Bridge random vibration analysis method based on mechanism enhanced physical information neural network

Abstract

The invention belongs to the field of bridge structure aseismic design and safety evaluation, and relates to a mechanism enhanced physical information neural network-based bridge random vibration analysis method, which comprises the following steps of: establishing a bridge structure motion equation configured with a nonlinear tuning inerter damper; a mechanism enhanced physical information neural network model comprising a three-head prediction module and a three-state gating mechanism is constructed, and the three-head prediction module predicts and decouples nonlinear restoring force into three independent subtasks of acceleration, speed and symbol. The three-state gating mechanism divides time steps into a security domain, a crossing domain and a singular point zero point domain according to the speed and adopts a differential supervision strategy; generating a sample based on a random near-fault seismic motion model to train the network; the trained mechanism enhanced physical information neural network is deeply fused with a direct probability integral method, and the random vibration response and the dynamic reliability of the bridge structure are efficiently and accurately calculated. The invention provides an efficient and reliable analysis tool for anti-seismic safety assessment of a bridge structure.

Description

Technical Field

[0001] This invention belongs to the field of seismic design and safety assessment of bridge structures, and relates to a method for analyzing random vibrations of bridges based on a mechanism-enhanced physical information neural network. Background Technology

[0002] As a crucial component of modern transportation infrastructure, the seismic safety of bridges directly impacts the reliable operation of transportation lifelines. Particularly in near-fault areas, seismic motions exhibit significant random characteristics and impulse effects, posing a severe threat to bridge structural safety. Traditional deterministic seismic response analysis methods, based on single or limited seismic motion records, cannot fully reflect the impact of the randomness of seismic motions (especially near-fault motions) on bridge response, making it difficult to accurately assess bridge seismic performance and scientifically quantify safety risks. Therefore, conducting random vibration analysis to comprehensively evaluate the seismic performance of bridges under random near-fault motions has become an urgent need.

[0003] Existing methods for analyzing the random vibration of bridge structures mainly include Monte Carlo Simulation (MCS) and Quasi-Monte Carlo Simulation (QMCS). These methods all suffer from low computational efficiency when performing random vibration analysis of bridges. Specifically, to accurately obtain the probability distribution characteristics of the structural response, repeated nonlinear time-history analyses of hundreds to tens of thousands of seismic motion samples are typically required. Each analysis necessitates iterative solutions to the dynamic equations at multiple time steps, resulting in extremely high computational costs. This severely restricts the application of existing random vibration analysis methods in bridge engineering.

[0004] To address the computational inefficiency of existing stochastic vibration analysis methods, the rapidly developing deep learning technology has provided a new solution for the efficient prediction of structural seismic responses in recent years. Among these, Physical-Informed Neural Networks (PINNs) have achieved some success in predicting the seismic responses of linear or smoothly nonlinear structures by embedding physical constraints into the neural network architecture or loss function. However, existing PINN methods face technical challenges for bridge structures equipped with strongly nonlinear damping devices (such as nonlinear tuned inertial dampers (NTIDs) containing Coulomb friction). Specifically, the strong nonlinearity and discontinuity of the sign function in Coulomb friction make it difficult for neural networks to accurately learn and predict its mechanical behavior. Although some methods attempt to approximate the sign function with a smooth function to achieve differentiability, this approach introduces significant prediction errors. Therefore, how to construct an efficient PINN model that can accurately capture the strongly nonlinear and discontinuous mechanical characteristics such as Coulomb friction, and combine it with stochastic vibration analysis theory to achieve efficient and accurate stochastic seismic response and dynamic reliability calculation of bridge structures under stochastic near-fault earthquake excitation, is a key technical problem that urgently needs to be solved in the field of bridge seismic design. Summary of the Invention

[0005] To address the aforementioned problems, this invention proposes a method for analyzing the stochastic vibration of bridges based on a Mechanism-Enhanced Physical Information Neural Network (ME-PINN). This method is applicable to the stochastic dynamic analysis of bridge structures equipped with nonlinear tuned inertial dampers (NTIDs) under stochastic near-fault ground motion excitation. This invention integrates a framework that combines a ME-PINN with direct probability integration, thereby achieving efficient and accurate stochastic vibration analysis of bridge structures equipped with strongly nonlinear damping devices. The main components include: establishing the equations of motion for the bridge structure equipped with NTIDs; constructing an ME-PINN model including modules such as a three-head prediction module and a three-state gating mechanism; training the ME-PINN model using samples generated from the stochastic near-fault ground motion model; and using direct probability integration in conjunction with the trained ME-PINN model to accurately and efficiently calculate the structure's stochastic vibration response and dynamic reliability.

