Training of bridge mechanical response prediction model, bridge mechanical response prediction method

Abstract

The one or more embodiments of the specification provide a training method of a bridge mechanical response prediction model and a prediction method of a bridge mechanical response. The training method comprises: obtaining a prediction output of the bridge mechanical response prediction model at at least one sampling point; calculating each loss component according to the prediction output; for each loss component, calculating a gradient of the loss component with respect to network parameters of the bridge mechanical response prediction model, and calculating an effective learning rate of the loss component according to the gradient; dynamically adjusting a weight coefficient of each loss component according to the effective learning rate; calculating a total loss according to each loss component and the corresponding weight coefficient, and updating the network parameters of the bridge mechanical response prediction model according to the total loss. Thus, under the condition of small samples, that is, in the case of only a small amount of labeled observation data or sparse simulation samples, each constraint term can also be kept in balanced convergence in the training, thereby reducing the dependence on a large-scale finite element simulation data set.

Description

Technical Field

[0001] This specification relates to one or more embodiments in the field of bridge structural health monitoring and safety assessment, and in particular to a method for training a bridge mechanical response prediction model and predicting the bridge mechanical response. Background Technology

[0002] In the field of bridge structural health monitoring and safety assessment, forward computation refers to the technique of accurately predicting the mechanical response (such as displacement field and stress field) of a structure given its geometry, material constitutive properties, load conditions, and boundary conditions. Currently, traditional numerical simulation based on the Finite Element Method (FEM) is the most commonly used analysis method in engineering practice. This method establishes a large-scale sparse equation system and uses direct methods or iterative preprocessing conjugate gradient methods for numerical solution, achieving high computational accuracy.

[0003] However, the traditional finite element method takes a long time to perform a single static nonlinear analysis of a complex bridge, and even longer for dynamic time history analysis, making it difficult to meet the requirements of online scanning and real-time state simulation under multiple working conditions. Summary of the Invention

[0004] In view of the above, one or more embodiments of this specification provide the following technical solutions: According to a first aspect of one or more embodiments of this specification, a training method for a bridge mechanical response prediction model is proposed, the method comprising: Obtain the prediction output of the bridge mechanical response prediction model at at least one sampling point, wherein the prediction output is the mechanical response of the bridge structure under a given working condition; Each loss component is calculated based on the predicted output, and each loss component corresponds to a constraint type. For each loss component, the gradient of the loss component with respect to the network parameters of the bridge mechanical response prediction model is calculated, and the effective learning rate of the loss component is calculated based on the gradient. The effective learning rate is used to measure the convergence speed of the corresponding loss component. The weight coefficients of each loss component are dynamically adjusted based on the effective learning rate of each loss component. The effective learning rate and weight coefficient of each loss component are negatively correlated. The total loss is calculated based on each loss component and its corresponding weight coefficient, and the network parameters of the bridge mechanical response prediction model are updated based on the total loss.

[0005] Optionally, calculating the effective learning rate of the loss component based on the gradient includes: A stochastic estimation method is used to estimate the trace of the neural tangent kernel matrix corresponding to the loss component by calculating the statistic of the square of the inner product of the gradient and the random probe vector for at least one random probe vector. The estimated trace is determined as the effective learning rate for the loss component.

[0006] Optionally, the method further includes: Multiple candidate points are selected, and the physical residual value and the local curvature value of the bridge structure response field at each candidate point are calculated based on the prediction output. A non-uniform sampling probability density is constructed based on the physical residual value and the local curvature value; New sampling points are selected from the candidate points according to the non-uniform sampling probability density for subsequent iterative training.

[0007] Optionally, the hidden layer of the bridge mechanical response prediction model employs a multi-scale periodic activation function, and / or, The input layer of the bridge mechanical response prediction model uses random Fourier feature encoding to pre-encode the coordinates of the sampling points.

[0008] Optionally, the input to the bridge mechanical response prediction model further includes a time variable, the mechanical response includes a dynamic time history response in the time dimension, and the loss component includes a time-domain dynamic constraint loss, which is used to constrain the dynamic time history response to satisfy the bridge structure dynamic control equations with or without damping.

[0009] Optionally, the loss component further includes frequency domain modal characteristic constraint loss, which is used to constrain the bridge modal parameters extracted from the dynamic time history response to satisfy generalized eigenvalue relationships and / or modal mass orthogonality relationships.

[0010] According to a second aspect of one or more embodiments of this specification, a method for predicting the mechanical response of a bridge is proposed, the method comprising: Acquire sampling point coordinates, time variables, and specified operating condition data; The sampling point coordinates, the time variable, and the specified working condition data are input into the trained bridge mechanical response prediction model to obtain the prediction output of the bridge mechanical response prediction model. The prediction output includes at least one of static displacement field, stress field, dynamic time history response, and bridge modal parameters. The bridge mechanical response prediction model was trained using the aforementioned method.

[0011] According to a third aspect of one or more embodiments of this specification, a training apparatus for a bridge mechanical response prediction model is provided, the apparatus comprising: Prediction output acquisition unit: acquires the prediction output of the bridge mechanical response prediction model at at least one sampling point, wherein the prediction output is the mechanical response of the bridge structure under a given working condition; Loss component calculation unit: Calculates each loss component based on the prediction output, and each loss component corresponds to a constraint type; Learning rate calculation unit: For each loss component, calculate the gradient of the loss component with respect to the network parameters of the bridge mechanical response prediction model, and calculate the effective learning rate of the loss component based on the gradient. The effective learning rate is used to measure the convergence speed of the corresponding loss component. Weight coefficient adjustment unit: dynamically adjusts the weight coefficient of each loss component according to the effective learning rate of each loss component, wherein the effective learning rate of the loss component and the weight coefficient are negatively correlated. Network parameter update unit: Calculates the total loss based on each loss component and its corresponding weight coefficient, and updates the network parameters of the bridge mechanical response prediction model based on the total loss.

