Indirect testing method for displacement response of bridge under moving load based on deep operator physical information neural network

Abstract

This invention discloses an indirect testing method for bridge displacement response under moving loads based on a deep operator physical information neural network, belonging to the field of bridge inspection technology, and includes: 1. acquiring vehicle acceleration response and bridge displacement response under different bridge conditions to construct a dataset; 2. constructing a deep operator network and training the deep operator network using the dataset to obtain a trained displacement prediction model for a simply supported beam under moving loads. This invention learns the mapping relationship between vehicle acceleration response and bridge displacement response under moving loads through a deep operator physical information neural network, thereby indirectly testing the bridge displacement response through vehicle acceleration response. This allows for bridge displacement testing without installing sensors on the bridge, and is used for bridge damage detection.

Description

Technical Field

[0001] This invention belongs to the field of bridge health monitoring, specifically a method for indirect testing of bridge displacement response under moving load based on a deep operator physical information neural network. Background Technology

[0002] The displacement response of a bridge under moving loads is a direct reflection of its stiffness, making it a crucial parameter for bridge safety assessment and an important aspect of bridge health monitoring. Existing bridge displacement response reconstruction technologies can be broadly categorized into three types. The first type is non-contact technology, employing methods such as millimeter-wave radar, laser ground scanning, and global navigation satellite systems to measure bridge displacement. However, this is significantly affected by environmental and weather conditions and involves high equipment costs. The second type is indirect measurement methods, reconstructing bridge displacement response based on acceleration or strain. However, these methods suffer from limitations such as the inability to reconstruct quasi-static displacements and difficulty in determining the neutral axis position. The third type is deep learning-based methods, including computer vision-based and data-driven approaches. These methods require a large number of training samples, exhibit poor generalization ability, and have high computational costs. Summary of the Invention

[0003] The present invention addresses the shortcomings of the prior art by providing an indirect testing method for bridge displacement response under moving loads based on a deep operator physical information neural network. This method aims to obtain the corresponding bridge displacement response using vehicle acceleration response without installing sensors on the bridge, thereby improving bridge damage detection efficiency and accuracy.

[0004] To achieve the above-mentioned objectives, the present invention adopts the following technical solution: The present invention provides an indirect testing method for bridge displacement response under moving load based on a deep operator physical information neural network, characterized by the following steps: Step 1: Obtain the vehicle-bridge coupling dataset Where A represents the vehicle's acceleration response, and ,in, Let I represent the vehicle acceleration at the i-th time step, and let I represent the total number of time steps. Indicates the bridge displacement response, and , This represents the bridge displacement at the nth bridge location sampling point at the i-th time step. This indicates the bridge location of the nth bridge location sampling point. This represents the duration of the i-th time step, and N represents the number of sampling points at the bridge location; Indicates bridge and vehicle parameters, and , Indicates the length of the bridge. This represents the product of the bridge's stiffness and the moment of inertia of its cross-section. Indicates vehicle speed. Indicates vehicle mass. This indicates the mass of the bridge per meter. Indicates the vehicle's suspension stiffness; Step 2: For After normalization, we obtain the normalized dataset. ;in, This represents the normalized vehicle acceleration response. This represents the normalized bridge and vehicle parameters. This represents the normalized bridge displacement response; Step 3: Construct a deep operator network, including a backbone network and branch networks, and perform... and The process is performed to obtain the predicted bridge displacement. ,in, This represents the predicted bridge displacement at the i-th time step at the n-th bridge location sampling point; Step 3.1: Establish the backbone network, which includes: Fourier embedding layers, C residual blocks, and fully connected layers, and... After processing, the deep spatiotemporal coordinate basis functions are obtained. ;in, Let S represent the amplitude of the basis function at the i-th time step of the n-th bridge location sampling point for the s-th inherent deformation mode of the bridge; S represents the total number of modes. Step 3.2: Establish a branch network using formula (7) and After processing, the deep spatiotemporal coordinate basis functions are obtained. modal projection coefficients ,in, Represents the projection coefficients of the s-th modal: (7) In equation (7), Conv represents the convolution operation. and These represent global max pooling and global average pooling operations, respectively; Concat represents the concatenation operation; Dense represents the fully connected output layer; and Gelu represents the activation function. Step 3.3: Obtain the predicted bridge displacement using equation (8). : (8) In equation (8), This represents the physical calibration coefficient of the s-th modal response.

