A method and application for predicting peak displacement envelope curve and track irregularity of sliding layer in high-speed railway system considering ground motion and pier height uncertainty

Abstract

This invention discloses a method and application for predicting peak displacement envelope curves and track irregularities in the sliding layer of a high-speed rail system, taking into account the uncertainty of ground motion and pier height. The method constructs a multimodal hybrid neural network model, TransTCN, that embeds domain prior knowledge. Feature extraction is performed on multimodal input features, including a time series based on ground motion fusion structural features and a spatial series of peak displacement outputs based on the elastic response spectrum method. Pier height and ground motion features are dimensionally fused as time series features to address temporal dependencies. Elastic response spectrum analysis results are used as spatial features to address spatial dependencies. The extracted features are then input into the constructed multimodal hybrid neural network model, TransTCN. This method predicts peak displacement envelope curves and track irregularities in the sliding layer of a high-speed rail system, taking into account the uncertainty of ground motion and pier height. This method improves prediction accuracy while reducing training sample requirements, enabling rapid assessment of regional high-speed rail system damage and possessing excellent practical value.

Description

Technical Field

[0001] The present invention relates to a method and application for predicting a peak displacement envelope curve of a sliding layer of a high-speed railway system and track irregularity taking into account uncertainty of earthquake motion and pier height. Background Art

[0002] As China's high-speed rail network expands into earthquake-prone areas, the complexity of track-bridge systems makes them more susceptible to damage during earthquakes, jeopardizing both structural and operational safety. Therefore, improving the resilience of high-speed rail lines to major natural disasters and ensuring their capacity is a key research priority. When an earthquake strikes the existing high-speed rail network, train operations are typically interrupted to ensure operational safety. This approach fails to adequately consider the inherent resilience of the track-bridge system, resulting in an overly conservative approach that significantly hinders post-disaster relief efforts. Therefore, rapidly assessing and repairing structural damage and track deformation across multiple track-bridge systems within a limited timeframe can effectively guide post-earthquake relief efforts and ensure operational safety.

[0003] Traditional methods for predicting and assessing the post-earthquake safety of track-bridge systems assess damage by calculating the nodal responses of key components and assessing driving safety by calculating track irregularities based on track surface deformation. However, these assessment methods rely on data derived from physics-based surrogate models, obtained through nonlinear finite element time-history analysis. Due to the random nature of earthquake excitation and the variability of structural characteristics, model-by-model time-history analysis is inefficient when designing or assessing multiple structures. Consequently, the computational costs of developing complex numerical models and performing the analyses have limited their practical adoption.

[0004] In addition, existing evaluation methods mostly use the response spectrum method (RSM) to estimate the average peak response of multi-degree-of-freedom (MDOF) systems. Although this method has good response prediction accuracy in practical applications, the RSM method cannot effectively handle structural nonlinearity. As the predicted structural degrees of freedom (DOFs) increase, the prediction error increases significantly, which will seriously threaten the correct assessment of structural damage and driving safety.

[0005] There are also reports using deep learning-based methods to implement nonlinear mapping of dynamic responses using data-driven models to improve computational efficiency. However, these methods are all neural network response prediction frameworks for the building sector and typically focus on predicting the response of a single key location or node, using inter-story drift angles or the responses of key beams and columns to assess the overall damage level of the building. However, for longitudinally long structures such as roads, bridges, and high-speed railways, single-target predictions are insufficient to reflect the overall damage to the structure. Therefore, the prediction objective naturally shifts to the peak displacement envelope or various overall linear indicators of the long structure. Furthermore, pier stiffness significantly affects the superstructure response of the track-bridge system, and damage is reflected in track surface deformation. Post-earthquake track structure deformation and damage can significantly impact post-earthquake train safety. Therefore, considering the uncertainty of pier height, comprehensively reflecting the earthquake damage of the high-speed rail network is crucial to timely assess operational risks. Furthermore, accurate response prediction for structures with complex degrees of freedom and high nonlinearity requires a large number of training samples. Therefore, reducing training requirements and achieving high-precision response prediction with a smaller sample size is a key challenge.

