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

Abstract

The invention discloses a bridge damping ratio identification method under a dimensionless control equation based on PINNs, and relates to the technical field of bridge damping ratio identification, the bridge damping ratio identification method comprises the following steps: carrying out network mapping on an input variable to obtain an output displacement; performing automatic differentiation on the output displacement to obtain each differential item; performing dimensionless processing on each parameter and differential item, and constructing a dimensionless free vibration differential equation containing an unknown damping ratio; loss is calculated, and a total loss function is constructed; and updating network parameters and trainable variables, and judging a training end point according to a network convergence mechanism. According to the bridge damping ratio identification method based on the PINNs under the dimensionless control equation, the rigidity and the linear density of the bridge are implied in dimensionless parameters, the magnitude order difference between all differential term coefficients is fundamentally eliminated, and bridge damping ratio identification based on the PINNs method is achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge damping ratio identification, and in particular to a bridge damping ratio identification method based on PINNs under a dimensionless control equation. Background Art

[0002] The damping ratio is a key dynamic parameter reflecting the energy dissipation capacity of a bridge structure and is considered an important indicator of structural health. Accurately identifying a bridge's damping ratio is crucial for understanding its dynamic characteristics and assessing structural damage. The vibration response of a bridge under excitation contains attenuation information related to its damping properties. Therefore, unknown damping ratios can be identified by analyzing measured vibration response data.

[0003] Based on sensor placement objectives, damping ratio identification methods can be categorized into direct measurement and indirect measurement. In direct measurement, sensors are mounted directly on the bridge structure to obtain vibration responses. Li et al. used a Bayesian spectral density algorithm to analyze long-term sensor data from a suspension bridge, revealing the impact of different environmental conditions on modal frequencies and damping ratios. Kim et al. determined the first-order modal damping ratio of a cable-stayed bridge by analyzing vibration data obtained from numerous sensors installed during operation. Magalhães et al. used a covariance-driven random subspace identification method to analyze over 2,500 operational data sets and accurately estimated multimodal parameters such as bridge structural damping. These direct measurement methods are widely used and have high identification accuracy. However, they typically rely on large amounts of vibration data, which is difficult to obtain in practical engineering due to environmental and cost constraints. Indirect measurement methods eliminate the requirement for direct sensor mounting on the bridge structure. González et al. used a simplified half-vehicle-bridge interaction model to identify the bridge damping ratio by mounting accelerometers on the vehicle axles. They validated this method for specific bridge spans and vehicle speeds. Yang et al. developed a bridge damping ratio identification method using a dual-axis mobile test vehicle equipped with accelerometers and laser sensors. This method extracts the attenuation characteristics of the scanning point response through Hilbert transform, realizing the identification of the damping ratio of a simply supported beam. Based on the vehicle-bridge coupling theory, Yang et al. used the peak attenuation method to analyze the acceleration signal of the stationary detection vehicle and extracted the first modal damping ratio of the bridge. The above-mentioned indirect measurement method requires fewer sensors, but is less accurate than the direct method and is easily interfered by factors such as road roughness. Considering that both direct and indirect measurement methods have their own limitations and complementary advantages, combining the advantages of the two methods, while retaining the high accuracy of the direct measurement method, reduces the direct measurement method's dependence on monitoring data, is very attractive.

[0004] Integrating physical laws with data-driven approaches is a viable approach to reducing monitoring data requirements. Physics-informed neural networks (PINNs) are a representative example of this approach. PINNs set the parameters in the governing equations as unknown, trainable variables and then identify these parameters based on partial monitoring data. PINNs can accurately identify unknown parameters using a small amount of monitoring data, thanks to the constraints imposed by physical laws on the solution space. Therefore, PINNs offer a new approach for accurately identifying bridge damping ratios in situations where data is scarce and measurement points are limited. As a physically constrained machine learning method, PINNs have achieved relatively mature research results in various fields, including fluid mechanics and solid mechanics. However, despite the relative maturity of its underlying theory, its application in structural health monitoring (SHM) requires continued expansion. Yin et al. found that directly applying PINNs to identify bridge damping ratios obscured information about the inertia and damping terms due to the excessively large coefficients of the stiffness terms in the vibration differential equations. To address this issue, they proposed a physics-informed neural network guided by function approximation (FA-PINN) that addresses this issue at the network architecture level. However, the stringent requirements of FA-PINN on data quality and fitting accuracy limit its application in actual engineering noise environments. Therefore, there is an urgent need to further explore and develop adaptable PINNs-based bridge damping ratio identification methods. Summary of the Invention

[0005] In response to the above-mentioned technical problems to be solved, the present invention provides a bridge damping ratio identification method based on PINNs under the dimensionless control equation, which implicitly incorporates the bridge stiffness and line density into the dimensionless parameters, fundamentally eliminating the order of magnitude differences between the coefficients of each differential term, and realizing the identification of the bridge damping ratio based on the PINNs method.