[0005] Step 4: Based on and Establish the total loss function ; Step 5: Use the Adam optimizer to iteratively train the deep operator neural network and calculate L to adjust the network parameters until L converges, thereby obtaining a trained displacement prediction model for simply supported beams under moving loads, which can be used to predict bridge displacements under moving loads.

[0006] The indirect testing method for bridge displacement response under moving load based on deep operator physical information neural network described in this invention is also characterized in that step 2 includes the following steps: Step 2.1: For each , Normalization is performed to obtain the product of the normalized bridge stiffness and the moment of inertia of the cross section. Normalized vehicle mass Normalized bridge mass per meter Normalized vehicle suspension stiffness This allows for the construction of preprocessed bridge and vehicle parameters. , This indicates the total travel time of the vehicle across the bridge, and ; Step 2.2: Normalize the vehicle acceleration response A to obtain the normalized vehicle acceleration response. ;in, This represents the normalized vehicle acceleration at the i-th time step; Step 2.3: For each and Perform normalization to obtain the duration of the i-th time step after normalization. The bridge location of the nth bridge location sampling point after normalization Thus constructing a spatiotemporal coordinate sequence ; Step 2.4: For After normalization, the normalized bridge displacement is obtained. ,in, This represents the bridge displacement at the i-th time step at the n-th bridge location sampling point after normalization, and serves as the data fitting label for the bridge displacement at the i-th time step at the n-th bridge location sampling point.

[0007] Furthermore, step 3.1 includes the following steps: Step 3.1.1: The Fourier embedding layer utilizes equation (1) to... Perform linear projection to obtain Fourier features. : (1) In equation (1), This indicates that a weight matrix to be learned is randomly generated and satisfies a normal distribution. Step 3.1.2: Each residual block includes: 1 gating switch and 2 spatiotemporal evolution feature branches, and these are sequentially applied to... The process yields the Cth deep spatiotemporal coordinate feature. ; Step 3.1.3: The fully connected layer pairs Processing is performed to output deep spatiotemporal coordinate basis functions. .

[0008] Furthermore, step 3.1.2 includes the following steps: Step 3.1.2.1: When c=1, for The initial coordinate features after nonlinear activation are used as the first Deep spatiotemporal coordinate features and with The input is fed into the gate switch of the c-th residual block, and then the c-th residual block is obtained using equation (2). Each modality participates in the weighting : (2) In equation (2), and These are the two weight matrices of the c-th gate switch. This indicates nonlinear activation processing; Step 3.1.2.2: The first spatiotemporal evolution feature branch of the c-th residual block uses equation (3) to... After processing, the first spatiotemporal evolution feature branch of the c-th residual block output is obtained. : (3) In equation (3), These are the two weight matrices of the first spatiotemporal evolution feature branch of the c-th residual block; Step 3.1.2.3: The second spatiotemporal evolution feature branch of the c-th residual block uses equation (4) to... After processing, the second spatiotemporal evolution feature branch of the c-th residual block output is obtained. : (4) In equation (4), These are the two weight matrices of the second spatiotemporal evolution feature branch of the c-th residual block; Step 3.1.2.4: Calculate the displacement mapping residual increment of the c-th residual block using equation (5). : (5) Step 3.1.2.5: Calculate the deep spatiotemporal coordinate characteristics of the c-th residual block using equation (6). ; (6) In equation (6), Indicates residual connection; Step 3.1.2.6: When c = 2, 3, ..., C, and Process the c-th residual block as input and output the deep spatiotemporal coordinate features of the c-th residual block. Thus, the deep spatiotemporal coordinate features of the Cth residual block are output from the Cth residual block. .