[0006] ME-PINN effectively addresses the inherent limitations of traditional PINN in handling strongly nonlinear problems by constructing a network architecture that includes a three-head prediction module and a three-state gating mechanism. This enables accurate and efficient prediction of the seismic time history response of bridge structures equipped with strongly nonlinear damping devices. The direct probability integral method derives the probability density integral equation of the random response based on the principle of probability conservation, and then solves the probability density integral equation and corresponding formulas to accurately calculate the random vibration response and dynamic reliability. This invention deeply integrates the mechanism-enhanced physical information neural network with the direct probability integral method, using the mechanism-enhanced physical information neural network to quickly predict the representative seismic time history response, ultimately achieving efficient and accurate calculation of the random vibration response and dynamic reliability of bridge structures equipped with strongly nonlinear damping devices.

[0007] The technical solution of this invention is as follows: The method for analyzing random vibration of bridges based on mechanism-enhanced physical information neural networks comprises the following steps: Step (1) Establish a finite element model of a bridge structure with a nonlinear tuned capacitive damper (NTID) and derive the equation of motion of the bridge structure containing NTID based on the finite element model. Step (2) Construct a mechanism-enhanced physical information neural network model (ME-PINN) based on the bridge structure motion equations containing NTID. ME-PINN includes a TCN-LSTM feature extraction module, a three-head prediction module, a physical assembly module, a single convolutional layer module (SCL), and a loss function module with a three-state gating mechanism; The forward propagation path sequentially passes through the TCN module to extract multi-scale temporal features, the LSTM module to capture long-term temporal dependencies, the three-head prediction module to decouple the NTID nonlinear restoring force prediction into three independent sub-tasks: acceleration prediction, velocity prediction, and sign prediction, the physical assembly module to assemble the complete restoring force time history according to the NTID constitutive relation, and the SCL module to calculate the bridge global state vector through discrete convolution operation. The backpropagation path optimizes network parameters through a loss function module that includes a three-state gating mechanism. The loss function module consists of two parts: basic physical loss and symbolic loss. The three-state gating mechanism divides the time step into three physical regions—the safe region, the traversal region, and the singularity zero-point region—based on the velocity magnitude and adopts a differentiated supervision strategy.

[0008] Step (3) Generate ground motion samples based on the random near-fault ground motion model to train the ME-PINN model; Spectral matching random near-fault ground motion models were used to generate slip-type and forward-direction pulse ground motion samples; The generated seismic motion samples are divided according to the ratio of training set to validation set; The Adam optimizer was used to train the ME-PINN model, and the convergence of the model was judged by the training loss and validation loss curves. After training, the model prediction accuracy was verified using actual recorded near-fault ground motion and spectral matching ground motion samples.

[0009] Step (4) After completing step (3), the direct probability integral method is combined with the trained ME-PINN model to efficiently and accurately calculate the stochastic seismic response and dynamic reliability results of the bridge structure.

[0010] The beneficial effects of this invention are: The mechanism-enhanced physical information neural network proposed in this invention effectively solves the problems of gradient vanishing and low prediction accuracy of traditional physical information neural networks when dealing with strong nonlinear discontinuous characteristics such as Coulomb friction by constructing modules such as a three-head prediction module and a three-state gating mechanism. It achieves accurate and efficient prediction of the seismic time history response of bridge structures equipped with strong nonlinear damping devices. This invention deeply integrates the mechanism-enhanced physical information neural network with the direct probability integral method, which significantly improves the computational efficiency of random vibration analysis. Compared with traditional methods, the calculation time is reduced from several hours to minutes, realizing efficient and accurate calculation of the random vibration response and dynamic reliability of bridge structures equipped with strong nonlinear damping devices. The efficient computational framework constructed in this invention, which integrates a mechanism-enhanced physical information neural network and a direct probability integral method, provides an intelligent analysis tool for seismic safety assessment and seismic design of bridge structures equipped with strong nonlinear damping devices. Attached Figure Description

[0011] Figure 1 This is a flowchart illustrating the implementation of the bridge random vibration analysis method based on mechanism-enhanced physical information neural network proposed in this invention.

[0012] Figure 2 This is a schematic diagram of a finite element model of a single-tower cable-stayed bridge with a nonlinear tuned inertial capacitive damper provided in an embodiment of the present invention.

[0013] Figure 3 This is a schematic diagram of the mechanism-enhanced physical information neural network architecture proposed in this invention.

[0014] Figure 4 The diagram shows representative near-fault slip-slip ground motions, where (a) to (c) are three representative samples generated by a random near-fault slip-slip ground motion model based on response spectrum matching.

[0015] Figure 5The diagram shows representative near-fault forward-direction ground motions, where (a) to (c) are three representative samples generated by a random near-fault forward-direction ground motion model based on response spectrum matching.

[0016] Figure 6 The training and validation loss curves of the mechanism-enhanced physical information neural network provided in the embodiments of the present invention.

[0017] Figure 7 The above is a comparison of the time history predictions of the nonlinear restoring force of NTID, where (a) to (d) are the prediction results under seismic excitation of TCU049-E, TCU049-V, TCU056-E and TCU102-V, respectively.

