output displacement is automatically differentiated to obtain the differential terms; Dimensionless treatment of each parameter and differential term is used to construct a dimensionless differential equation for free vibration with unknown damping ratio; Calculate the loss and construct the total loss function; Update network parameters and trainable variables, and determine the training endpoint based on the network convergence mechanism; The process of constructing the dimensionless free vibration differential equation is as follows: The differential equations of motion of the beam under each mode are expressed as follows: ； In the formula, EI The flexural stiffness of the beam; Let be the linear density of the beam; For the first n The viscous damping coefficient per unit length of the beam under first-order modal conditions. Indicates the spatial variables of the beam x and time variables t Lateral vibration displacement; Input to the model x , t and output w ( x , t Dimensionlessization: ； In the formula: , and These are, respectively, dimensionless spatial variables, dimensionless time variables, and dimensionless output displacements; L For spatial scale, the value is equal to the bridge span. Normalize to the interval between 0 and 1; T Time scale; D The displacement scale is set to the maximum value of the vibration response, normalizing the displacement to the interval between -1 and 1. D To scale the dimensionless displacement used for network computation; The partial differential terms are then transformed into: ； ； By simplifying the differential equation of motion using the partial differential terms, we obtain the dimensionless differential equation of free vibration, which is: ； in, The output displacement is dimensionless. n Indicates the modal order. Represents dimensionless spatial variables. Represents a dimensionless time variable. express n Damping ratio of bridge under first-order modes.

2. The bridge damping ratio identification method based on PINNs under dimensionless governing equations according to claim 1, characterized in that, The input variables are first seeded randomly before network mapping, and the training grid points for unsupervised learning are divided according to the bridge length and the testing time. The input variable corresponding to each grid point is ( x , t ), x For spatial variables, t The time variable is used as the input variable, and the network output is obtained through a non-linear mapping. w , w This represents the displacement perpendicular to the longitudinal direction of the bridge.

3. The bridge damping ratio identification method based on PINNs under dimensionless governing equations according to claim 1, characterized in that, The automatic differentiation generates parameter gradients through computational graphs: ； in, This is the matrix right multiplication operator. For loss function, k This represents the layer index in the computation graph. This represents the parameter vector of PINNs. v k For the first K Intermediate variables in the layer; Indicates the first i The output variables of the layer, For the first i Layer function f right Jacobian matrix, Transpose it; The gradient is formalized as a discrete integral computed over the path space of the graph: ； in, Indicates from parameter vector To loss function The set of all backpropagation paths, For path Jacobian matrix multiplication on top of each other, Indicates the first j The output variables of the layer, This represents the parent node index, identifying the input source layer of the current operator. This represents the child node index, identifying the output layer of the current operator.

4. The bridge damping ratio identification method based on PINNs under dimensionless governing equations according to claim 1, characterized in that, The total loss function is expressed as: ； In the formula, the subscripts c, d, and b correspond to the physical domain, data domain, and boundary domain, respectively. Represents the physical loss function. Represents the data loss function. Represents the boundary loss function. This represents the displacement output of the physical domain. This represents the displacement output in the data domain. This represents the displacement output of the boundary domain. These are the weighting coefficients of the physical loss function. These are the weighting coefficients of the data loss function. These are the weighting coefficients of the boundary loss function. Represents the residual points of the boundary domain. Represents the residual points in the physical domain. Represents the residual points in the data domain. The vibration acceleration signal collected by the sensor is used as monitoring data.

5. The bridge damping ratio identification method based on PINNs under dimensionless governing equations according to claim 4, characterized in that, The network convergence mechanism first determines the minimum number of iterations to ensure the network escapes local optima, and then uses parameter stability analysis to determine the network's termination.

6. The bridge damping ratio identification method based on PINNs under dimensionless governing equations according to claim 5, characterized in that, During model training, the damping ratio parameter is inverted by the network. With iteration steps s Continuously updated to form a time series ,in N The total number of iterations; the mean and standard deviation of the damping ratio parameter are calculated using a sliding time window: ； in, This represents the mean of the results at the current iteration step. Indicates the length of the sliding time window. This represents the damping ratio identification value at the current iteration step. This represents the standard deviation of the results at the current iteration step. Network convergence is determined when both the minimum number of iterations and the stability of the inversion parameters are satisfied. A minimum number of iterations is set; when this minimum number is reached, the standard deviation convergence condition is activated, indicating that the model's learning of various types of information has saturated. Training terminates when the convergence condition is met, and the final result is taken. As a result of network identification.

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