[0012] According to a fourth aspect of one or more embodiments of this specification, a device for predicting the mechanical response of a bridge is provided, the device comprising: Data acquisition unit: Acquires sampling point coordinates, time variables, and specified operating condition data; Response prediction unit: Input the sampling point coordinates, the time variable and the specified working condition data into the trained bridge mechanical response prediction model to obtain the prediction output of the bridge mechanical response prediction model. The prediction output includes at least one of static displacement field, stress field, dynamic time history response and bridge modal parameters. The bridge mechanical response prediction model was trained using the aforementioned method.

[0013] According to a fifth aspect of one or more embodiments of this specification, an electronic device is provided, comprising: a processor; a memory for storing processor-executable instructions; wherein the processor performs the steps of the aforementioned method by executing the executable instructions.

[0014] According to a sixth aspect of one or more embodiments of this specification, a computer-readable storage medium is provided that stores computer instructions thereon, which, when executed by a processor, implement the steps of the aforementioned method.

[0015] According to a seventh aspect of one or more embodiments of this specification, a computer program product is provided, comprising a computer program / instructions that, when executed by a processor, implement the steps of the aforementioned method.

[0016] As can be seen from the above embodiments, by employing the scheme provided in this specification, by calculating the gradient of each loss component with respect to the network parameters, determining the effective learning rate for measuring the convergence speed of the corresponding loss component based on the gradient, and then dynamically adjusting the weight coefficients of each loss component according to the principle that the effective learning rate is negatively correlated with the weight coefficients, it is possible to maintain balanced convergence of different physical constraints during the training process. This reduces the dependence on large-scale finite element simulation datasets. Even under small sample conditions, i.e., with only a small amount of labeled observation data or sparse simulation samples, the gradient contributions of each constraint term can be balanced in the total loss, preventing one constraint term from dominating training due to excessively large gradients or another constraint term from stagnating due to excessively small gradients, thereby training a stable and convergent bridge mechanical response prediction model. Attached Figure Description

[0017] Figure 1 This is a flowchart illustrating a training method for a bridge mechanical response prediction model, provided as an exemplary embodiment.

[0018] Figure 2 This is a flowchart of a sampling point update method provided in an exemplary embodiment.

[0019] Figure 3 This is a flowchart of another training method for a bridge mechanical response prediction model provided in an exemplary embodiment.

[0020] Figure 4 This is a flowchart of a method for predicting the mechanical response of a bridge, provided in an exemplary embodiment.

[0021] Figure 5 This is a schematic diagram of the structure of a device provided in an exemplary embodiment.

[0022] Figure 6 This is a block diagram of a training device for a bridge mechanical response prediction model, provided in an exemplary embodiment.

[0023] Figure 7 This is a block diagram of a bridge mechanical response prediction device provided in an exemplary embodiment. Detailed Implementation

[0024] In the field of bridge structural health monitoring and safety assessment, forward computation refers to the technique of accurately predicting the mechanical response (such as displacement field and stress field) of a structure given its geometry, material constitutive properties, load conditions, and boundary conditions. Currently, traditional numerical simulation based on the finite element method (FEM) is the most commonly used analysis method in engineering practice. This method establishes a large-scale sparse equation system and uses direct methods or iterative preprocessing conjugate gradient methods for numerical solution, achieving high computational accuracy.

[0025] However, the traditional finite element method takes a long time to perform a single static nonlinear analysis of a complex bridge, and even longer for dynamic time history analysis, making it difficult to meet the requirements of online scanning and real-time state simulation under multiple working conditions.

[0026] To improve computational efficiency, researchers have developed fast proxy methods based on reduced-order models (ROMs), projecting high-dimensional physical systems into low-dimensional subspaces using techniques such as Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD). However, these methods rely on low-rank linear approximations, and their approximation accuracy easily degrades under highly nonlinear conditions such as overload, support detachment, local damage, and abrupt changes in cable stiffness, limiting their engineering applicability.

[0027] In recent years, data-driven deep neural network surrogate models have been introduced into this field. These methods utilize models such as convolutional neural networks, graph neural networks, or Transformers, employing end-to-end supervised training with a large amount of finite element simulation data to directly establish a black-box mapping from working parameters to structural response. However, these methods heavily rely on large-scale labeled samples, while labeled full-field response data is extremely scarce in actual bridge engineering. This results in poor generalization ability of the model when facing new bridge types or working conditions beyond the training distribution, failing to meet practical needs.

[0028] This specification provides a training scheme for a bridge mechanical response prediction model and a prediction scheme for bridge mechanical response. By monitoring the gradient of each loss component with respect to the model network parameters, the effective learning rate used to measure the convergence speed of the corresponding loss component is dynamically quantified, and the weight coefficients of each loss component are adaptively adjusted according to the magnitude of the learning rate. This ensures that multiple objectives maintain balanced convergence during the training process, thereby solving the problem of training instability caused by gradient imbalance in bridge mechanical response prediction models under small sample conditions and improving the accuracy of bridge mechanical response prediction.

[0029] The training of the bridge mechanical response prediction model and the prediction scheme for the bridge mechanical response provided in this specification can be executed by a server, by a terminal device in conjunction with the server, or by a distributed computing cluster or cloud service platform. The server is a physical server containing an independent host, or a virtual server hosted by a host cluster. The terminal device can be a PC, mobile phone, tablet, laptop, PDA (Personal Digital Assistant), etc., and one or more embodiments in this specification do not limit this.

[0030] In practical deployment scenarios, an architecture combining offline training and online prediction is typically adopted. The training phase demands significant computing resources, requiring extensive iterative computations and weight updates. High-performance computing clusters can be used to train the bridge mechanical response prediction model. After training, the model parameters are exported and deployed to the prediction endpoint. The prediction endpoint can be a cloud-based inference server, receiving real-time data and returning prediction results; it can also be an edge computing node deployed at the bridge site or a nearby data center to achieve low-latency real-time response; or it can be a terminal device (such as a monitoring workstation, portable detection terminal, etc.) directly used by engineering technicians. Those skilled in the art can flexibly choose the model training and deployment method based on the computing resources, real-time requirements, and data transmission costs of the actual application scenario.