[0009] Furthermore, step 4 includes the following steps: Step 4.1: Establish the data fitting loss using equation (9) : (9) Step 4.2: Establish boundary condition loss using equation (10) : (10) In equation (10), This represents the predicted bridge displacement at the i-th time step of the left end of the bridge. This represents the predicted bridge displacement at the i-th time step of the right end of the bridge. Let represent the second derivative of the predicted bridge displacement at the i-th time step from the left end of the bridge. Let represent the second derivative of the predicted bridge displacement at the i-th time step of the right end of the bridge; Step 4.3: Establish the initial condition loss using equation (11) : (11) In equation (11), This indicates the first time step at the nth bridge location sampling point. Predicting bridge displacement, This indicates the first time step at the nth bridge location sampling point. The first derivative of the predicted bridge displacement; Step 4.4: Use equation (12) to establish the dimensionless physical equation for loss. : (12) In equation (12), Indicates the scaling factor. Let represent the moving load exerted by the vehicle on the bridge at the i-th time step, and obtain it from equation (13); (13) In equation (13), Indicates to The normalized vertical displacement of the vehicle at the i-th time step. It is obtained from equation (14); (14) In equation (14), Represents gravitational acceleration; Step 4.5: Construct the total loss function using equation (15) ; (15) In equation (15), The weights for the data fitting loss. The weights for the boundary condition loss. The weights for the initial conditional loss. This represents the weight of the loss in the physical equations.

[0010] The present invention provides an electronic device, including a memory and a processor, characterized in that the memory is used to store a program supporting the processor in performing the method described therein, and the processor is configured to execute the program stored in the memory.

[0011] The present invention provides a computer-readable storage medium on which a computer program is stored, characterized in that the computer program is executed by a processor to perform the steps of the method described thereon.

[0012] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This invention acquires the acceleration response of vehicles as they pass over bridges by installing acceleration sensors on the vehicles. A deep operator neural network is then established to reconstruct the bridge displacement under moving loads using the vehicle acceleration response, enabling precise testing of the bridge's displacement response. By monitoring the difficult-to-obtain bridge displacement response using easily obtainable vehicle acceleration responses, no sensors need to be installed on the bridge, avoiding the difficulties, wear and tear on instruments, and high cost of installation equipment associated with previous bridge damage monitoring methods.

[0013] 2. This invention employs a deep operator neural network to learn function mappings, enabling the model to predict bridge responses at any point in time and space once trained, without needing to rerun the entire complex model. Furthermore, the trained model can generalize to unseen physical parameters and stimulus inputs, exhibiting advantages such as strong generalization ability and low dependence on sample data.

[0014] 3. This invention employs a physical information neural network to provide physics learning, enabling the model to learn underlying physical laws while fitting data, thus improving the model's interpretability. The automatic differentiation technique of the physical information neural network significantly accelerates the calculation efficiency of higher-order derivatives, offering faster computation speed and lower computational cost compared to other deep learning methods. Attached Figure Description

[0015] Figure 1 This is a schematic diagram of the vehicle-bridge coupling model in this invention; Figure 2 This is a structural diagram of the deep operator physical neural network in this invention; Figure 3 This is a structural diagram of the backbone network in this invention; Figure 4 This is a structural diagram of the branch network in this invention; Figure 5 This is a historical graph of the training loss of the examples used in this invention; Figure 6 This is a diagram showing the displacement at the mid-span of the bridge predicted by the example used in this invention. Figure 7 This is a displacement diagram at one-quarter of the bridge predicted by the example used in this invention. Detailed Implementation

[0016] In this embodiment, an indirect testing method for bridge displacement response under moving load based on a deep operator physical information neural network consists of two parts. The first part is the establishment of a relational model. This involves simulating the bridge response under moving load using finite element analysis to obtain vehicle acceleration response data and bridge displacement response data. Finally, a mapping relationship between vehicle acceleration response and bridge displacement response is established using a deep operator neural network, training a simple supported beam displacement prediction model under moving load. The second part is the prediction process. Using the model established by this method, the corresponding bridge displacement response can be obtained simply by measuring the vehicle acceleration response value. Specifically, the method includes the following steps: Step 1: Collection of sample data.