[0018] Figure 8 The following is a comparison of the earthquake time history response predictions for the main beam, where (a) to (d) show the predicted displacement, velocity, and acceleration results under seismic excitation by TCU049-E, TCU049-V, TCU056-E, and TCU102-V, respectively.

[0019] Figure 9 The application iterative sequence sampling strategy provided in this embodiment of the invention determines the convergence curve of the optimal number of representative points.

[0020] Figure 10 The figures are comparison diagrams of the extreme response probability density functions of the main girder of the single-tower cable-stayed bridge provided in the embodiments of the present invention, wherein Figures (a) to (c) are the probability density functions of the extreme response of the main girder displacement, velocity and acceleration, respectively.

[0021] Figure 11 The above is a comparison of the time-varying mean and standard deviation curves of the main girder response of a single-tower cable-stayed bridge provided in an embodiment of the present invention, where (a) ~ (c) are the time-varying mean and standard deviation of the displacement, velocity, and acceleration response of the main girder, respectively.

[0022] Figure 12 The dynamic reliability curve of the lateral displacement angle at the top of the tower of a single-tower cable-stayed bridge provided in the embodiment of the present invention. Detailed Implementation

[0023] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0024] Figure 1 The flowchart of the efficient random vibration analysis method for bridges based on mechanism-enhanced physical information neural networks of the present invention is shown below. The specific implementation steps are as follows.

[0025] Step (1): Establish a finite element model of the bridge structure with nonlinear tuned capacitive damper NTID, and derive the equation of motion of the bridge structure containing NTID based on the finite element model.

[0026] The purpose of this step is to obtain the mass matrix M, stiffness matrix K, and damping matrix C of the bridge structure in order to establish the SCL module of the ME-PINN model. The bridge structure system matrix can be obtained using, but is not limited to, the following methods: (a) establishing a finite element model of the bridge structure and extracting the system matrix; (b) directly calculating the system matrix based on existing bridge design parameters; (c) obtaining the bridge structure system matrix through inversion from measured data. In this invention, a bridge structure model with a nonlinear tuned capacitive damper (NTID) is established using the finite element method, such as... Figure 2 As shown. Based on this finite element model, the equations of motion for a bridge structure with NTID configuration under random near-fault ground motion were established, which can be expressed as: (1) Where M, C, and K represent the mass matrix, damping matrix, and stiffness matrix of the bridge structure excluding NTID, respectively; U( t ), , These represent the displacement, velocity, and acceleration vectors of the bridge structure, respectively. denoted as random seismic excitation, where θ is a random vector characterizing the randomness of near-fault earthquakes; and These represent the unit influence vectors at the location where NTID is applied and the location of seismic excitation, respectively. NTID , , These represent the relative displacement, velocity, and acceleration between the NTID connection nodes, respectively. Represents the state vector of NTID. This indicates the nonlinear restoring force provided by the NTID device; , , These represent the coefficient of inertia, the coefficient of viscous friction, and the amplitude of Coulomb friction, respectively. k This indicates the stiffness of the spring in the NTID device. c The linear damping coefficient of NTID is represented; sign(·) is the sign function.

[0027] Step (2): Construct a mechanism-enhanced physical information neural network model.

[0028] The mechanism-enhanced physical information neural network model adopts a dual-path architecture design, including a forward propagation path and a backward propagation path. Its complete architecture is as follows: Figure 3As shown, the model consists of five core functional modules: a TCN-LSTM feature extraction module, a three-head prediction module, a physical assembly module, an SCL module, and a loss function module with a three-state gating mechanism for network training. The first four modules form the forward propagation path for rapid prediction of seismic responses; the loss function module forms the backpropagation path to guide parameter optimization during network training.

[0029] The forward propagation path includes the following five cascaded functional modules in sequence according to the signal flow direction: The TCN feature extraction module, as the first-level module, accepts the seismic excitation time history as input. This module consists of three sequentially cascaded time series blocks. Each time series block contains two convolutional layers, a ReLU activation layer, a Dropout regularization layer (dropout rate of 0.3), and residual connections. The dilation factors of the three time series blocks are set to 1, 2, and 4 sequentially, and the number of filters is set to 64, 128, and 196 sequentially. The kernel size of each convolutional block is 3. The first time series block receives the original seismic excitation input, and its output is passed to the second time series block. The output of the second time series block is passed to the third time series block, and the output of the third time series block serves as the final output of the TCN module.

[0030] LSTM Feature Encoding Module: As the second-level module, it receives multi-scale temporal features output by the TCN feature extraction module. This module consists of three vertically stacked LSTM layers. The first LSTM layer receives the TCN output features, and its hidden state is passed to the second LSTM layer. The hidden state of the second LSTM layer is passed to the third LSTM layer, and the output of the third LSTM layer serves as the encoded temporal feature representation.