[0031] Figure 1 This is a flowchart illustrating a training method for a bridge mechanical response prediction model, provided as an exemplary embodiment.

[0032] Please refer to Figure 1 The training method for the bridge mechanical response prediction model may include the following steps: Step 102: Obtain the prediction output of the bridge mechanical response prediction model at at least one sampling point, wherein the prediction output is the mechanical response of the bridge structure under a given working condition.

[0033] In some embodiments, the bridge mechanical response prediction model refers to a neural network model that uses the partial differential equation constraints of the bridge structure as a component of the loss function to predict the mechanical response of the bridge structure under known working conditions, i.e., the forward calculation of the bridge structure. Specifically, the bridge mechanical response prediction model can be a Physical Information Neural Network (PINNs).

[0034] In some embodiments, the sampling points can be pre-selected physical points within the bridge computational domain, or they can be sample points obtained through data fitting. For physical points, their coordinates are typically generated according to certain rules; for sample points obtained through data fitting, their coordinates can correspond to the actual sensor measurement point locations or the observation node locations in finite element simulations.

[0035] In some embodiments, after receiving the coordinates of the sampling points and the specified working parameters, the bridge mechanical response prediction model performs forward propagation calculations to obtain the prediction output at each sampling point. In static analysis scenarios, the prediction output may include displacement fields, stress fields, etc.

[0036] Step 104: Calculate each loss component based on the predicted output, with each loss component corresponding to a constraint type.

[0037] In some embodiments, the loss component is a part of the total loss function that corresponds to a single physical constraint or data fitting objective, such as partial differential equation residual loss, boundary condition loss, data fitting loss, etc.

[0038] In some embodiments, based on the prediction output of step 102 described above, the specific values ​​of each loss component can be calculated. There are typically two loss components, each corresponding to a different constraint type. For example, the loss components may include partial differential equation residual loss, boundary condition loss, and data fitting loss, etc.

[0039] Step 106: For each loss component, calculate the gradient of the loss component with respect to the network parameters of the bridge mechanical response prediction model, and calculate the effective learning rate of the loss component based on the gradient. The effective learning rate is used to measure the convergence speed of the corresponding loss component.

[0040] In some embodiments, the effective learning rate is a quantitative indicator reflecting the sensitivity of a loss component to network parameter updates, and can be used to measure the convergence speed of the corresponding loss component. Specifically, for each loss component, its gradient with respect to the network parameters can be calculated, and the square of the magnitude of this gradient can be used as its effective learning rate.

[0041] In some embodiments, a larger effective learning rate indicates that, under the current parameter state, the loss component is more sensitive to small changes in the parameters, meaning the corresponding constraint term "learns faster." Conversely, a small effective learning rate indicates that the corresponding constraint term is not sensitive to parameter updates, resulting in slow convergence.

[0042] Step 108: Dynamically adjust the weight coefficients of each loss component based on the effective learning rate of each loss component. The effective learning rate and weight coefficients of the loss components are negatively correlated.

[0043] In some embodiments, after calculating the effective learning rate of each loss component, the weight coefficients of each loss component can be dynamically adjusted according to the principle of "reducing the weight coefficient for components with high learning rates and increasing the weight coefficient for components with low learning rates," meaning that the effective learning rate and the weight coefficient are negatively correlated. Therefore, for loss components with high effective learning rates, their weight coefficients are reduced to suppress their dominant role, while for loss components with low effective learning rates, their weight coefficients are increased to enhance their influence.

[0044] Step 110: Calculate the total loss based on each loss component and its corresponding weight coefficient, and update the network parameters of the bridge mechanical response prediction model based on the total loss.

[0045] In some embodiments, the calculated loss components can be weighted and summed with the weight coefficients determined in the preceding steps to obtain the total loss, and the network parameters of the bridge mechanical response prediction model can be updated using gradient descent or other algorithms. This completes one training iteration until the bridge mechanical response prediction model converges or reaches a preset number of training rounds.

[0046] As can be seen from the above description, by employing the scheme provided in this specification, calculating the gradient of each loss component with respect to the network parameters, determining the effective learning rate to measure the convergence speed of the corresponding loss component based on the gradient, and adjusting the weight coefficients of each loss component according to the principle that the effective learning rate is negatively correlated with the weight coefficients, it is possible to maintain balanced convergence of different physical constraints during the training process. This reduces the dependence on large-scale finite element simulation datasets. Even under small sample conditions, i.e., with only a small amount of labeled observation data or sparse simulation samples, the gradient contributions of each constraint term can be balanced in the total loss, preventing one constraint term from dominating training due to excessively large gradients or another constraint term from stagnating due to excessively small gradients, thereby training a stable and convergent bridge mechanical response prediction model.

[0047] In some embodiments, the effective learning rate of the loss component can be calculated by directly calculating the square of the gradient magnitude. However, for deep neural networks with a large number of parameters, while mathematically feasible, directly calculating the gradient of each loss component and taking the square of its magnitude is computationally expensive. To reduce computational cost, a stochastic estimation method can be used when calculating the effective learning rate of the loss component. This involves estimating the trace of the tangent kernel matrix corresponding to the loss component by calculating the statistics of the square of the gradient of at least one random probe vector and the inner product of that random probe vector, and then determining the estimated trace as the effective learning rate of the loss component.

[0048] The Neural Tangent Kernel (NTK) is a kernel function that describes the training dynamics of a neural network in the infinite width limit. The trace of its matrix can approximately reflect the convergence speed of different loss components.