[0017] Obtain vehicle-bridge coupling dataset Where A represents the vehicle's acceleration response, and ,in, Let I represent the vehicle acceleration at the i-th time step, and let I represent the total number of time steps. Indicates the bridge displacement response, and , This represents the bridge displacement at the nth bridge location sampling point at the i-th time step. This indicates the bridge location of the nth bridge location sampling point. This represents the duration of the i-th time step, and N represents the number of sampling points at the bridge location; Indicates bridge and vehicle parameters, and , Indicates the length of the bridge. This represents the product of the bridge's stiffness and the moment of inertia of its cross-section. Indicates vehicle speed. Indicates vehicle mass. This indicates the mass of the bridge per meter. This indicates the vehicle's suspension stiffness.

[0018] In this embodiment, 14 different bridge working conditions were designed. For each working condition, three or four elements were randomly selected and stiffness reductions were performed to simulate local damage to the bridge. A total of 210 vehicle-bridge coupling datasets D were obtained. The specific parameter settings for each bridge working condition are shown in the table below:

[0019] Step 2: For After normalization, we obtain the normalized dataset. ;in, This represents the normalized vehicle acceleration response. This represents the normalized bridge and vehicle parameters. This represents the normalized bridge displacement response.

[0020] Step 2.1: Take the product of bridge stiffness and section moment of inertia as a reference value. Vehicle weight reference value Reference value for bridge mass per meter Vehicle suspension stiffness reference value To each , Normalization is performed to obtain the product of the normalized bridge stiffness and the moment of inertia of the cross section. Normalized vehicle mass Normalized bridge mass per meter Normalized vehicle suspension stiffness This allows for the construction of preprocessed bridge and vehicle parameters. , This indicates the total travel time of the vehicle across the bridge, and .

[0021] Step 2.2: Find the maximum value of the vehicle acceleration response in each sample, and normalize the vehicle acceleration response A to obtain the normalized vehicle acceleration response. ;in, This represents the normalized vehicle acceleration at the i-th time step.

[0022] Step 2.3: For each and Perform normalization to obtain the duration of the i-th time step after normalization. The bridge location of the nth bridge location sampling point after normalization Thus constructing a spatiotemporal coordinate sequence .

[0023] Step 2.4: For After normalization, the normalized bridge displacement is obtained. ,in, This represents the bridge displacement at the i-th time step at the n-th bridge location sampling point after normalization, and serves as the data fitting label for the bridge displacement at the i-th time step at the n-th bridge location sampling point.

[0024] Step 3: Construct a deep operator network, including a backbone network and branch networks, and perform... and The process is performed to obtain the predicted bridge displacement. ,in, This represents the predicted bridge displacement at the i-th time step at the n-th bridge location sampling point; the network structure diagram is as follows. Figure 1 As shown.

[0025] Step 3.1: Establish the backbone network, which includes: Fourier embedding layers, C residual blocks, and fully connected layers, and... After processing, the deep spatiotemporal coordinate basis functions are obtained. ;in The amplitude of the basis function at the nth time step of the nth bridge location sampling point represents the s-th inherent deformation mode of the bridge. S represents the total modal order. In this example, C=6 and S=128. The structure diagram of the backbone network is shown below. Figure 2 As shown.

[0026] Step 3.1.1: The Fourier embedding layer utilizes equation (1) to... Perform linear projection to obtain Fourier features. : (1) In equation (1), This indicates that a weight matrix to be learned is randomly generated and satisfies a normal distribution.

[0027] Step 3.1.2: Each residual block includes: 1 gating switch and 2 spatiotemporal evolution feature branches, and these are sequentially applied to... The process yields the Cth deep spatiotemporal coordinate feature. ; Step 3.1.2.1: When c=1, for The initial coordinate features after nonlinear activation are used as the first Deep spatiotemporal coordinate features and with The input is fed into the gate switch of the c-th residual block, and then the c-th residual block is obtained using equation (2). Each modality participates in the weighting : (2) In equation (2), and These are the two weight matrices of the c-th gate switch. This indicates nonlinear activation processing.