[0006] Therefore, how to fully consider the inherent resilience of the track-bridge system, assess the extent of damage to the track-bridge system and the degree of track surface deformation as quickly as possible under the uncertainty of earthquake motion and pier height, overcome the computational burden of purely physics-based proxy models and the modeling difficulties caused by changes in structural parameters, and at the same time significantly improve the fitting accuracy of pure data-driven models and reduce the amount of training data required, remains an urgent problem to be solved. Summary of the Invention

[0007] In order to solve the above problems and achieve the above objectives, the present invention first provides a method for predicting the peak displacement envelope curve of the sliding layer and track irregularity of a high-speed railway system taking into account the uncertainty of ground motion and pier height, which specifically includes the following steps:

[0008] 1) Construct a multimodal hybrid neural network model TransTCN that embeds domain prior knowledge;

[0009] 2) Multimodal input feature extraction is performed on the time series based on the fusion of seismic motion characteristics with pier height characteristics and the spatial series of peak displacement output based on the elastic response spectrum method. The seismic motion characteristics and pier height characteristics are dimensionally fused as time series features to handle temporal dependencies; the elastic response spectrum analysis results are used as spatial features to handle spatial dependencies.

[0010] 3) The multimodal input features extracted in step 2) are input into the multimodal hybrid neural network model TransTCN constructed in step 1) for processing, so as to realize the prediction of the peak displacement envelope curve of the sliding layer and the track irregularity of the high-speed railway system considering the uncertainty of the earthquake motion and the pier height.

[0011] In the aforementioned method for predicting the peak displacement envelope curve of the sliding layer and track irregularity of a high-speed railway system considering the uncertainty of ground motion and pier height, in step 1), the multimodal hybrid neural network model TransTCN is a long sequence prediction framework combining the Transformer and TCN networks, which includes a Transformer module and a TCN module;

[0012] The Transformer module is used to capture the temporal characteristics of earthquake motion and the spatial structure characteristics of the response spectrum method. It includes an embedding layer, a multi-head self-attention layer, and a fully connected feedforward network layer.

[0013] The TCN module is used to capture complete temporal dependencies, which includes multiple dilated causal convolutional layers and a cross-path layer, and uses rectified linear units as activation functions and batch normalization as convolution filters.

[0014] In the aforementioned method for predicting the peak displacement envelope curve of the sliding layer and track irregularity of a high-speed railway system considering the uncertainty of ground motion and pier height, in step 2), the time series based on the ground motion fusion structural characteristics is a sequence obtained by fusing seismic data processing and pier height characteristics. The specific fusion process is as follows:

[0015] The earthquake motion sequence is expressed as E = {e1, e2, ..., e t}∈R t , where e t is the seismic acceleration at each moment, t is the time sequence number;

[0016] The pier height sequence is expressed as P = {p1, p2, ..., p s-1}∈R s-1 , p s-1 is the height of the pier, s is the number of bridge spans;

[0017] The predicted output of the peak displacement envelope or track irregularity of the structure after the earthquake is O = {O1, O2, …, Oα}∈Rα, where O k is the peak displacement of the output point, α is the number of nodes at the predicted location of the structure;

[0018] Using multiplication fusion, the structural characteristic time series of a single fused ground motion is expressed as:

[0019] PE in =P T E (1);

[0020] Where: P is the pier height sequence; E is the earthquake motion sequence; PE in represents the input features; T represents the transposition operation.