[0031] The three-head prediction module, as the third-level module, receives the encoded features output by the LSTM feature encoding module. This module contains three parallel fully connected neural network branches, namely: Acceleration prediction head: consists of two fully connected layers, with ReLU activation in the middle layer and linear activation in the output layer, predicting NTID relative acceleration. a pred ( t ); Velocity prediction head: consists of two fully connected layers, with ReLU activation in the middle layer and linear activation in the output layer, predicting NTID relative velocities. v pred ( t ); Symbolic prediction head: Contains two fully connected layers. The middle layer uses ReLU activation, and the output layer uses the tanh activation function to constrain the output to the [-1,1] interval, predicting the direction of friction force. s pred ( t).

[0032] The three prediction heads receive the same input features simultaneously and output three prediction values ​​independently and in parallel.

[0033] Physical Assembly Module: As the fourth-level module, it receives the three outputs from the three-head prediction module. This module is based on the NTID constitutive relation. F pred ( t ) = b · a pred ( t ) + c v · v pred ( t ) + f 0· s pred ( t The acceleration, velocity, and sign predictions are assembled into a complete nonlinear restoring force time history. F pred ( t ),in b The inertia coefficient, c v The coefficient of viscous friction is... f 0 represents the amplitude of the Coulomb friction force.

[0034] SCL module: As the fifth-level module, it receives the restoring force time history output by the physical assembly module. F pred ( t ) and the original seismic excitation a g ( t This module first combines the two into a global activation vector. Then, discrete convolution calculation is performed through a single-layer convolutional network, and its formula can be expressed as: (2) Among them, V i For the first i The global state vector of the bridge at each time step; F j The global activation vector has a time step index range of . j ∈ { i – m +1, i – m +2,…, i –1};A k,1 For the first k Order of influence coefficient matrix k = 1, 2,…, m ,m This represents the number of truncated terms in the influence coefficient matrix. For the influence coefficient matrix A... k,1 The calculation formula is: (3) (4) Where 0 and I represent the zero matrix and identity matrix, respectively; parameters β and γ It's Newmark- β The time integration parameters of the method are set to 0.25 and 0.5 respectively; the parameter Δ t The time step representing the seismic motion is set to 0.02 s in this invention. Q1, Q2, and T are intermediate calculation matrices, and H, S1, S2, and W are state-space transformation auxiliary matrices. a x , x = 0, 1, 2,…, 5, which are auxiliary coefficients for calculating the influence coefficient matrix. The influence coefficient matrix A is calculated using formula (4). k,1 The model remains constant during training. Based on formula (2), the global state vector V of the bridge can be calculated. i V i = [ , ] T Indicates the critical nodes of the bridge at time... t i displacement and speed Response. Based on the obtained velocity response. The acceleration response can be calculated based on the forward difference formula. ,Right now: (5) The backpropagation path optimizes network parameters through a loss function module that incorporates a three-state gating mechanism. The loss function module consists of two parts: basic physical loss and symbolic loss. The basic physical loss includes acceleration loss. L a and speed loss L v These are used to monitor the output accuracy of the acceleration prediction head and the velocity prediction head, respectively, and their calculation formulas are as follows: (6) (7) in, t i For the first i At that moment, N The total time step, and These are the acceleration and velocity values ​​calculated by the SCL module, respectively. To address the challenge of velocity sign prediction, the three-state gating mechanism adjusts the velocity magnitude... The time step is divided into three physical regions and a differentiated supervision strategy is adopted: security domain ( m / s) direct supervised symbol prediction; traversing domain ( m / s) using acceleration symbols as physical information to guide symbolic prediction; singularity zero-point domain ( (m / s) Soft constraints are used to allow for a certain prediction tolerance. The symbolic losses for the three regions are as follows: (8) (9) (10) in, L s,safe , L s,cross , L s,zero The symbolic losses are respectively for the safe domain, the traversal domain, and the singularity zero-point domain. v 0=10 -2 m / s is the normalization parameter. v soft = 10 -3 m / s is the soft constraint tolerance parameter. M safe , M cross , M zero These are binary masks (with values ​​of 0 or 1) for the safe region, the crossing region, and the singularity / zero region, respectively. The total symbol loss is... L s = L s,safe + L s,cross + L s,zero Ultimately, the total loss function is optimized through weighted aggregation to achieve the overall model optimization: (11) in, ω a , ω v , ω s These are the weighting coefficients for acceleration loss, velocity loss, and symbol loss, respectively.

[0035] Step (3): Generate training samples based on the spectral matching random near-fault ground motion model, train the mechanism-enhanced physical information neural network, and verify the model accuracy through deterministic earthquake time history response prediction.

[0036] Two spectral-matched stochastic near-fault ground motion models were used to generate fling-step and forward-directivity pulse ground motion samples, respectively, for a total of 800 samples (400 fling-step and 400 forward-directivity). The acceleration response spectra of the generated ground motion samples are consistent with the target design response spectrum, with a duration of 30 s and a time step of 0.02 s. Representative ground motion samples are shown below. Figure 4 and Figure 5 As shown.