[0028] Step 3.1.2.2: The first spatiotemporal evolution feature branch of the c-th residual block uses equation (3) to... After processing, the first spatiotemporal evolution feature branch of the c-th residual block output is obtained. : (3) In equation (3), These are the two weight matrices of the first spatiotemporal evolution feature branch of the c-th residual block.

[0029] Step 3.1.2.3: The second spatiotemporal evolution feature branch of the c-th residual block uses equation (4) to... After processing, the second spatiotemporal evolution feature branch of the c-th residual block output is obtained. : (4) In equation (4), These are the two weight matrices of the second spatiotemporal evolution feature branch of the c-th residual block.

[0030] Step 3.1.2.4: Calculate the displacement mapping residual increment of the c-th residual block using equation (5). : (5) Step 3.1.2.5: Calculate the deep spatiotemporal coordinate characteristics of the c-th residual block using equation (6). ; (6) In equation (6), This indicates a residual connection.

[0031] Step 3.1.2.6: When c = 2, 3, ..., C, and Process the c-th residual block as input and output the deep spatiotemporal coordinate features of the c-th residual block. Thus, the deep spatiotemporal coordinate features of the Cth residual block are output from the Cth residual block. .

[0032] Step 3.1.3: The fully connected layer pairs Processing is performed to output deep spatiotemporal coordinate basis functions. ; Step 3.2: Establish a branch network using formula (7) and After processing, the deep spatiotemporal coordinate basis functions are obtained. modal projection coefficients ,in, The projection coefficients of the s-th modal are shown in the diagram below. Figure 3 As shown: (7) In equation (7), Conv represents the convolution operation. and These represent global max pooling and global average pooling operations, respectively. Concat represents the concatenation operation, Dense represents the output layer of a fully connected layer, and Gelu represents the activation function.

[0033] Multi-scale convolution is employed to extract local features of vehicle acceleration response. Several dilated convolutions with different receptive fields are used to simultaneously extract high-frequency impacts and low-frequency oscillations in the vehicle acceleration response. Three dilated convolutional layers with different receptive fields are designed. The first layer has 24 kernels, transforming the original 1-dimensional signal sequence into a 24-dimensional feature vector sequence. The kernel size is 3, indicating that the length of the data point observed by the kernel on the time axis is 3. The dilation rate is 1, indicating that the sampling interval between kernel elements is 1, i.e., standard gapless convolution. This first layer is used to capture high-frequency local features in the vehicle acceleration response. The second layer has 32 kernels with a kernel size of 5, used to capture more complex combinations of response features and observes a slightly wider temporal window. With a dilation rate of 4, meaning the convolutional kernel samples every three data points, the receptive field of the kernel is effectively increased, allowing it to capture the mid-frequency components in the vehicle acceleration response, i.e., exhibiting a certain regular local vibration trend within a short period of time. The third layer has 48 convolutional kernels, used to process the most macroscopic features in the vehicle acceleration response. With a kernel size of 7 and a dilation rate of 8, it covers a wider time range, capturing the low-frequency components and overall contours in the vehicle acceleration response, and processing longer time series data, such as the long waveform displacement trend generated during the process of a vehicle entering and leaving a bridge.

[0034] Average pooling is used to obtain the overall average vibration energy or vibration trend of the vehicle acceleration response, while max pooling is used to capture the vibration peak value in the vehicle acceleration response.

[0035] Step 3.3: Obtain the predicted bridge displacement using equation (8). : (8) In equation (8), This represents the physical calibration coefficient of the s-th modal response.

[0036] Step 4: Based on and Establish the total loss function ; Step 4.1: Establish the data fitting loss using equation (9) Used to measure the predicted bridge response With normalized bridge displacement Differences between them: (9) in, The magnitude is Left and right, and the initialization exist Nearby, to avoid The gradient generated during backpropagation is too small, causing the optimizer to be unable to effectively update the network weights. Therefore, a method is adopted. As an amplification factor, it brings the label data and the output domain of the network initialization to the same order of magnitude, ensuring the numerical stability of the gradient and greatly accelerating the convergence speed of the model.