[0021] In the aforementioned method for predicting the peak displacement envelope curve of the sliding layer and track irregularity of a high-speed railway system considering the uncertainty of ground motion and pier height, the spatial sequence of peak displacements output based on the elastic response spectrum method in step 2) is the result of integrating domain prior knowledge into the neural network using the response spectrum method. The specific analysis process is as follows:

[0022] For a multi-span high-speed railway track bridge system composed of arbitrary structures, its motion equation is expressed as A spring-mass system with damping in degrees of freedom:

[0023]

[0024] Where:

[0025] M is the mass matrix of the multi-span high-speed railway track bridge system;

[0026] C is the damping matrix of the multi-span high-speed railway track bridge system;

[0027] K is the stiffness matrix of the multi-span high-speed railway track bridge system;

[0028] is the acceleration vector of the multi-span high-speed railway track bridge system;

[0029] is the velocity vector of the multi-span high-speed railway track bridge system;

[0030] x(t) is the displacement vector of the multi-span high-speed railway track bridge system;

[0031] P(t) is the equivalent load acting on the system;

[0032] Where C is the classic Rayleigh damping, and the equation can be naturally decoupled;

[0033] Under a given earthquake motion sequence, P(t) can be expressed as:

[0034]

[0035] Where: ι represents the influence vector, represents the ground acceleration;

[0036] For a linear elastic system with multiple degrees of freedom, the system response is expressed as an effective combination of multiple modal components:

[0037]

[0038] Where:

[0039] u(t) is the total response of the system;

[0040] u n(t) is the nth-order modal response;

[0041] is the static response of the nth mode;

[0042] A n (t) is the pseudo acceleration response of the nth mode,

[0043] N is the number of modes,

[0044] n represents the modal order;

[0045] Among them A n (t) is:

[0046]

[0047] Where:

[0048] A n (t) is the pseudo acceleration response of the nth mode;

[0049] represents the square of the natural frequency of the nth mode;

[0050] D n (t) is the displacement response of the system;

[0051] nth-order modal static response for

[0052]

[0053] Where:

[0054] Γ n represents the modal participation factor,

[0055] φ n nth-order mode shape vector;

[0056] Since the modal static response is related to the modal participation factor, modal frequency and mode shape vector;

[0057] Combining equations (4)-(6), the total response u(t) equation of the system is transformed into:

[0058]

[0059] Where Γ n and φ n represent the modal participation factor and the nth-order mode shape vector respectively;

[0060] Obtaining the maximum structural response during the earthquake excitation period is a key step in the seismic design or seismic assessment of the structure, that is, solving,

[0061]

[0062] Where:

[0063] u no =Γ n φ n D n (t),

[0064] represents the peak response of the nth mode, obtained from the response spectrum calculation;

[0065] Taking into account the correlation between the peak modal responses to prevent significant errors, the perfect square combination method is used to predict the maximum response of the structure when the modal responses have significant cross-correlation:

[0066]

[0067] Where, ρ in represents the correlation between the i-th and n-th modal responses; the formula can be expanded as follows:

[0068]

[0069] In the CQC rule, the modal correlation coefficient ρ in Calculated by the following formula:

[0070]

[0071] Where:

[0072] u o is the maximum response of the structure,

[0073] N is the number of modes,

[0074] u io is the peak response of the ith mode,

[0075] ζ i is the damping ratio of the i-th mode, which indicates the ability of this mode to dissipate vibration energy.

[0076] ζ n is the damping ratio of the nth mode,

[0077] represents the natural frequency ratio of the i-th mode to the n-th mode,

[0078] ω i and ω n are the natural frequencies of the i-th mode and the n-th mode, respectively.

[0079] In the aforementioned method for predicting the peak displacement envelope curve of the sliding layer and track irregularity of a high-speed railway system considering the uncertainty of ground motion and pier height, in step 3), the features extracted in step 2) are input into the multimodal hybrid neural network model TransTCN constructed in step 1), and the specific processing is as follows:

[0080] After the time series of earthquake motion fusion structural features and the spatial series of peak displacement output based on the elastic response spectrum method are input into the multimodal hybrid neural network model TransTCN, they first pass through the embedding layer of the Transformer module to encode the position of the input sequence and convert it into a vector sequence. Then, they are processed by the multi-head self-attention layer to obtain the attention score of each sequence point and output the attention tensor of the current sequence position.