[0037] The generated seismic motion samples were divided into training and validation sets in a 7:3 ratio. The Adam optimizer was used for training, with a learning rate of 0.001, a batch size of 32, and a training period of 500 epochs. The weight coefficients of the loss function were set to... ω a = 1.0、 ω v = 1.0、 ω s = 2.0. The training and validation loss curves are as follows: Figure 6 As shown, the two curves stabilize and remain consistent after about 250 epochs, indicating that the model has reached full convergence and there is no overfitting.

[0038] After training, the model's prediction accuracy was validated using actual recorded near-fault ground motion and spectral matching ground motion samples. For example... Figure 7 and Figure 8 As shown, the time histories of the NTID nonlinear restoring force predicted by the ME-PINN model, as well as the time histories of the main beam displacement, velocity, and acceleration, are compared with those of the explicit time-domain method (ETDM) and Newmark- β The calculation results obtained by the method show good agreement, verifying the prediction accuracy of the model. Furthermore, the ME-PINN model only requires 0.1 s to predict the time history response of a single earthquake, compared to 13.98 s for ETDM and Newmark- β The method takes 51.96 seconds, a significant improvement in computational efficiency.

[0039] Step (4): Deeply integrate the trained mechanism-enhanced physical information neural network with the direct probability integral method to efficiently and accurately calculate the random vibration response and dynamic reliability of the bridge structure.

[0040] This step, based on the ME-PINN model trained in step (3), integrates it as an efficient surrogate model into the Direct Probability Integral Method (DPIM) framework, replacing the traditional finite element time history analysis method to achieve efficient calculation of random vibration response and dynamic reliability. The Direct Probability Integral Method establishes a probability density integral equation based on the principle of probability conservation. Its core lies in calculating the probability density function of the random response through representative point selection and probability assignment. For the random vibration response of a bridge structure equipped with a nonlinear tuned inertial capacitive damper under random near-fault ground motion, its probability density function... p Y ( y , t The formula for calculating ) is: (12) in, y Indicates the seismic response of bridges. N tol θ represents the number of representative points required by the DPIM method. q For the first q One representative point, P q Assign a probability to the corresponding representative point. g (θ q , t ) is the physical mapping function. σ ( t ) represents the smoothing parameter.

[0041] Similarly, based on the direct probability integral method, the time-varying mean of the random vibration response E [ Y ( t Standard deviation Std [ Y ( t The formula for calculating )] can be expressed as: (13) (14) Power reliability R ( t This can be represented as: (15) in, Y thd The failure threshold For extreme response, H [·] represents the Heaviside function.

[0042] An iterative sequence sampling strategy is used to adaptively determine the optimal number of representative points. N tolThis strategy first generates a pool of candidate representative points based on the GF-difference point selection method, then iteratively adds representative points and calculates the extreme response. When the relative error of five consecutive iterations is less than 5 × 10⁻⁶, the strategy is successful. -5 The process ends at that time. (As...) Figure 9 As shown, the optimal number of representative points determined in this embodiment is 835. The ME-PINN model trained in step (3) is used as an efficient surrogate model to replace the traditional time history analysis method, and the seismic time history response of each representative point is obtained by rapidly predicting the seismic ground motion samples corresponding to the 835 representative points. g (θ q , t Based on the prediction results, the random vibration response and dynamic reliability of the bridge structure are calculated using the above formulas. The flowchart illustrating the efficient and accurate random vibration and dynamic reliability analysis of bridge structures by integrating ME-PINN and DPIM is shown below. Figure 1 As shown.

[0043] like Figure 10 and Figure 11 As shown, the extreme response probability density function and time-varying mean and standard deviation curves of the main beam calculated using the ME-PINN and DPIM fusion framework are compared with the explicit time-domain method combined with the quasi-Monte Carlo method (ETDM+QMCS, 10 3 (sample) and Newmark- β The method combines the Monte Carlo method (Newmark- β +MCS, 10 4 The results (for the sample) are in high agreement. The root mean square error of the time-varying mean and standard deviation are both less than 0.05, and the KL divergence of the probability density function is also less than 0.05, verifying the computational accuracy of the proposed method.

[0044] like Figure 12 As shown, a lateral displacement angle threshold of 0.004 at the tower top is used for dynamic reliability assessment. The calculation results of the ME-PINN and DPIM fusion framework are consistent with the benchmark method. In terms of computational efficiency, the proposed method requires only 83.5 s, compared to 10350 s for ETDM+QMCS and Newmark- β The computation time of +MCS is 519600 s, which reduces the computation time by approximately 124 times and 6221 times respectively, significantly improving the computational efficiency of random vibration analysis.