[0049] Assume the network parameters are Various types of losses The neural tangent nucleus (NTK) matrix is ​​as follows: in, Represents the residual loss of partial differential equations. Represents boundary loss. Represents data loss. Represents the coordinates of the sampling point. Representing the The NTK kernel matrix corresponding to the loss components. Representing the Residual or output function.

[0050] To reduce computational overhead, the Hutchinson random trace estimation method can be used, which estimates the trace value using a random probe vector: in, This represents the NTK trace estimate for the current iteration. Represents the number of random probes. Representing the A random probe vector, The value ranges from 1 to , Representing the Residual function relative to network parameters Jacobian matrix, Represents the identity matrix.

[0051] Therefore, when calculating the effective learning rate, it is possible to first generate independent random probe vectors For each Calculate gradient and Take the inner product, then square it, and finally calculate... The mean of the squared values ​​is used as the NTK trace estimate, and the estimated trace value is used as the effective learning rate of the corresponding loss component.

[0052] In some embodiments, the weighting coefficients of each loss term can be dynamically adjusted based on the calculated effective learning rate: in, Representing the first iteration The weighting coefficients of the loss components. This represents the reference effective learning rate. Representing the Set of class constraint points or sampling points Representative set The number of elements.

[0053] In some embodiments, after calculating the weight coefficients of each loss component, the total loss can be calculated by weighted summation to update the network parameters of the bridge mechanical response prediction model. Taking a static analysis scenario as an example, the formula for calculating the total loss is as follows: in, Represents the total static loss. Represents the residual loss of partial differential equations. The weighting coefficients represent the residual loss of the partial differential equation in the current iteration. Represents boundary condition loss. This represents the weighting coefficient of the boundary condition loss in the current iteration. Represents the data fitting loss. The weight coefficients representing the data fitting loss in the current iteration are: This represents the constraint loss at the interface of a component connection or computational subdomain. It is used to constrain conditions such as displacement continuity and internal force / stress balance at the boundaries of different structural components or computational subdomains of a bridge. Examples include beam-cable connections, beam-pier / support connections, and segmented modeling interfaces. If no separate interface is defined, this loss component can be set to zero or incorporated into the boundary condition loss. This represents the weight coefficient for the connection or calculation of the interface constraint loss of the component, which remains unchanged in different iteration rounds.

[0054] In some embodiments, the residual loss of partial differential equations The calculation formula is as follows: in, This represents the number of sampling points for the partial differential equation. This represents the residual after the governing equations are applied to the predicted output. Representing the One sampling point.

[0055] In some embodiments, boundary condition loss The calculation formula is as follows: in, Represents the number of boundary points. This represents the boundary values ​​after the boundary operator is applied to the predicted output. This represents the target value given the boundary conditions. Representing the One sampling point.

[0056] In some embodiments, data fitting loss The calculation formula is as follows: in, Represents the number of data samples. Representative sampling point The predicted output, Representative sampling point Sparse measured or finite element simulation labeled response, Representing the One sampling point.

[0057] Therefore, when the effective learning rate of a certain loss component is high, it indicates that the loss component is more sensitive to the update of network parameters and converges faster in the current training phase. Using the above formula, the weight coefficient of this loss component is calculated to be relatively small, thus suppressing its contribution to the total loss. Conversely, when the effective learning rate of a certain loss component is low, the weight coefficient of this loss component is calculated to be relatively large using the above formula, and its weight coefficient is adjusted accordingly to enhance the influence of this constraint on parameter updates. Through this dynamic adjustment mechanism, the training imbalance phenomenon of large gradient loss components suppressing small gradient loss components is avoided, thereby ensuring that all physical constraint terms maintain balanced convergence during training under small sample conditions, achieving stable and efficient model convergence.

[0058] In some embodiments, the distribution of bridge sampling points has a significant impact on the model training effect, and the selection of sampling points can be a fixed set of sampling points determined at the initial time. However, in small-sample training scenarios for bridge structures, a uniformly random sampling method can easily lead to insufficient sampling point density in high-stress gradient regions (such as near supports, cable-beam connections, and locations of abrupt changes in cross-section), thereby limiting the prediction accuracy of the model in these critical regions.

[0059] Therefore, this embodiment introduces an adaptive sampling point update scheme during training, which can be executed in each training round or every preset number of iteration steps. Please refer to [link / reference]. Figure 2 This includes the following steps: Step 202: Select multiple candidate points and calculate the physical residual value and the local curvature value of the bridge structure response field at each candidate point based on the prediction output.

[0060] In some embodiments, a candidate point set covering the entire bridge computational domain can be generated or maintained. This candidate point set can be a dense grid of points or a randomly sampled set of points. For each candidate point, the physical residual value and local curvature value can be calculated using the current bridge mechanical response prediction model (i.e., the model network parameters of the current iteration step).

[0061] Among them, the physical residual value is the residual norm of the governing equation at the candidate point, and the local curvature value is the Frobenius norm of the displacement field Hessian matrix. The Hessian matrix reflects the curvature change of the displacement field in space. The larger its value, the more drastic the response change near the candidate point (e.g., stress concentration region).

[0062] Step 204: Construct a non-uniform sampling probability density based on the physical residual value and the local curvature value.

[0063] In some embodiments, a non-uniform sampling probability density can be constructed based on the physical residual value and the local curvature value: in, This represents the sampling probability density of the current iteration. The residuals representing the static control equation residuals after the residual operators are applied to the predicted output. The Hessian curvature matrix representing the predicted displacement field. As curvature weight, Basic coverage items, For vector 2 norm, Let Frobenius be the matrix norm.

[0064] Step 206: Select new sampling points from the candidate points according to the non-uniform sampling probability density for subsequent iterative training.

[0065] In some embodiments, sampling with or without replacement can be performed from the candidate point set according to a non-uniform sampling probability density, and a preset number of candidate points (e.g., 500 new points each time) can be selected as new sampling points for subsequent iterative training of the bridge mechanical response prediction model.