[0081] The TCN module then captures the complete temporal dependency. The attention tensor obtained by the Transformer module is normalized and filtered. The output features are enriched with temporal information through multiple dilated causal convolutional layers. The rectified linear unit is used as the activation function, and normalization is used as a convolution filter to further extract tensor features at the corresponding sequence position.

[0082] Huber Loss is used as the loss function. By fine-tuning the parameter δ, an effective combination of mean square error and mean absolute error is achieved. The output result is adjusted. The adjusted network can be used to realize the prediction of the peak displacement envelope curve of the sliding layer and track irregularity of the high-speed railway system considering the uncertainty of ground motion and pier height.

[0083] As a preferred technical solution, the aforementioned high-speed railway system sliding layer peak displacement envelope curve and track irregularity prediction method considering the uncertainty of ground motion and pier height,

[0084] The processing process of inputting the time series of structural characteristics based on earthquake motion fusion and the spatial series of peak displacement output based on elastic response spectrum method into the Transformer module is as follows:

[0085] Input TransTCN model features XS = {xs1, xs2, ..., xs t} is converted into three trainable linear mapping feature matrices by the embedding layer, named as: query key Sum

[0086] Among them, xs t is the characteristic of time series t, n model ,m model denote the number of queries and the length of the input sequence in the attention module, respectively, k and d vare the feature dimensions of the query, key, and value matrices respectively;

[0087] The scaled point product attention and softmax function are used to consider its bidirectional temporal dependency on the historical sequence. The self-attention matrix is ​​expressed as

[0088]

[0089] in, It is the scaling factor of the optimized training to adjust the calculation of the inner product to prevent the inner product from being too large. In self-attention, d k and d v Equal; T is the transpose symbol;

[0090] Multiple queries key Sum Perform different linear transformation projections, and then calculate the attention on each segment in parallel. Combine h independent scaled dot product attention results through vector concatenation and linear transformation and obtain the output value through linear transformation, satisfying:

[0091] MultiHead(Q,K,V)=Concatenate(head1,…,head h )W o (14);

[0092] Where:

[0093] h represents the number of heads,

[0094] represents the multi-head attention weight,

[0095] head h represents the attention head,

[0096] Concatenate represents the concatenation operation;

[0097] For the hi-th attention head, a single attention is calculated by different weight matrices:

[0098]

[0099] Where: represents the self-attention weight, Represents the multi-head attention weight, Attention is the self-attention calculation, d model is the characteristic dimension of the entire model.

[0100] As a preferred technical solution, the aforementioned high-speed railway system sliding layer peak displacement envelope curve and track irregularity prediction method considering the uncertainty of ground motion and pier height, the complete time dependency capture process through the TCN module is as follows:

[0101] One-dimensional time series input With the filter f:{1,2,…,ζ-1}, the dilated convolution operation F on the elements θ in the one-dimensional sequence can be expressed as:

[0102]

[0103] Where:

[0104] θ represents the time step to be processed; ci is the time step currently being processed; cd is the dilation factor used to control the size of the interval; ζ is the size of the filter; * represents the convolution operation; It is a one-dimensional time series input;

[0105] The receptive field of TCN can be calculated as:

[0106]

[0107] Where: R field is the receptive field of TCN; N stack is the number of stacked TCN layers, that is, the number of stacked convolutional layers; is the size of the convolution kernel, cd is the expansion factor of each convolution layer;

[0108] The multimodal hybrid neural network model TransTCN uses a residual module to avoid model degradation. The calculation process of the residual block is as follows:

[0109]

[0110] Where: Represents the feature representation learned by the DCC layer; It is the input and the output features obtained after the convolution operation of the DCC layer. These features can effectively represent the temporal information of the input sequence.