[0037] Step 4.2: Establish boundary condition loss using equation (10) This is used to constrain the physical state at both ends of the bridge. For a simply supported beam model, the displacement and bending moment at both ends of the bridge are zero, and the second derivative of the displacement is used to represent the bending moment: (10) In equation (10), This represents the predicted bridge displacement at the i-th time step of the left end of the bridge. This represents the predicted bridge displacement at the i-th time step of the right end of the bridge. Let represent the second derivative of the predicted bridge displacement at the i-th time step from the left end of the bridge. Let represent the second derivative of the predicted bridge displacement at the i-th time step of the right end of the bridge.

[0038] Step 4.3: Establish the initial condition loss using equation (11) Used to constrain At time 0, the bridge's displacement and velocity are both 0. (11) In equation (11), This indicates the first time step at the nth bridge location sampling point. Predicting bridge displacement, This indicates the first time step at the nth bridge location sampling point. The first derivative of the predicted bridge displacement.

[0039] Step 4.4: Use equation (12) to establish the dimensionless physical equation for loss. This is the core of the physical information neural network, which forces the model output to follow the dynamic equations of the Euler-Bernoulli beam: (12) In equation (12), Indicates the scaling factor. Let represent the moving load exerted on the bridge by the vehicle at the i-th time step, and be obtained from equation (13).

[0040] (13) In equation (13), Indicates to The normalized vertical displacement of the vehicle at the i-th time step. It is obtained from equation (14); (14) In equation (14), The acceleration due to gravity is represented by a state-space discretization method to calculate the vertical displacement of the vehicle. Compared with the simple explicit Euler method, this matrix exponent-based discretization method has high numerical stability, can obtain the exact solution of the differential equation at the step node, and has high computational efficiency.

[0041] Step 4.5: Construct the total loss function using equation (15) ; (15) In equation (15), The weights for the data fitting loss. The weights for the boundary condition loss. The weights for the initial conditional loss. This represents the weight of the loss in the physical equations.

[0042] Step 5: Employ a two-stage loss function weighting strategy. The first stage does not involve calculating the loss from the physical equations, focusing instead on learning data fitting, boundary conditions, and initial conditions. The second stage begins learning the physical equations and uses the Sigmoid function for weight transition. The Adam optimizer is used for iterative training of the deep operator neural network, with a learning rate set to... Set the total training cycle Phase 1 Weight transition phase Phase Two The weights of the loss function in the first stage are set as follows: The weights of the loss function in the second stage are set as follows: The network parameters are adjusted by calculating L until L converges, resulting in a trained displacement prediction model for a simply supported beam under moving loads, which is used to predict bridge displacements under moving loads. The history of the loss function during training is as follows: Figure 4 As shown.

[0043] In this embodiment, an electronic device includes a memory and a processor. The memory stores a program that supports the processor in executing the above-described method, and the processor is configured to execute the program stored in the memory.

[0044] In this embodiment, a computer-readable storage medium stores a computer program, which is executed by a processor to perform the steps of the above method.

[0045] Specifically, for example Figure 1 Taking a simply supported beam bridge as an example, the total length of the bridge is 20m, the vehicle speed is 2m / s, the moment of inertia of the bridge section is 0.15m⁴, the elastic modulus is 27.5GPa, the mass per meter of the bridge is 2500kg / m, and the vehicle mass is 1000kg. The finite element method is used in Matlab for simulation, and the Newmark-β method is used to solve the dynamic response. The bridge element is a two-node Euler plane beam element, and the vehicle uses a quarter-vehicle model, i.e., a single-spring mass block model. The vehicle-bridge coupling model is as follows: Figure 5 As shown.

[0046] Obtain the vehicle acceleration response to be predicted from the MATLAB finite element model. Response to vehicle acceleration Input the trained model to obtain the predicted bridge displacement response. Calculate the relative position of the mid-span error: A quarter span of the bridge Bridge displacement diagram as follows Figure 6 , Figure 7As shown, the prediction results are relatively accurate, and the working conditions of the target bridge are outside the working conditions of the training samples, indicating that the model has strong generalization ability.