[0066] In some embodiments, for bridge structures with complex geometric boundaries, candidate points generated by random sampling may fall outside the physical domain of the bridge (e.g., the air domain below the bridge deck or voids inside the foundation). To avoid invalid sampling, a signed distance function (SDF) can be used to describe the geometric boundaries of the bridge structure. By filtering with SDF or projecting correction, newly added sampling points can be constrained to the interior or boundary of the valid physical domain.

[0067] Therefore, by adopting the above sampling point update strategy, newly added sampling points can be dynamically migrated to key regions with high residuals and high curvature. This improves the physical constraint density and training efficiency of key regions without significantly increasing the total number of sampling points, thereby enhancing the utilization of computing resources and the accuracy of local response prediction of the model.

[0068] In some embodiments, the bridge mechanical response prediction model can adopt a standard fully connected network structure, and its hidden layer activation function is usually a hyperbolic tangent or linear rectified unit. However, the standard activation function has the problem of spectral bias: the network tends to preferentially learn the low-frequency components in the input signal, and its ability to capture high-frequency abrupt features such as stress concentration and local vibration is weak. This problem is particularly prominent in bridge structural health monitoring scenarios that require precise prediction of stress field. To overcome the above-mentioned spectral bias, this embodiment can improve the network structure of the bridge mechanical response prediction model.

[0069] In some embodiments, the input layer of the bridge mechanical response prediction model employs Random Fourier Feature Embedding (RFFE) to pre-encode the coordinates of the sampling points. In other words, RFFE is introduced into the input layer of the bridge mechanical response prediction model to pre-encode the coordinates of the input sampling points, thus mitigating the network's tendency to degrade to low frequencies. Specifically, random feature mapping can be applied to the input sampling point coordinates to generate high-frequency feature representations. This encoding method introduces high-frequency basis functions into the model network by mapping the input to a linear combination of high-dimensional random Fourier basis functions, enabling the model network to achieve multi-scale spatial resolution at the input layer.

[0070] The form of random Fourier feature encoding is: in, Represents a high-dimensional periodic feature map. Represents the coordinates of the sampling point. Represents operating parameters, Representing the A set of random frequency matrices or row vectors.

[0071] The value ranges from 1 to , Represents the number of random frequency groups. Represents the standard deviation of random frequency sampling. Represents the identity matrix.

[0072] In some embodiments, the hidden layers of the bridge mechanical response prediction model may employ multi-scale periodic activation functions; that is, in each hidden layer (or several specific layers) of the model network, a multi-scale periodic activation function replaces the standard nonlinear activation function. Layered network propagation can be represented as: in, Representing the Layer output, Representing the The weight matrix of the layer, Representing the The layer's bias vector, The multi-scale periodic activation function can be defined as: Represents the output of the hidden layer. This represents the number of superposition terms of sinusoidal basis functions in a multi-scale periodic activation function. Representing the Trainable amplitude coefficients of sinusoidal basis functions Representing the The frequency coefficients of the sinusoidal basis functions Representing the The phase parameters of each sinusoidal basis function.

[0073] The frequency coefficients of each wavelet are distributed in a geometric series: , This represents the geometric growth rate of the frequency coefficient.

[0074] Therefore, by superimposing multi-frequency sinusoidal basis functions, the model network can simultaneously capture the macroscopic deformation modes and local high-gradient responses of the bridge structure in the same forward propagation.

[0075] Of course, multi-scale activation functions can also use wavelet basis functions of different scales to replace sinusoidal basis functions, and can also achieve multi-frequency expression capabilities, without any restrictions.

[0076] In some embodiments, the two improvements to the model network structure described above (random Fourier feature encoding and multi-scale periodic activation function) can be used selectively or simultaneously. The input layer employs random Fourier feature encoding, enabling the model network to perceive high-frequency spatial variations through high-frequency precoding in the input layer, which can alleviate spectral bias to some extent. The hidden layer employs a multi-scale periodic activation function, allowing the model network to obtain wide-spectrum representation capabilities through the superposition of multi-frequency sinusoidal bases in the hidden layer, thus improving the ability to capture high-frequency responses. The combination of these two techniques enables the model network to have good fitting capabilities across the entire frequency band (from low to high frequencies), significantly improving the prediction accuracy in stress concentration regions.

[0077] The aforementioned embodiments primarily describe static analysis scenarios for bridge structures, where the predicted outputs of the bridge mechanical response prediction model are displacement and stress fields. However, in actual bridge health monitoring, in addition to static response, dynamic time history responses (such as vibration time histories under earthquakes and vehicle loads) and modal parameters (natural frequencies, mode shapes) are also important indicators for assessing the safety status of bridge structures. Therefore, this embodiment, based on the aforementioned training method, extends the predicted output of the bridge mechanical response prediction model from static to dynamic, achieving integrated static and dynamic prediction.

[0078] In this embodiment, the input to the bridge mechanical response prediction model also includes a time variable, and the mechanical response output by the model includes a dynamic time-history response in the time dimension. Accordingly, the loss component includes a time-domain dynamic constraint loss, which is used to constrain the dynamic time-history response to satisfy the bridge structure dynamic control equations, whether damped or undamped.

[0079] Specifically, the inputs to the bridge mechanical response prediction model also include time variables. The output mechanical response includes the dynamic time history response with a time dimension. This refers to the displacement (or velocity, acceleration) curve of the structure at the sampling point location over time. The bridge mechanical response prediction model may or may not use the aforementioned multi-scale periodic activation function and RFFE encoding; only the time... It can be added to the encoding process as an additional input dimension.

[0080] In addition, a time-domain dynamic constraint loss needs to be added to the loss function. This loss is used to constrain the dynamic time-history response output of the model to satisfy the damped structural dynamic control equations. Taking the common Rayleigh damped model as an example, the dynamic control equations are: in, Represents dynamic residual, The dynamic time history response representing the model output, Represents material density, Represents the Rayleigh damping equivalent coefficient or damping operator. This represents the stress tensor derived from the predicted displacement field. This represents physical forces or external loads.