[0111] As a preferred technical solution, the aforementioned high-speed railway system sliding layer peak displacement envelope curve and track irregularity prediction method considering the uncertainty of ground motion and pier height, the multimodal hybrid neural network model TransTCN output layer uses Huber Loss as the loss function, which is defined as a piecewise function:

[0112]

[0113] Where: is the predicted value, is the true value, δ is the adjustment coefficient, which determines the transition threshold between the loss function from quadratic to linear;

[0114] By continuously adjusting δ to make the predicted output consistent with the actual output, the adjusted network can be used to predict the peak displacement envelope curve of the sliding layer and track irregularity of the high-speed railway system considering the uncertainty of ground motion and pier height.

[0115] The aforementioned method for predicting the peak displacement envelope curve of the sliding layer and track irregularity of a high-speed railway system considering the uncertainty of ground motion and pier height also includes the step of training the multimodal hybrid neural network model TransTCN constructed in step 1). The ground motion records and uncertainty parameter sampling process used in the training are as follows:

[0116] The ground motion records used for training, validation, and testing are matched with the design response spectra that fully consider the influence of ground motion propagation path, site conditions, spectral characteristics, and soil shear wave velocity. Then, a training data sample library is obtained based on the Latin hypercube stratified sampling method.

[0117] The uncertainty parameter of the earthquake motion E=[e1,e2] T and the uncertain parameters of pier height PH=[P1,P2,…,P S-1 ] T Construct S+1 dimensional probability space R U , represents the total space of uncertainties of all random variables including ground motion and pier height parameters;

[0118] Let ξ = s + 1, which represents the probability space dimension of the combination of bridge structure and ground motion;

[0119] For a ξ-dimensional random variable The random variables include all the bridge pier height parameters and ground motion parameters, which together define R u Each sample point in ;

[0120] Decompose it into P mutually disjoint orthogonal subspaces where a single subspace ξ j The dimensions satisfy:

[0121]

[0122] It means that the sum of the dimensions of each subspace is equal to the dimension ξ of the entire probability space, and the dimension of each subspace does not exceed ξ;

[0123] At the same time, the orthogonality condition should be met:

[0124]

[0125] Where: is an empty set, ∩ and are the intersection operation and the union operation of all subspaces, respectively. Equations (21) and (22) indicate that different subspaces are independent of each other, that is, there is no overlap between them, and the union of all subspaces is equal to the original probability space R U ;

[0126] Since the subspaces are mutually orthogonal, The sample points extracted from the space can represent the uncertainty of the random variables in the space. The stratified sampling set without replacement for each orthogonal subspace becomes the sample library. The final sample set is the Latin hypercube uniformly distributed sampling points in the probability space. The probability density satisfies the linear relationship. For different probability distributions, the sample points of the random parameters in the specific probability distribution space can be obtained by using the equal probability density transformation, and finally the required seismic motion-structure random parameter sample set is obtained.

[0127] The aforementioned method for predicting the peak displacement envelope curve of the sliding layer and track irregularity of a high-speed railway system, which takes into account the uncertainty of earthquake motion and pier height, is used to predict the peak displacement envelope curve of the sliding layer and track irregularity of a high-speed railway system with a typical high-speed railway simply supported beam bridge as the benchmark structure.

[0128] The beneficial effects of the present invention are as follows:

[0129] 1) The present invention adopts a multimodal deep learning technology that integrates domain prior knowledge with structural features. Based on the prior domain knowledge of the response spectrum method and the characteristic tensor of the pier height and seismic motion time history, a long sequence prediction framework TransTCN that combines the Transformer and TCN networks is constructed. It can effectively extract time series features and structural features, improve prediction accuracy while meeting the training sample requirements, overcome the computational burden of purely physics-based proxy models and the modeling difficulties caused by changes in structural parameters, and significantly improve the fitting accuracy of pure data-driven models and reduce the training data volume requirements.

[0130] 2) The long-sequence prediction framework TransTCN constructed in this paper can accurately capture the complex spatiotemporal dependencies between time-step features. We compared various classic neural network-based models and conducted ablation experiments on reliable training samples verified by shaking table tests. This demonstrated the effectiveness and applicability of the RSM-based PDK method in improving network generalization performance. TransTCN can effectively extract both temporal and structural features, improving prediction accuracy. Compared with well-performing classic neural network models, the proposed model achieved a minimum 7.1% improvement in prediction accuracy. Under the same accuracy conditions, the training sample requirement decreased by 300%, providing new insights into regional earthquake risk assessment and performance-based seismic design methods.