Claims

1. A method for indirectly testing bridge displacement response under moving load based on a deep operator physical information neural network, characterized in that, Includes the following steps: Step 1: Obtain the vehicle-bridge coupling dataset Where A represents the vehicle's acceleration response, and ,in, Let I represent the vehicle acceleration at the i-th time step, and let I represent the total number of time steps. Indicates the bridge displacement response, and , This represents the bridge displacement at the nth bridge location sampling point at the i-th time step. This indicates the bridge location of the nth bridge location sampling point. This represents the duration of the i-th time step, and N represents the number of sampling points at the bridge location; Indicates bridge and vehicle parameters, and , Indicates the length of the bridge. This represents the product of the bridge's stiffness and the moment of inertia of its cross-section. Indicates vehicle speed. Indicates vehicle mass. This indicates the mass of the bridge per meter. Indicates the vehicle's suspension stiffness; Step 2: For After normalization, we obtain the normalized dataset. ;in, This represents the normalized vehicle acceleration response. This represents the normalized bridge and vehicle parameters. This represents the normalized bridge displacement response; Step 3: Construct a deep operator network, including a backbone network and branch networks, and perform... , and The process is performed to obtain the predicted bridge displacement. ,in, This represents the predicted bridge displacement at the i-th time step at the n-th bridge location sampling point; Step 3.1: Establish the backbone network, which includes: Fourier embedding layers, C residual blocks, and fully connected layers, and... After processing, the deep spatiotemporal coordinate basis functions are obtained. ;in, Let S represent the amplitude of the basis function at the i-th time step of the n-th bridge location sampling point for the s-th inherent deformation mode of the bridge; S represents the total number of modes. Step 3.2: Establish a branch network using formula (7) and After processing, the deep spatiotemporal coordinate basis functions are obtained. modal projection coefficients ,in, Represents the projection coefficients of the s-th modal: (7) In equation (7), Conv represents the convolution operation. and These represent global max pooling and global average pooling operations, respectively; Concat represents the concatenation operation; Dense represents the fully connected output layer; and Gelu represents the activation function. Step 3.3: Obtain the predicted bridge displacement using equation (8). : (8) In equation (8), Represents the physical calibration coefficient of the s-th modal response; Step 4: Based on and Establish the total loss function ; Step 5: Use the Adam optimizer to iteratively train the deep operator neural network and calculate L to adjust the network parameters until L converges, thereby obtaining a trained displacement prediction model for simply supported beams under moving loads, which can be used to predict bridge displacements under moving loads.

2. The indirect testing method for bridge displacement response under moving load based on a deep operator physical information neural network according to claim 1, characterized in that, Step 2 includes the following steps: Step 2.1: For each , Normalization is performed to obtain the product of the normalized bridge stiffness and the moment of inertia of the cross section. Normalized vehicle mass Normalized bridge mass per meter Normalized vehicle suspension stiffness This allows for the construction of preprocessed bridge and vehicle parameters. , This indicates the total travel time of the vehicle across the bridge, and ; Step 2.2: Normalize the vehicle acceleration response A to obtain the normalized vehicle acceleration response. ;in, This represents the normalized vehicle acceleration at the i-th time step; Step 2.3: For each and Perform normalization to obtain the duration of the i-th time step after normalization. The bridge location of the nth bridge location sampling point after normalization Thus constructing a spatiotemporal coordinate sequence ; Step 2.4: For After normalization, the normalized bridge displacement is obtained. ,in, This represents the bridge displacement at the i-th time step at the n-th bridge location sampling point after normalization, and serves as the data fitting label for the bridge displacement at the i-th time step at the n-th bridge location sampling point.