[0006] In order to solve the above technical problems, the technical solution proposed by the present invention is: A bridge damping ratio identification method based on PINNs under dimensionless governing equations includes the following steps: The input variables are mapped through the network to obtain the output displacement; The output displacement is automatically differentiated to obtain the differential terms; Dimensionless treatment of parameters and differential terms, and construction of dimensionless free vibration differential equations with unknown damping ratios; 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.

[0007] As a further improvement of the above technical solution: Preferably, the input variables are first set with random seeds before network mapping, and the training grid points of unsupervised learning are divided according to the bridge length and test time. The input variables corresponding to each grid point are ( x , t ), x is a spatial variable, t is a time variable, and the input variable is nonlinearly mapped to obtain the network output w , w Represents the displacement perpendicular to the longitudinal direction of the bridge.

[0008] Preferably, the automatic differentiation generates parameter gradients by computing a graph: ;

[0009] in, is the matrix right multiplication operator, is the loss function, k represents the layer index in the computation graph, represents the parameter vector of PINNs, v k For the k Layer intermediate variables; represents the output variable of the i-th layer, is the i-th layer function f right The Jacobian matrix of Transpose it;

[0010] The gradient is formalized as a discrete integral over the computation graph path space: ; in, Indicates that the parameter To the loss function The set of all back-propagation paths of For path Multiply the Jacobian matrices on , Indicates the j The output variable of the layer, Represents the parent node index, identifying the input source layer of the current operator. Represents the child node index, identifying the output layer of the current operator.

[0011] Preferably, the dimensionless free vibration differential equation is: ;

[0012] in, is the dimensionless output displacement, n represents the modal order, represents the dimensionless spatial variable, represents the dimensionless time variable.

[0013] Preferably, the total loss function is expressed as: ; Where, 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, Indicates 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 region, represents the residual point in the physical domain, represents the residual point in the data domain, The vibration acceleration signal collected by the sensor serves as supervision data.

[0014] Preferably, the network convergence mechanism is to first formulate a 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 parameter stability analysis.

[0015] Preferably, during the model training process, the damping ratio parameter of the network inversion is With the number of iterations s Continuously updated to form a time series ,in N is the total number of iterations; the sliding time window is used to calculate the mean and standard deviation of the damping ratio parameter: ;

[0016] in, Indicates the mean value of the results under the current iteration steps, represents the length of the sliding time window, Indicates the damping ratio identification value under the current iteration step, Indicates the standard deviation of the results under the current number of iterations. The network converges when both the minimum number of iterations and the stability of the inversion parameters are met. Set the minimum number of iterations. When the minimum number of iterations is reached, the standard deviation convergence condition is activated, and the model's learning of various types of information is saturated. When the convergence condition is met, the training is terminated and the final value is taken. As the network identification result.

[0017] The bridge damping ratio identification method based on PINNs under dimensionless control equations provided by the present invention has the following advantages over the prior art: The bridge damping ratio identification method of the present invention, based on PINNs under dimensionless governing equations, accurately identifies the bridge damping ratio using a small amount of vibration data, thereby improving the applicability and efficiency of bridge health monitoring. The bridge damping ratio identification method of the present invention first addresses the physical residual imbalance problem of PINNs identification of bridge damping ratios by nondimensionalizing the governing equations. Secondly, based on numerical simulations, it analyzes the effects of sensor configuration, bridge parameters, and noise levels on the accuracy of damping ratio identification. The bridge damping ratio identification method of the present invention can accurately identify the first-order damping ratio of a bridge under data-scarce conditions. Furthermore, the method exhibits strong robustness to sensor configuration, bridge parameters, and noise interference with an SNR ≥ 20 dB. Furthermore, the engineering applicability of the proposed method was verified through field bridge tests. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 This is a flow chart of the bridge damping ratio identification method of the present invention.

[0019] Figure 2 Schematic diagram of the on-site testing scheme of the present invention.