[0045] Table 1 compares the accuracy and efficiency of the ME-PINN combined with direct probability integration method provided in the embodiments of the present invention with the benchmark method.

[0046] Table 1

[0047] Reference example: High-efficiency random vibration and dynamic reliability analysis of a single-tower cable-stayed bridge equipped with a nonlinear tuned inertial capacitive damper.

[0048] Based on the mechanism-enhanced physical information neural network proposed in this invention, an efficient random vibration analysis method for bridges is presented, taking a single-tower cable-stayed bridge equipped with a nonlinear tuned inertial damper (NTID) subjected to random near-fault ground motion as an example. The bridge has a two-span arrangement, a main girder length of 460 m, and a tower height of 151.8 m. The finite element model is as follows: Figure 2 As shown. For this example, stochastic vibration analysis and dynamic reliability analysis are performed under stochastic near-fault seismic excitation. The specific calculation steps are as follows: (1) Establish the OpenSees finite element model of the single-tower cable-stayed bridge and install 3 NTIDs accordingly, with their parameters set as follows: b = 1.64×10 5 kg, f 0 = 2.76 × 10 5 N、 c v = 7.6 × 10 5 N·s / m, k = 6.32×10 5 N / m, c = 1.80×10 3 N·s / m, modal damping ratio ζ = 0.05.

[0049] (2) 800 ground motion samples were generated based on a random near-fault ground motion model using response spectrum matching. The target response spectrum of the ground motion samples was designed according to the JTG / T 2231-01-2020 standard, with a ground motion duration of 30 s and a time step of 0.02 s. Representative ground motion time history samples are shown below. Figure 4 and Figure 5 As shown.

[0050] (3) The 800 samples were divided into a training set and a validation set in a 7:3 ratio to train the ME-PINN model. The training parameters were set as follows: Adam optimizer, learning rate 0.001, batch size 32, training period 500 epochs, and loss function weights. ω a = ω v = 1.0、 ω s = 2.0. The training loss and validation loss converged after approximately 250 epochs. Figure 6 This indicates that the ME-PINN model has converged sufficiently and is not overfitted. The model accuracy is verified using actual ground motion records from the PEER database, such as... Figure 7 and Figure 8 As shown, the ME-PINN prediction results are compared with ETDM and Newmark- β The calculation results of the method are highly consistent, and a single prediction only takes 0.1 s.

[0051] (4) Using the trained ME-PINN model combined with the direct probability integral method, a rapid response prediction is made for the representative seismic time history response of a single-tower cable-stayed bridge with NTID configuration. The time-varying mean, standard deviation, and probability density function of the main girder response of the single-tower cable-stayed bridge are calculated based on equations (8)-(10). Figure 10 and Figure 11 As shown, the calculation results are combined with ETDM and QMCS (10 3 (sample) and Newmark- β Combined with MCS (10 4 The samples are highly compatible, with the root mean square error of the time-varying mean and standard deviation both less than 0.05, and the KL divergence of the PDF both less than 0.05.

[0052] (5) The lateral displacement ratio criterion at the top of the tower is adopted. Y thd = 0.004) for dynamic reliability calculation. Based on equation (11), the following results were obtained: For the displacement angle criterion, under sliding-slip, forward-direction, and non-pulse ground motion, the reliability of the single-tower cable-stayed bridge with NTID configuration is 0.452, 0.292, and 0.819, respectively, which are 12.4%, 18.2%, and 6.8% higher than that of the uncontrolled structure (0.402, 0.247, and 0.767), respectively. Figure 12 As shown.

[0053] In summary, this invention proposes an efficient random vibration analysis method for bridges based on a mechanism-enhanced physical information neural network. It presents a unified and efficient framework that deeply integrates the mechanism-enhanced physical information neural network with the direct probability integral method. By constructing a three-head prediction module and a three-state gating mechanism, it effectively solves the inherent difficulties of traditional neural networks in handling strong nonlinearity. This enables efficient and accurate calculation of the random vibration response and dynamic reliability of bridge structures equipped with strong nonlinear damping devices under random near-fault ground motion, and provides an efficient and reliable analytical tool for seismic safety assessment of bridge structures.