[0131] 3) The present invention integrates prior knowledge of the target domain to make up for the deficiencies of deep learning in interpretability, effectively reducing data requirements, and embeds prior information such as control equations, physical laws, engineering theories, and expert experience into the data-driven model. While achieving data fitting, the prior knowledge is used to constrain the prediction results to conform to physical mechanisms. In earthquake engineering, constrained models have stronger robustness and significantly reduce the demand for training data.

[0132] 4) The long sequence prediction framework model TransTCN constructed by the present invention, which combines the Transformer and TCN networks, integrates the advantages of TCN temporal convolution and the advantages of Transformer multi-head self-attention mechanism, and accurately captures the complex spatiotemporal dependencies between multiple time step features. The multimodal hybrid neural network model can effectively predict the seismic peak response and track irregularity of the track structure under the uncertainty of the bridge pier height of the high-speed rail system, and has significant advantages in calculation accuracy compared with other similar technologies; the spatial feature input of the superimposed response spectrum method can significantly improve the performance of the model, and has potential advantages in dealing with regional earthquake loss assessment under structural randomness. The model of the present invention has significantly lower data requirements than other models, has better convergence and faster convergence speed, and realizes the prediction of the seismic peak response and track irregularity of the track structure under the uncertainty of the bridge pier height of the high-speed rail system, providing an effective calculation tool for real-time identification of regional high-speed rail system damage and regional structural seismic risk assessment, which helps to overcome the limitations of traditional finite element methods in large-sample earthquake damage calculations. BRIEF DESCRIPTION OF THE DRAWINGS

[0133] Figure 1 This is a schematic diagram of the fusion of ground motion and structural characteristics;

[0134] Figure 2 Schematic diagram of the TCN and transformer network structure; ((a) is the TCN residual block, (b) is the DCC network layer diagram, (c) is the transformer sub-model, and (d) is the multi-head attention structure);

[0135] Figure 3 This is a schematic diagram of the overall architecture of TransTCN;

[0136] Figure 4 Schematic diagram of the CRTSII track-bridge system structure and verification results; ((a) is a finite element diagram of the high-speed railway track-bridge system, (b) is the seismic experiment and seismic input, and (c) is the experimental verification results).

[0137] Figure 5 To select the response spectrum and average spectrum of the earthquake motion set;

[0138] Figure 6The structural seismic response is obtained by performing nonlinear time history analysis on the finite element model established for the input of random parameter sample sets;

[0139] Figure 7 Prepare diagrams for training data;

[0140] Figure 8 shows the performance comparison of the models under different pier height distributions; (a) shows the performance comparison of the models under peak-shaped pier height distribution, (b) shows the performance comparison of the models under W-shaped pier height distribution, (c) shows the performance comparison of the models under valley-shaped pier height distribution, and (d) shows the performance comparison of the models under uniform pier height distribution).

[0141] Figure 9 shows the performance comparison of the models under different data inputs. (Figure (a) shows the R of different models under different data inputs. 2 Indicator comparison, (b) is the MAE indicator comparison of different models under different data inputs, and (c) is the MSE indicator comparison of different models under different data inputs). DETAILED DESCRIPTION

[0142] The following will be combined with the embodiments and drawings to clearly and completely describe the technical solutions of the present invention. Obviously, the embodiments described are only preferred embodiments of the present invention, not all embodiments, and are not intended to limit the present invention in any other form. Any person skilled in the art may use the disclosed technical content to make changes or modifications. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention shall still fall within the scope of protection of the technical solution of the present invention.