[0020] Figure 3 This is a diagram of the original signal and the filtered signal used in the experimental verification of the present invention.

[0021] Figure 4 This is a diagram of the test results in the experimental verification of the present invention. DETAILED DESCRIPTION

[0022] The following is a detailed description of the specific embodiments of the present invention. It should be understood that the specific embodiments described herein are only used to illustrate and explain the present invention and are not intended to limit the present invention.

[0023] like Figure 1 As shown, the present invention is based on the bridge damping ratio identification method of PINNs under the dimensionless control equation, which includes the following steps: Step S1: Input variables are mapped via the network to obtain output variables.

[0024] First, set up a random seed and divide the unsupervised learning training grid points according to the bridge length and test time. The input variable corresponding to each grid point is ( x , t ), x is a spatial variable, t is a time variable, and the input variable is nonlinearly mapped to obtain the network output w , w Represents the displacement perpendicular to the longitudinal direction of the bridge.

[0025] The nonlinear mapping is completed by a 4-layer fully connected feedforward neural network with 64-128-128-64 nodes, and each layer uses the hyperbolic tangent activation function.

[0026] Step S2, output displacement w The differential terms are obtained through automatic differentiation (AD).

[0027] The displacement obtained by nonlinear mapping w Differentiation calculations are required under AD. This is because the displacement solution to the beam's free vibration equation is implicit when solving an inverse problem with unknown parameters, and the differential terms required for the governing equations cannot be obtained through analytical differentiation. Furthermore, numerical differentiation calculations are subject to truncation and roundoff errors. Therefore, AD, which is suitable for implicit problems and offers high precision, is the optimal choice for differential calculations in the method of the present invention.

[0028] AD achieves accurate propagation of derivatives through the topological structure of the computation graph. Let the parameter vector of PINNs be ,in d is the total dimension of trainable parameters, is a set of real numbers, the objective function ,in m The dimension of the output physical quantity, such as spatial coordinates, time, or the joint dimension of multi-dimensional physical fields. F Can be decomposed into K A composite map of differentiable operators: (1) in, represents function composition, Indicates the K the (last) differentiable operator, Represents the objective function F The first differentiable operator, processing the parameter vector and generate initial intermediate output, Indicates the i A differentiable operator, represents the parameter vector of PINNs. Indicates the i Local mapping of layer operators, For the i The output dimension of the layer , Indicates the i The output variable of the layer. The forward propagation is generated recursively by the local mapping: (2) in, is the parent node set of the node, that is, the calculation v iRequired direct input variables; E Represents the edge set in the computational graph, which is used to describe the data dependency between nodes (variables or operators); Indicates the j The output variable of the layer, Represents the parent node index, identifying the input source layer of the current operator. Indicates the child node index, identifying the output layer of the current operator. In the reverse differential propagation mechanism, the accompanying variable is defined ,in is the loss function. Backward differential propagation follows the reverse accumulation rule of the chain rule: (3) in, For the i Layer Function f right The Jacobian matrix, its transpose Ensure that matrix multiplication dimensions are compatible ( ). For nodes v j The set of child nodes of v j The successor node of the parent node.

[0029] AD generates parameter gradients through the computation graph: (4) in, is the matrix right multiplication operator. Furthermore, the gradient can be formalized as a discrete integral on the computation graph path space: (5) in, Indicates that the parameter To the loss function The set of all back-propagation paths of For path The Jacobian matrix multiplication on the path is the gradient from the parameter to loss complete transmission route.

[0030] This invention utilizes a reverse-mode approach to efficiently compute the gradients of high-dimensional outputs (various losses) with respect to low-dimensional parameters (damping ratios). Furthermore, within the adopted TensorFlow framework, the 'GradientTape' mechanism is not only naturally adapted to reverse-mode AD but also automatically tracks and optimizes all trainable parameters in the computation graph, simplifying the implementation of parameter inversion under complex physical constraints.

[0031] Step S3: dimensionlessly process the parameters and differential terms to construct a dimensionless free vibration differential equation containing an unknown damping ratio.

[0032] When PINNs identify unknown parameters, they need to embed the governing equations of the physical problem to be solved for unsupervised learning. This paper identifies the bridge damping ratio in the context of free vibration of a simply supported beam. The differential equations of motion of the beam under each mode are expressed as: (6) Where, EI is the flexural stiffness of the beam; is the linear density of the beam; For the n The viscous damping coefficient per unit length of the beam under the first mode is, Represents the beam in space variable x and time variables t The lateral vibration displacement.