Claims

1. A method for analyzing the random vibration of bridges based on a mechanism-enhanced physical information neural network, characterized in that, The steps are as follows: Step (1) Establish a finite element model of a bridge structure with a nonlinear tuned capacitive damper (NTID) and derive the equation of motion of the bridge structure containing NTID based on the finite element model. Step (2) Construct a mechanism-enhanced physical information neural network model ME-PINN based on the equation of motion of the bridge structure containing NTID; ME-PINN includes a TCN-LSTM feature extraction module, a three-head prediction module, a physical assembly module, a single convolutional layer module SCL, and a loss function module with a three-state gating mechanism; The forward propagation path sequentially passes through the TCN module to extract multi-scale temporal features, the LSTM module to capture long-term temporal dependencies, the three-head prediction module to decouple the NTID nonlinear restoring force prediction into three independent sub-tasks: acceleration prediction, velocity prediction, and sign prediction, the physical assembly module to assemble the complete restoring force time history according to the NTID constitutive relation, and the SCL module to calculate the bridge global state vector through discrete convolution operation. The backpropagation path optimizes network parameters through a loss function module containing a three-state gating mechanism. The loss function module consists of two parts: basic physical loss and symbolic loss. The three-state gating mechanism divides the time step into three physical regions—safe region, crossing region, and singularity zero-point region—based on the velocity magnitude and adopts a differentiated supervision strategy. Step (3) Generate ground motion samples based on the random near-fault ground motion model to train the ME-PINN model; Spectral matching random near-fault ground motion models were used to generate slip-type and forward-direction pulse ground motion samples; The generated seismic motion samples are divided according to the ratio of training set to validation set; The Adam optimizer was used to train the ME-PINN model, and the convergence of the model was judged by the training loss and validation loss curves. After training, the model prediction accuracy was verified using actual recorded near-fault ground motion and spectral matching ground motion samples. Step (4) After completing step (3), the direct probability integral method is combined with the trained ME-PINN model to calculate the stochastic seismic response and dynamic reliability of the bridge structure.

2. The bridge stochastic vibration analysis method based on mechanism-enhanced physical information neural network according to claim 1, characterized in that, Step (1) is as follows: The mass matrix M, stiffness matrix K, and damping matrix C of the bridge structure are obtained to establish the SCL module of the ME-PINN model. A bridge structure model with a nonlinear tuned capacitive damper (NTID) is established using the finite element method. Based on this finite element model, the equations of motion for the bridge structure with NTID under random near-fault ground motion are established, as follows: (1) Where M, C, and K represent the mass matrix, damping matrix, and stiffness matrix of the bridge structure excluding NTID, respectively; U( t ), , These represent the displacement, velocity, and acceleration vectors of the bridge structure, respectively. denoted as random seismic excitation, where θ is a random vector characterizing the randomness of near-fault earthquakes; and These represent the unit influence vectors at the location where NTID is applied and the location of seismic excitation, respectively; u NTID , , These represent the relative displacement, velocity, and acceleration between the NTID connection nodes, respectively. Represents the state vector of NTID. This indicates the nonlinear restoring force provided by the NTID device; , , These represent the coefficient of inertia, the coefficient of viscous friction, and the amplitude of Coulomb friction, respectively. k This indicates the stiffness of the spring in the NTID device. c The linear damping coefficient of NTID is represented; sign(·) is the sign function.