[0081] Temporal dynamics constraint loss It can be defined as the mean of the squared residuals of the governing equations at all spatiotemporal sampling points: in, The number of spatiotemporal sampling points. It is a set of spatiotemporal sampling points.

[0082] In this embodiment, during model training, the total loss function is based on the aforementioned static loss (partial differential equation residual loss). Boundary condition loss Data fitting loss Based on the equal loss, add time-domain dynamic constraint loss. This is then multiplied by the weighting coefficients determined by the aforementioned adaptive weighting mechanism. Thus, during training, the model not only learns static equilibrium relationships but also dynamic evolution laws, ensuring that its output time-history response satisfies the physical control equations.

[0083] In this embodiment, for structures without damping or using other damping models, only the expression for the damping term in the formula needs to be modified accordingly, while the rest of the training process remains unchanged.

[0084] In some embodiments, based on the above-mentioned time-domain dynamic constraints, frequency-domain modal characteristic constraint loss can also be introduced to achieve synchronous learning of modal parameters (natural frequencies, mode shapes). This frequency-domain modal characteristic constraint loss is used to constrain the bridge modal parameters extracted from the dynamic time history response to satisfy the generalized eigenvalue relationship and / or the mode shape mass orthogonality relationship.

[0085] In this embodiment, the dynamic time history response output by the model The frequency domain representation can be obtained by performing a Fast Fourier Transform (FFT). In the frequency domain, the structure's first... First natural frequency Corresponding to The peak position of the power spectral density function. For each natural frequency. Extract the spatial distribution of displacement at this frequency as the mode shape vector. : in, This represents the Fourier transform with respect to the time variable. Representing the Predicted mode shape.

[0086] In this embodiment, the frequency domain modal characteristic constraints include generalized eigenvalue equation constraints and modal mass orthogonality constraints. The generalized eigenvalue equation constraint ensures that the extracted natural frequencies and mode shapes satisfy the characteristic equation for undamped free vibration of the structure. The loss corresponding to this constraint can be defined as the sum of squares of the residuals of the characteristic equation at all natural frequencies and corresponding mode shape sampling points. The modal mass orthogonality constraint ensures that different mode shapes are orthogonal with respect to the mass matrix and the stiffness matrix. Therefore, the frequency domain modal characteristic constraint loss... The following formula can be used for calculation: in, and Represents the Lamé elastic constant. This represents the modal order of the modal loss included in the frequency domain. It is the identity matrix. Represents the orthogonal penalty weight. Represents a structural domain. Represents the Kroneck symbol (when) hour, =1, otherwise (0). In the formula, the first term is the constraint of the generalized eigenvalue equation, and the second term is the orthogonality constraint of the modal mass.

[0087] In some embodiments, in the scenario of hydrostatic-dynamic hybrid analysis, the total loss of the integrated hydrostatic-dynamic analysis is... as follows: in, For temporal dynamic constraint loss The weighting coefficients, Frequency domain modal feature constraint loss The weighting coefficients are determined by the effective learning rate of each loss component.

[0088] In some embodiments, during the training phase of the bridge mechanical response prediction model, the partial differential equation residual loss, boundary condition loss, data fitting loss, time-domain dynamic residual loss, and frequency-domain modal residual loss are calculated in the same training batch. Based on the trace estimation results of the neural tangent kernel, the weight coefficients of the above losses are dynamically refreshed in each iteration step or every preset number of iteration steps. Then, backpropagation is performed on the weighted total loss to update the network parameters of the model.

[0089] In some embodiments, during the inference phase, for any given new operating condition parameters... The trained bridge mechanical response prediction model performs one forward propagation: This allows for the simultaneous output of the bridge's static displacement field, stress field, natural frequencies, mode shapes, and dynamic time history response, without the need to rebuild the finite element model or solve large sets of equations. For network parameters Bridge mechanical response prediction model, This is the corresponding predicted output.

[0090] Figure 3 This is a flowchart of another training method for a bridge mechanical response prediction model provided in an exemplary embodiment.

[0091] Please refer to Figure 3 The bridge mechanical response prediction model can receive three types of input data, including sampling point coordinates, time variables, and specified load condition parameters (such as material elastic modulus, Poisson's ratio, load condition parameters, boundary condition identifiers, etc.). These input data are first pre-encoded using Random Fourier Feature Encoding (RFFE) to map the original input to a high-dimensional random Fourier basis function space, outputting a high-frequency feature vector. This encoded feature vector replaces the original input and is fed into the model's backbone network.

[0092] The backbone network of the bridge mechanical response prediction model adopts a multi-layer fully connected structure. Its hidden layers use multi-scale periodic activation functions. By superimposing multiple sinusoidal basis functions of different frequencies, the network can simultaneously capture the macroscopic low-frequency deformation modes and local high-frequency abrupt responses of the bridge structure in the same forward propagation. The backbone network can output the full-field mechanical response of the bridge, including the static displacement field, static stress field, dynamic time history response, natural frequencies, and mode shape parameters. Among them, the natural frequencies and mode shapes can be extracted by performing a fast Fourier transform on the output dynamic time history response.

[0093] After obtaining the model prediction output through forward propagation, various types of loss components can be calculated. These include: partial differential equation residual loss, boundary condition loss, data fitting loss, temporal dynamic constraint loss, and frequency domain modal feature constraint loss. The calculation of each of these loss components can be achieved using the formulas in the aforementioned embodiments.

[0094] After calculating the scalar values ​​of each loss component, the gradient of each loss component with respect to the network parameters is calculated. The Hutchinson stochastic estimation method is then used to approximate the trace of the neural tangent kernel matrix corresponding to each loss component, and the estimated trace is taken as the effective learning rate for that loss component. Then, the weight coefficients of each loss component are dynamically adjusted according to the principle that the effective learning rate is negatively correlated with the weight coefficients. Finally, each loss component is multiplied by its corresponding weight coefficient and summed to obtain the total loss. The network parameters are then updated based on backpropagation using the total loss.