[0033] w The direction is perpendicular to the longitudinal direction of the bridge, that is, the lateral vibration displacement. The properties of the two are the same, the only difference is that, w It refers to the displacement of this property, and w ( x , t ) refers to the specific solution, in the space variable x and time variables t Displacement in coordinates, i.e. the longitudinal direction of the bridge x In time t The lateral vibration displacement.

[0034] The partial differential equation PDE loss constructed by formula (6) is: (7) in, Represents the residual point in the physical domain.

[0035] In actual bridges, the EI value is usually higher than and The loss of the PDE is five to six orders of magnitude, which makes it particularly sensitive to the calculation of the fourth-order partial derivative, leading to problems such as loss value explosion and the network's inability to correctly identify unknown parameters. Therefore, in order to solve the problem of large differences in the coefficients of the differential terms in equation (6), the present invention performs dimensionless processing on the free vibration equation of the beam.

[0036] The input of the model x 、 t and output w ( x , t )Dimensionless: (8) Where: 、 and They are dimensionless spatial variables, dimensionless time variables and dimensionless output displacement respectively; L is the spatial scale, which is equal to the bridge span. Normalized to the range of 0 to 1; T is the time scale; D The displacement scale is set to the maximum value of the vibration response, and the displacement is normalized to the range of -1 to 1. D The dimensionless displacement used for network calculations is scaled. It should be noted that there are no strict restrictions on the displacement scale; its core purpose is to enhance weak signal characteristics and reduce numerical calculation errors by scaling the bridge vibration response. Therefore, in practical engineering, the displacement scale can be flexibly set to optimize network identification results.

[0037] The partial differential terms are transformed into:

[0038] (9) Substituting formula (9) into formula (6) and simplifying it, we can get: (10) From formula (10), we can see that the time scale T must include In order to eliminate the influence of the fourth-order partial derivative coefficient, we set , , is the natural angular frequency of the beam in the nth mode. The damping ratio of each mode satisfies ,in, express n The bridge damping ratio under the first-order mode. Therefore, formula (10) is further simplified to: (11) in, is the dimensionless output displacement, n represents the modal order.

[0039] Equation (11) is the dimensionless free vibration differential equation of the beam. Embedding this equation into PINNs as a physical law guiding network training allows identification of the bridge damping ratio. The proposed model does not require information about bridge stiffness and line density.

[0040] Step S4, calculating the loss.

[0041] (1) Calculate the physical residual based on equation (11) obtained in step S3, and further obtain the physical loss.

[0042] (2) According to the simply supported boundary conditions, the displacement and bending moment at both ends of the beam are 0. Combining the dimensionless parameters and differential terms in step S3, the boundary residual is calculated to further obtain the boundary loss.

[0043] (3) Acceleration sensors are placed on the beam to collect the free vibration response signal of the beam, and the collected acceleration response is used as the data constraint of the network for supervised learning.

[0044] In order to calculate the data residual, the output displacement obtained in step S1 needs to be w Take the second derivative with respect to time: , and then compared with the collected acceleration signal to further obtain the data loss.

[0045] In this embodiment, high-precision first-order damping ratios can be identified using at least two accelerometers. The sensor layout can be selected as follows: the midpoint of the bridge span + any other point. Based on the spatial symmetry of the acceleration vibration response under the first-order mode, the response data is mirrored at symmetrical locations of non-midpoint acquisition points. This means that supervised data from three measurement points is obtained using two sensors for network training.

[0046] Step S5: construct a total loss function.

[0047] The total loss function is constructed based on the various sub-losses obtained in step S4. The total loss function of the model in the present invention is expressed as: (12) Where, 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, Indicates 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 region, represents the residual point in the physical domain, represents the residual point in the data domain, The vibration acceleration signal collected by the sensor serves as the supervisory data. To align the network output with the sensor data, the second-order time derivative of the network output is required.

[0048] 、 and The weight coefficients corresponding to the three loss functions are as follows. Appropriate loss function weights can ensure that the network fully learns the characteristics of each field. However, the manual parameter adjustment process of the weight coefficients is time-consuming and lacks generalization, making it difficult to adapt to the needs of different scenarios. Therefore, the loss function weight coefficients of the model of the present invention are set as adaptive weight coefficients. Define the trainable logarithmic weight parameters 、 and , the exponential transformation is used to ensure the non-negativity of the weights and prevent gradient explosion. The weight coefficient of the loss function can be expressed as: (13) 、 and The initial values ​​are all set to 0, corresponding to the initial weight .