3. The indirect testing method for bridge displacement response under moving load based on a deep operator physical information neural network according to claim 2, characterized in that, Step 3.1 includes the following steps: Step 3.1.1: The Fourier embedding layer utilizes equation (1) to... Perform linear projection to obtain Fourier features. : (1) In equation (1), This indicates that a weight matrix to be learned is randomly generated and satisfies a normal distribution. Step 3.1.2: Each residual block includes: 1 gating switch and 2 spatiotemporal evolution feature branches, and these are sequentially applied to... The process yields the Cth deep spatiotemporal coordinate feature. ; Step 3.1.3: The fully connected layer pairs Processing is performed to output deep spatiotemporal coordinate basis functions. .

4. The indirect testing method for bridge displacement response under moving load based on a deep operator physical information neural network according to claim 3, characterized in that, Step 3.1.2 includes the following steps: Step 3.1.2.1: When c=1, for The initial coordinate features after nonlinear activation are used as the first Deep spatiotemporal coordinate features and with The input is fed into the gate switch of the c-th residual block, and then the c-th residual block is obtained using equation (2). Each modality participates in the weighting : (2) In equation (2), and These are the two weight matrices of the c-th gate switch. This indicates nonlinear activation processing; Step 3.1.2.2: The first spatiotemporal evolution feature branch of the c-th residual block uses equation (3) to... After processing, the first spatiotemporal evolution feature branch of the c-th residual block output is obtained. : (3) In equation (3), These are the two weight matrices of the first spatiotemporal evolution feature branch of the c-th residual block; Step 3.1.2.3: The second spatiotemporal evolution feature branch of the c-th residual block uses equation (4) to... After processing, the second spatiotemporal evolution feature branch of the c-th residual block output is obtained. : (4) In equation (4), These are the two weight matrices of the second spatiotemporal evolution feature branch of the c-th residual block; Step 3.1.2.4: Calculate the displacement mapping residual increment of the c-th residual block using equation (5). : (5) Step 3.1.2.5: Calculate the deep spatiotemporal coordinate characteristics of the c-th residual block using equation (6). ; (6) In equation (6), Indicates residual connection; Step 3.1.2.6: When c = 2, 3, ..., C, and Process the c-th residual block as input and output the deep spatiotemporal coordinate features of the c-th residual block. Thus, the deep spatiotemporal coordinate features of the Cth residual block are output from the Cth residual block. .

5. The indirect testing method for bridge displacement response under moving load based on a deep operator physical information neural network according to claim 2, characterized in that, Step 4 includes the following steps: Step 4.1: Establish the data fitting loss using equation (9) : (9) Step 4.2: Establish boundary condition loss using equation (10) : (10) In equation (10), This represents the predicted bridge displacement at the i-th time step of the left end of the bridge. This represents the predicted bridge displacement at the i-th time step of the right end of the bridge. Let represent the second derivative of the predicted bridge displacement at the i-th time step from the left end of the bridge. Let represent the second derivative of the predicted bridge displacement at the i-th time step of the right end of the bridge; Step 4.3: Establish the initial condition loss using equation (11) : (11) In equation (11), This indicates the first time step at the nth bridge location sampling point. Predicting bridge displacement, This indicates the first time step at the nth bridge location sampling point. The first derivative of the predicted bridge displacement; Step 4.4: Use equation (12) to establish the dimensionless physical equation for loss. : (12) In equation (12), Indicates the scaling factor. Let represent the moving load exerted by the vehicle on the bridge at the i-th time step, and obtain it from equation (13); (13) In equation (13), Indicates to The normalized vertical displacement of the vehicle at the i-th time step. It is obtained from equation (14); (14) In equation (14), Represents gravitational acceleration; Step 4.5: Construct the total loss function using equation (15) ; (15) In equation (15), The weights for the data fitting loss. The weights for the boundary condition loss. The weights for the initial conditional loss. This represents the weight of the loss in the physical equations.

6. An electronic device, comprising a memory and a processor, characterized in that, The memory is used to store a program that supports a processor in executing the method of any one of claims 1-5, the processor being configured to execute the program stored in the memory.

7. A computer-readable storage medium storing a computer program thereon, characterized in that, The computer program is executed by the processor to perform the steps of the method according to any one of claims 1-5.

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