[0095] During training, an adaptive sampling point supplementation operation can also be performed. Specifically, under the current model parameters, for each candidate point in the pre-defined candidate point set within the bridge computational domain, the physical residual value and local curvature value are calculated, and a non-uniform sampling probability density is constructed based on these two indicators. Then, new sampling points are selected from the candidate points according to this probability density, and the newly added sampling points are projected and corrected to the interior of the bridge physical domain using a signed distance function. The newly added sampling points are supplemented into the sample set for loss calculation in subsequent iterations.

[0096] Therefore, by adopting the above method, it is possible to train a bridge mechanical response prediction model with static, dynamic and modal prediction capabilities with only a small amount of labeled observation data or sparse simulation samples, that is, without the need for large-scale labeled samples, and with high prediction accuracy in key areas such as stress concentration.

[0097] In some embodiments, a method for predicting the mechanical response of a bridge is also provided; please refer to [reference needed]. Figure 4 The method for predicting the mechanical response of the bridge may include the following steps: Step 402: Obtain the coordinates of the sampling points, time variables, and specified working condition data.

[0098] In this embodiment, the target working condition data to be predicted can be obtained first. Specifically, the sampling point coordinates refer to the set of spatial location points within the bridge structure computational domain that need to be predicted for response. For example, the sampling points can be the coordinates of all nodes in the bridge finite element model, or the coordinates of measurement points on key sections selected according to prediction requirements (such as mid-span sections, support sections, cable-beam connection points, etc.). The set of sampling point coordinates can be predefined and stored, or it can be dynamically specified by the user during prediction.

[0099] Time variables can be used when it is necessary to predict the dynamic time history response. For scenarios where only static response prediction is required, the time variable can be omitted or set to a constant (e.g., t=0). When it is necessary to predict the dynamic time history response, the time variable is usually defined as a time series, such as from 0 seconds to 10 seconds, with a sampling interval of 0.01 seconds, for a total of 1001 time points.

[0100] The specified load case data includes the bridge structure's load parameters, material parameters, and boundary conditions under the current predicted scenario. Load parameters may include the magnitude and location of vehicle loads, time history curves of wind loads, and seismic acceleration time histories; material parameters may include the elastic modulus, Poisson's ratio, and density of concrete and steel; boundary conditions may include the constraint type (hinged, fixed, or elastic) and constraint location of the supports. This load case data can be obtained from bridge design documents, real-time monitoring systems, or user input.

[0101] Step 404: Input the sampling point coordinates, the time variable, and the specified working condition data into the trained bridge mechanical response prediction model to obtain the prediction output of the bridge mechanical response prediction model. The prediction output includes at least one of static displacement field, stress field, dynamic time history response, and bridge modal parameters.

[0102] After obtaining the input data, it is assembled into the input tensor required by the model, and the input tensor is then fed into the trained bridge mechanical response prediction model. This bridge mechanical response prediction model is trained using the training method provided in the aforementioned embodiments, which will not be described in detail here.

[0103] The bridge mechanical response prediction model performs a forward propagation calculation and outputs the predicted mechanical response value for each sampling point at each time point. Specifically, it may include at least one of the static displacement field, stress field, dynamic time history response, and bridge modal parameters.

[0104] Because the bridge mechanical response prediction model has already learned the static and dynamic response laws of the bridge structure through physical constraints during the training phase, only one forward propagation is needed during inference to obtain all the required mechanical responses. Compared with the traditional finite element method, it does not require rebuilding a large-scale sparse equation system and solving iteratively; compared with purely data-driven deep learning methods, it does not rely on a large number of labeled samples to obtain physically consistent prediction results.

[0105] Figure 5 This is a schematic structural diagram of a device provided in an exemplary embodiment. Please refer to... Figure 5 At the hardware level, the device includes a processor 502, an internal bus 504, a network interface 506, memory 508, and non-volatile memory 510, and may also include other hardware required for its functions. One or more embodiments of this specification can be implemented in software, for example, the processor 502 reads the corresponding computer program from the non-volatile memory 510 into memory 508 and then runs it. Of course, in addition to software implementation, one or more embodiments of this specification do not exclude other implementation methods, such as logic devices or a combination of hardware and software, etc. That is to say, the execution subject of the following processing flow is not limited to each logic unit, but can also be hardware or logic devices.

[0106] Please refer to Figure 6 The training device 600 for predicting bridge mechanical response can be applied to, for example... Figure 5 The device shown is used to implement the technical solution described in this specification. The training device 600 for the bridge mechanical response prediction model may include: The prediction output acquisition unit 601 acquires the prediction output of the bridge mechanical response prediction model at at least one sampling point, wherein the prediction output is the mechanical response of the bridge structure under a given working condition. The loss component calculation unit 602 calculates each loss component based on the prediction output, and each loss component corresponds to a constraint type. The learning rate calculation unit 603 calculates the gradient of each loss component with respect to the network parameters of the bridge mechanical response prediction model, and calculates the effective learning rate of the loss component based on the gradient. The effective learning rate is used to measure the convergence speed of the corresponding loss component. The weight coefficient adjustment unit 604 dynamically adjusts the weight coefficient of each loss component according to the effective learning rate of each loss component. The effective learning rate of the loss component and the weight coefficient are negatively correlated. The network parameter update unit 605 calculates the total loss based on each loss component and its corresponding weight coefficient, and updates the network parameters of the bridge mechanical response prediction model based on the total loss.

[0107] Optionally, calculating the effective learning rate of the loss component based on the gradient includes: A stochastic estimation method is used to estimate the trace of the neural tangent kernel matrix corresponding to the loss component by calculating the statistic of the square of the inner product of the gradient and the random probe vector for at least one random probe vector. The estimated trace is determined as the effective learning rate for the loss component.