[0049] During the network training phase, the logarithmic weight coefficients are optimized via backpropagation: (14) The model can achieve a balance in the magnitude of each loss term by adaptively adjusting the weight coefficient of the loss function. Specifically, if the gradient of a loss term is dominant, its weight coefficient will automatically decrease to suppress the dominant effect of the loss, and vice versa. The learning rate hyperparameter used in the model of the present invention is .

[0050] Step S6: Update network parameters and trainable variables, and determine the training endpoint based on the network convergence mechanism.

[0051] The network parameters and the damping ratio to be identified are updated in the direction of reducing the total loss, and then the training is terminated based on the network convergence mechanism.

[0052] Reasonable training termination conditions can ensure sufficient model convergence while controlling computing resources. This paper designs a hybrid convergence determination mechanism that integrates qualitative and quantitative characteristics. This mechanism first establishes a minimum number of iterations to ensure that the network escapes the local optimal solution, and then uses parameter stability analysis to determine the network's termination.

[0053] During the model training process, the damping ratio parameter of the network inversion With the number of iterations s Continuously updated to form a time series ,in N is the total number of iterations. Using a sliding time window (length W = 1000) to calculate the mean and standard deviation of the damping ratio parameter: (15) in, Indicates the mean value of the results under the current iteration steps, represents the length of the sliding time window, Indicates the damping ratio identification value under the current iteration step, Indicates the standard deviation of the result under the current iteration step. When the minimum iteration step and the stability of the inversion parameters are met at the same time, the network converges. In this embodiment, the minimum iteration step satisfies The minimum number of iterations is manually selected through a small amount of simple pre-training. Simple pre-training can be to conduct a certain number of training tests under set parameters to determine reasonable and stable related indicators.

[0054] The stability of the inversion parameters is determined by the standard deviation of the final 1000-step inversion results after the minimum number of iterations, that is, When the minimum number of iterations is reached, the standard deviation converges to is activated to ensure that the model has saturated all kinds of information. When the convergence conditions are met, the training is terminated and the final As the network identification result.

[0055] Experimental verification A large bridge was selected as a field test bridge. The target bridge is a 13-span, two-lane, simply supported concrete beam bridge with a total length of 331.46 meters. Each span is 25.0 meters long and 8.5 meters wide. With no traffic loads on the bridge deck and no irregular vibration sources near the bridge site, the DH5907N dynamic signal analysis system was used to measure the micro-vibration response of the bridge span structure caused by random loads such as wind load, ground pulsation, and water flow at the bridge site. Parameters such as the natural frequency and damping ratio of the structure were measured. The analysis system has an effective measurement frequency range of 0-39.6 Hz, which covers the testing requirements of the structural fundamental frequency. The modal parameters of the structure were extracted using the cross-power spectrum function between each response measurement point and a reference point. The natural frequency of the bridge's first-order mode was determined to be 6.25 Hz, and the damping ratio was 3.21%.

[0056] The method of the present invention is based on the free vibration response of a simply supported beam. Figure 2 As shown in the figure, two vehicles are driven side by side to excite the bridge vibration. The process is divided into three stages: Figure 2 Figure 1 shows the on-site test plan. (a) shows the loading plan, (b) shows a photo of the loading vehicle, and (c) shows a schematic diagram of the loading vehicle. The first phase is the adjustment phase, in which the vehicle accelerates to 30-40 km / h before entering the test beam section. The second phase is the excitation phase, in which the vehicle passes over the test beam section to excite bridge vibration. The third phase is the test phase. After the vehicle leaves the test beam section, the free vibration response of the beam is collected using three accelerometers located at 1 / 4, 1 / 2, and 3 / 4 of the span.

[0057] Based on the first-order natural frequency of the bridge, a zero-phase Butterworth bandpass filter with a bandwidth of 3 Hz is used to extract the first-order modal acceleration response. The original acceleration response and the extracted response are shown in Figure 2. Figure 3 As shown in the figure, given the small amplitude of the bridge vibration response caused by vehicle excitation, a smaller displacement scale can be manually selected to avoid numerical instability during training. Regarding data acquisition, only valid signals from the 1-second free vibration decay segment are required to complete the modeling. In this experiment, the vibration response in the 0.5-1.5 second range was selected as the data constraint for network training.