[0143] Example 1

[0144] To overcome the computational burden of purely physics-based proxy models and the modeling difficulties caused by structural parameter changes, and to significantly improve the fitting accuracy of purely data-driven models and reduce the amount of training data required, this embodiment proposes a multimodal TransTCN fusion network embedded with domain prior knowledge to achieve a method for predicting peak displacement response and track irregularity of high-speed railways under uncertain pier height conditions.

[0145] This example uses a typical high-speed railway simply supported beam bridge as the research object, and realizes the prediction of the peak displacement envelope curve of the sliding layer and track irregularity of the high-speed railway system under the uncertainty of earthquake motion and pier height. It also compares with other existing similar technologies and demonstrates the superiority of the model proposed in this example in terms of computational efficiency and accuracy. The proposed method can provide technical support for the quantification of driving safety uncertainty and regional driving risk assessment in high-speed railway networks.

[0146] First, this embodiment constructs a multimodal hybrid neural network model TransTCN that embeds domain prior knowledge; the domain prior knowledge is prior information including relevant control equations, physical laws, engineering theories and expert experience in the domain; the constructed multimodal hybrid neural network model TransTCN is a long sequence prediction framework that combines Transformer and TCN networks, which includes a Transformer module and a TCN module; the TCN module is used to capture complete temporal dependencies, which includes multiple dilated causal convolutional layers and a cross-path layer, uses rectified linear units as activation functions, and batch normalization as convolution filters.

[0147] Then, feature extraction is performed on the multimodal input features of the time series based on the fusion of structural characteristics of earthquake motion and the spatial series based on the output peak displacement of the elastic response spectrum method. The pier height characteristics and earthquake motion characteristics are used as time series features through dimension fusion to deal with the temporal dependency. The elastic response spectrum analysis results are used as spatial features to deal with the spatial dependency.

[0148] Finally, the extracted features are input into the constructed multimodal hybrid neural network model TransTCN to realize the prediction of the peak displacement envelope curve of the sliding layer and track irregularity of the high-speed railway system considering the uncertainty of seismic motion and pier height.

[0149] The specific process is as follows:

[0150] 1. Data Preparation

[0151] 1.1 Seismic data processing and pier height feature fusion

[0152] In typical earthquake damage classification or response prediction, the input sequence is simply the earthquake motion time series or various time-frequency decomposition results of the earthquake motion, and the predicted output is the nodal response or nodal damage state. When using the same method to predict earthquake damage in buildings with long, longitudinal structures, such as high-speed rail, there is a risk of overlooking or underestimating the structural damage state. Furthermore, building a neural network framework for multiple nodal responses simultaneously imposes a significant computational burden. The response spectrum method, which predicts the peak response of multi-degree-of-freedom (MDOF) systems by combining modal responses, has gained widespread acceptance in practical applications due to its favorable compromise between simplicity, practicality, and efficiency. However, this method has significant limitations in calculating the nonlinear response of structures.

[0153] In order to effectively integrate the seismic motion characteristics and the bridge pier height characteristics, this embodiment processes the bridge pier height characteristics and the seismic motion characteristics respectively as follows:

[0154] The earthquake motion sequence is expressed as E = {e1, e2, ..., e t}∈R t , where et is the seismic acceleration at each moment, and t is the time sequence number.

[0155] The heights of multiple piers of the target bridge are expressed as P = {p1, p2, ..., p s-1}∈R s-1 , p s-1 is the height of the pier, and s is the number of bridge spans.

[0156] The output of the prediction target (taking the peak displacement envelope of the structure after the earthquake as an example) is O k ={O1,O2,…,Oα}∈Rα, where O k is the peak displacement of the output point, and α is the number of nodes at the predicted structure position.

[0157] Then, the multiplication fusion method is used to improve the interaction between features, thereby improving the model's ability to recognize complex features. The feature processing process is as follows: Figure 1 As shown in Figure 2, the characteristic input of a single fused ground motion is expressed as:

[0158] PE in =PTE (1)

[0159] Where: P is the pier height sequence; E is the earthquake motion sequence; PE in represents the input features; T represents the transposition operation.