[0108] Optionally, the device further includes: A new sampling point unit 606 is added to select multiple candidate points and calculate the physical residual value and the local curvature value of the bridge structure response field at each candidate point based on the prediction output; a non-uniform sampling probability density is constructed based on the physical residual value and the local curvature value; new sampling points are selected from the candidate points according to the non-uniform sampling probability density for subsequent iterative training.

[0109] Optionally, the hidden layer of the bridge mechanical response prediction model employs a multi-scale periodic activation function, and / or, The input layer of the bridge mechanical response prediction model uses random Fourier feature encoding to pre-encode the coordinates of the sampling points.

[0110] Optionally, the input to the bridge mechanical response prediction model further includes a time variable, the mechanical response includes a dynamic time history response in the time dimension, and the loss component includes a time-domain dynamic constraint loss, which is used to constrain the dynamic time history response to satisfy the bridge structure dynamic control equations with or without damping.

[0111] Optionally, the loss component further includes frequency domain modal characteristic constraint loss, which is used to constrain the bridge modal parameters extracted from the dynamic time history response to satisfy generalized eigenvalue relationships and / or modal mass orthogonality relationships.

[0112] Please refer to Figure 7 The bridge mechanical response prediction device 700 can also be applied to, for example... Figure 5 The device shown implements the technical solution described in this specification. The bridge mechanical response prediction device 700 may include: Data acquisition unit 701 acquires sampling point coordinates, time variables, and specified working condition data; The response prediction unit 702 inputs the sampling point coordinates, the time variable and the specified working condition data into the trained bridge mechanical response prediction model to obtain the prediction output of the bridge mechanical response prediction model. The prediction output includes at least one of static displacement field, stress field, dynamic time history response and bridge modal parameters. The bridge mechanical response prediction model was trained using the aforementioned method.

[0113] Based on the same concept as the methods described above, this specification also provides an electronic device, including: a processor; a memory for storing processor-executable instructions; wherein the processor performs the steps of the method as described in any of the above embodiments by executing the executable instructions.

[0114] Based on the same concept as the methods described above, this specification also provides a computer-readable storage medium having computer instructions stored thereon that, when executed by a processor, implement the steps of the methods as described in any of the above embodiments.

[0115] Based on the same concept as the methods described above, this specification also provides a computer program product, including a computer program / instructions that, when executed by a processor, implement the steps of the methods as described in any of the above embodiments.

Claims

1. A training method for a bridge mechanical response prediction model, characterized in that, The method includes: Obtain the prediction output of the bridge mechanical response prediction model at at least one sampling point, wherein the prediction output is the mechanical response of the bridge structure under a given working condition; Each loss component is calculated based on the predicted output, and each loss component corresponds to a constraint type. For each loss component, the gradient of the loss component with respect to the network parameters of the bridge mechanical response prediction model is calculated, and the effective learning rate of the loss component is calculated based on the gradient. The effective learning rate is used to measure the convergence speed of the corresponding loss component. The weight coefficients of each loss component are dynamically adjusted based on the effective learning rate of each loss component. The effective learning rate and weight coefficient of each loss component are negatively correlated. The total loss is calculated based on each loss component and its corresponding weight coefficient, and the network parameters of the bridge mechanical response prediction model are updated based on the total loss.

2. The method according to claim 1, characterized in that, The step of calculating the effective learning rate of the loss component based on the gradient includes: A stochastic estimation method is used to estimate the trace of the neural tangent kernel matrix corresponding to the loss component by calculating the statistic of the square of the inner product of the gradient and the random probe vector for at least one random probe vector. The estimated trace is determined as the effective learning rate for the loss component.

3. The method according to claim 1, characterized in that, The method further includes: Multiple candidate points are selected, and the physical residual value and the local curvature value of the bridge structure response field at each candidate point are calculated based on the prediction output. A non-uniform sampling probability density is constructed based on the physical residual value and the local curvature value; New sampling points are selected from the candidate points according to the non-uniform sampling probability density for subsequent iterative training.

4. The method according to claim 1, characterized in that, The hidden layer of the bridge mechanical response prediction model employs a multi-scale periodic activation function, and / or, The input layer of the bridge mechanical response prediction model uses random Fourier feature encoding to pre-encode the coordinates of the sampling points.

5. The method according to claim 1, characterized in that, The input to the bridge mechanical response prediction model also includes a time variable. The mechanical response includes a dynamic time history response in the time dimension. The loss component includes a time-domain dynamic constraint loss, which is used to constrain the dynamic time history response to satisfy the bridge structure dynamic control equations with or without damping.

6. The method according to claim 5, characterized in that, The loss component also includes frequency domain modal feature constraint loss, which is used to constrain the bridge modal parameters extracted from the dynamic time history response to satisfy the generalized eigenvalue relationship and / or the mode shape mass orthogonality relationship.

7. A method for predicting the mechanical response of a bridge, characterized in that, The method includes: Acquire sampling point coordinates, time variables, and specified operating condition data; The sampling point coordinates, the time variable, and the specified working condition data are input into the trained bridge mechanical response prediction model to obtain the prediction output of the bridge mechanical response prediction model. The prediction output includes at least one of static displacement field, stress field, dynamic time history response, and bridge modal parameters. The bridge mechanical response prediction model is trained using the method described in any one of claims 1 to 6.

8. An electronic device, characterized in that, include: processor; A memory for storing processor-executable instructions; wherein the processor implements the steps of the method as described in any one of claims 1-7 by executing the executable instructions.

9. A computer-readable storage medium, characterized in that, It stores computer instructions that, when executed by a processor, implement the steps of the method as described in any one of claims 1-7.

10. A computer program product, characterized in that, Includes a computer program / instructions that, when executed by a processor, implement the steps of the method as described in any one of claims 1-7.

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