[0058] Table 1 shows the results of training using data from different measurement points. When the network was trained using vibration data from three measurement points, the error between the recognition results and the target bridge test data was approximately 1%. It is worth noting that in sequence 2, the data at 0.75L was taken from the data at 0.25L. In sequence 2, the model only used data from two measurement points (the data at 0.25L was also used at 0.75L). This is because the acceleration response of a simply supported beam in the first mode is symmetrically distributed along the longitudinal direction. Based on this characteristic, vibration data from symmetrical measurement points can be complemented through mirror mapping, thereby enhancing the data constraint effect. In actual engineering, the vibration responses at symmetrical locations may differ due to factors such as noise and filtering. Using this mirror mapping method not only reduces the number of sensor deployments but also helps to learn the symmetrical characteristics of the vibration system.

[0059] Table 1 Test results

[0060] Experimental results show that the method of the present invention only requires two sensors to effectively identify the first-order damping ratio of a simply supported beam.

[0061] The above examples are merely preferred embodiments of the present invention and are not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, they are not intended to limit the present invention. Therefore, any simple modifications, equivalent variations, and modifications to the above examples that do not depart from the technical solution of the present invention and are based on the technical essence of the present invention shall fall within the scope of protection of the technical solution of the present invention.

Claims

1. A bridge damping ratio identification method based on PINNs under dimensionless governing equations, characterized by: The following steps are involved: The input variables are mapped through the network to obtain the output displacement; The output displacement is automatically differentiated to obtain the differential terms; Dimensionless treatment of parameters and differential terms, and construction of dimensionless free vibration differential equations with unknown damping ratios; 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.

2. The bridge damping ratio identification method based on PINNs under dimensionless control equations according to claim 1 is characterized in that: The input variables are first set with random seeds before network mapping, and the training grid points of unsupervised learning are divided according to the bridge length and test time. The input variables corresponding to each grid point are ( x , t ), x is a spatial variable, t is a time variable, and the input variable is nonlinearly mapped to obtain the network output w , w Represents the displacement perpendicular to the longitudinal direction of the bridge.

3. The bridge damping ratio identification method based on PINNs under dimensionless control equations according to claim 1 is characterized in that: The automatic differentiation generates parameter gradients through the computation graph: ； in, is the matrix right multiplication operator, is the loss function, k represents the layer index in the computation graph, represents the parameter vector of PINNs, v k For the k Layer intermediate variables; represents the output variable of the i-th layer, is the i-th layer function f right The Jacobian matrix of Transpose it; The gradient is formalized as a discrete integral over the computation graph path space: ； in, Indicates that the parameter To the loss function The set of all back-propagation paths of For path Multiply the Jacobian matrices on , Indicates the j The output variable of the layer, Represents the parent node index, identifying the input source layer of the current operator. 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 control equations according to claim 1 is characterized in that: The dimensionless free vibration differential equation is: ； in, is the dimensionless output displacement, n represents the modal order, represents the dimensionless spatial variable, represents the dimensionless time variable.

5. The bridge damping ratio identification method based on PINNs under dimensionless control equations according to claim 4 is characterized in that: The total loss function is expressed as: ； Where, 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, Indicates 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 region, represents the residual point in the physical domain, represents the residual point in the data domain, The vibration acceleration signal collected by the sensor serves as supervision data.

6. The bridge damping ratio identification method based on PINNs under dimensionless control equations according to claim 5 is characterized in that: The network convergence mechanism is to first formulate a 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 parameter stability analysis.

7. The bridge damping ratio identification method based on PINNs under dimensionless control equations according to claim 6 is characterized in that: During the model training process, the damping ratio parameter of the network inversion With the number of iterations s Continuously updated to form a time series ,in N is the total number of iterations; the sliding time window is used to calculate the mean and standard deviation of the damping ratio parameter: ； in, Indicates the mean value of the results under the current iteration steps, represents the length of the sliding time window, Indicates the damping ratio identification value under the current iteration step, Indicates the standard deviation of the results under the current number of iterations; the network converges when both the minimum number of iterations and the stability of the inversion parameters are met; set the minimum number of iterations. When the minimum number of iterations is reached, the standard deviation convergence condition is activated, and the model's learning of various types of information is saturated. When the convergence condition is met, the training is terminated and the final value is taken. As the network identification result.

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