3. The method for analyzing random vibration of bridges based on a mechanism-enhanced physical information neural network according to claim 1, characterized in that, In step (2), the forward propagation path includes the following five cascaded functional modules in sequence according to the signal flow direction: The TCN feature extraction module, as the first-level module, accepts the seismic excitation time history as input. It consists of three sequentially cascaded time series blocks. Each time series block contains two convolutional layers, a ReLU activation layer, a Dropout regularization layer, and residual connections. The dilation factors of the three time series blocks are set to 1, 2, and 4 in sequence, and the number of filters is set to 64, 128, and 196 in sequence. The kernel size of each convolutional block is 3. The first time series block receives the original seismic excitation input, and its output is passed to the second time series block. The output of the second time series block is passed to the third time series block, and the output of the third time series block is used as the final output of the TCN module. LSTM Feature Encoding Module: As the second-level module, it receives multi-scale temporal features output by the TCN feature extraction module. This module consists of three layers of LSTM networks stacked vertically. The first layer of LSTM receives the TCN output features, and its hidden state is passed to the second layer of LSTM. The hidden state of the second layer of LSTM is passed to the third layer of LSTM. The output of the third layer of LSTM is used as the encoded temporal feature representation. Three-head prediction module: As the third-level module, it receives the encoded features output by the LSTM feature encoding module; This module contains three parallel fully connected neural network branches, namely: Acceleration prediction head: consists of two fully connected layers, with ReLU activation in the middle layer and linear activation in the output layer, predicting NTID relative acceleration. a pred ( t ); Velocity prediction head: consists of two fully connected layers, with ReLU activation in the middle layer and linear activation in the output layer, predicting NTID relative velocities. v pred ( t ); Symbolic prediction head: Contains two fully connected layers. The middle layer uses ReLU activation, and the output layer uses the tanh activation function to constrain the output to the [-1,1] interval, predicting the direction of friction force. s pred ( t ); Three prediction heads simultaneously receive the same input features and independently and in parallel output three predicted values. Physical Assembly Module: As a fourth-level module, it receives the three outputs from the three-head prediction module; this module is based on the NTID constitutive relation. F pred ( t ) = b · a pred ( t ) + c v · v pred ( t ) + f 0· s pred ( t The acceleration, velocity, and sign predictions are assembled into a complete nonlinear restoring force time history. F pred ( t ),in b The inertia coefficient, c v The coefficient of viscous friction is... f 0 represents the amplitude of the Coulomb friction force; SCL module: As the fifth-level module, it receives the restoring force time history output by the physical assembly module. F pred ( t ) and the original seismic excitation a g ( t This module first combines the two into a global activation vector. Then, discrete convolution calculation is performed through a single-layer convolutional network, and its formula can be expressed as: (2) Among them, V i For the first i The global state vector of the bridge at each time step, F j The global activation vector has a time step index range of . j ∈ { i – m +1, i – m +2,…, i –1};A k,1 For the first k Order of influence coefficient matrix k = 1, 2,…, m , m The number of truncated terms in the influence coefficient matrix; for the influence coefficient matrix A k,1 The calculation formula is: (3) (4) Where 0 and I represent the zero matrix and identity matrix, respectively, and the parameter... β and γ It's Newmark- β The time integration parameter of the method; parameter Δ t The time step of the ground motion is represented by Q1, Q2, and T; H, S1, S2, and W are intermediate calculation matrices, and state space transformation auxiliary matrices are H, S1, S2, and W. a x , x = 0, 1, 2,…, 5, which are the auxiliary coefficients for calculating the influence coefficient matrix; the influence coefficient matrix A is calculated by formula (4). k,1 The model remains constant during training; the global state vector V of the bridge is finally calculated based on formula (2). i V i = [ , ] T Indicates the critical nodes of the bridge at time... t i displacement and speed Response; based on the obtained velocity response The acceleration response is calculated based on the forward difference formula. ,Right now: (5)。 4. The method for analyzing random vibration of bridges based on a mechanism-enhanced physical information neural network according to claim 1, characterized in that, In step (2), the backpropagation path includes: The loss function module consists of two parts: basic physical loss and symbolic loss; the basic physical loss includes acceleration loss. L a and speed loss L v These are used to monitor the output accuracy of the acceleration prediction head and the velocity prediction head, respectively, and the calculation formulas are as follows: (6) (7) in, t i For the first i At that moment, N The total time step, and These are the acceleration and velocity values ​​calculated by the SCL module, respectively; for velocity sign prediction, the three-state gating mechanism determines the velocity magnitude based on the velocity value. The time step is divided into three physical regions and a differentiated supervision strategy is adopted: the security domain is... m / s, directly supervised symbolic prediction; crossing the domain periphery m / s, using acceleration symbols as physical information to guide symbolic prediction; singularity zero-point domain is m / s, using soft constraints to allow for a certain prediction tolerance; the symbolic losses for the three regions are as follows: (8) (9) (10) in, L s,safe , L s,cross , L s,zero The symbolic losses are respectively for the safe domain, the traversal domain, and the singularity-zero domain; v 0 = 10 -2 m / s is the normalization parameter. v soft = 10 -3 m / s is the soft constraint tolerance parameter. M safe , M cross , M zero Binary masks for the safe region, the crossing region, and the singularity-zero region, respectively; the total symbolic loss is... L s = L s,safe + L s,cross + L s,zero Ultimately, the total loss function is optimized through weighted aggregation to achieve the overall model optimization. (11) in, ω a , ω v , ω s These are the weighting coefficients for acceleration loss, velocity loss, and symbol loss, respectively.

5. The method for analyzing random vibration of bridges based on a mechanism-enhanced physical information neural network according to claim 1, characterized in that, Step (4) is as follows: Based on the ME-PINN model trained in step (3), it is integrated into the Direct Probability Integration Method (DPIM) framework as an efficient surrogate model to replace the finite element time history analysis method, thereby achieving efficient calculation of random vibration response and dynamic reliability; for the random vibration response of a bridge structure equipped with a nonlinear tuned inertial capacitive damper under random near-fault ground motion, its probability density function p Y ( y , t The formula for calculating ) is: (12) in, y Indicates the seismic response of bridges. N tol θ represents the number of representative points required by the DPIM method. q For the first q One representative point, P q Assign a probability to the corresponding representative point. g (θ q , t ) is the physical mapping function. σ ( t ) represents the smoothing parameter; Based on the direct probability integral method, the time-varying mean of random vibration response E [ Y ( t Standard deviation Std [ Y ( t The formula for calculating )] can be expressed as: (13) (14) Power reliability R ( t ) is represented as: (15) in, Y thd The failure threshold For extreme response, H [·] represents the Heaviside function.

6. The method for analyzing random vibration of bridges based on a mechanism-enhanced physical information neural network according to claim 5, characterized in that, An iterative sequence sampling strategy is used to adaptively determine the optimal number of representative points. N tol The strategy first generates a pool of candidate representative points based on the GF-difference point selection method, then iteratively adds representative points and calculates the extreme response. When the relative error of five consecutive iterations is less than 5 × 10⁻⁶, the strategy is successful. -5 The process will end at that time.

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