[0160] 1.2 Response spectrum method

[0161] The seismic design and assessment of structural systems are usually based on the peak control of the seismic response of forces or displacements. The response spectrum method (RSM), which uses the modal superposition principle to calculate the potential peak response of multi-degree-of-freedom systems, has great development potential. However, the use of the RSM method in major structural risk assessments still has limitations. For example, the time of occurrence of the peak modal response will not be completely matched, and the peak response can only be approximated by modal combination, which obviously will result in significant errors and ignores the historical information related to plastic deformation. To this end, in this embodiment, it is incorporated into the neural network as expert knowledge to appropriately address the above limitations. The details are as follows:

[0162] 1.2.1 Domain Prior Knowledge Based on Response Spectrum Method

[0163] For a multi-span high-speed railway track bridge system with any structure, its motion equation can be expressed as A spring-mass system with damping in degrees of freedom:

[0164]

[0165] Where:

[0166] M is the mass matrix of the multi-span high-speed railway track bridge system;

[0167] C is the damping matrix of the multi-span high-speed railway track bridge system;

[0168] K is the stiffness matrix of the multi-span high-speed railway track bridge system;

[0169] is the acceleration vector of the multi-span high-speed railway track bridge system;

[0170] is the velocity vector of the multi-span high-speed railway track bridge system;

[0171] x(t) is the displacement vector of the multi-span high-speed railway track bridge system;

[0172] P(t) is the equivalent load acting on the system.

[0173] Where C is the classic Rayleigh damping, and the equation can be naturally decoupled;

[0174] Under a given earthquake motion sequence, P(t) can be expressed as:

[0175]

[0176] Where: ι represents the influence vector, Represents ground acceleration.

[0177] Therefore, for a linear elastic system with multiple degrees of freedom, the system response can be expressed as a valid combination of multiple modal components as follows:

[0178]

[0179] Where: u(t) is the total response of the system; u n (t) is the nth-order modal response; is the static response of the nth mode; A n (t) is the pseudo acceleration response of the nth mode, N is the number of modes, and u represents the modal order.

[0180] Among them A n (t) is:

[0181]

[0182] In this formula, A n (t) is the pseudo acceleration response of the nth mode; represents the square of the natural frequency of the nth mode; D n (t) is the displacement response of the system.

[0183] nth-order modal static response for

[0184]

[0185] Where: Γ n represents the modal participation factor; φ n nth-order mode shape vector;

[0186] Since the modal static response is related to the modal participation coefficient, modal frequency and modal vibration shape vector, combining equations (4)-(6), the total system response n(t) equation is transformed into:

[0187]

[0188] Obtaining the maximum structural response during an earthquake excitation period is a key step in the seismic design or seismic assessment of a structure, that is, solving:

[0189]

[0190] Among them, u no =Γ n φ n D n (t), represents the peak response of the nth mode, which can be easily obtained from the response spectrum calculation.

[0191] Obtaining the maximum response from a time history analysis requires a lot of computational resources, but the peak modal response obtained by combining the response spectrum can be used to approximate the maximum peak response of the structure. It is worth noting that the modal peak responses usually do not occur at the same time point. The simplest approximation rule is Assuming that the modal peaks occur at the same time and direction, an approximate response is obtained by summing the absolute modal responses. This method is called the absolute sum (ABSSUM) modal combination rule, and it generally provides an upper limit on the actual response. However, in structural design and evaluation, the ABSSUM rule is considered too conservative.

[0192] In addition, some studies have proposed a root sum of squares (SRSS) rule based on the statistically independent modal response assumption, which can significantly reduce the calculation error of the ABSSUM rule when the modal frequencies of the structure are well separated:

[0193]

[0194] However, for structures with close modal periods, such as the high-speed railway track bridge system used in this embodiment, the correlation between the peak modal responses needs to be considered to prevent significant errors.

[0195] Therefore, this embodiment uses the perfect square combination (CQC) rule to more accurately predict the maximum response of the structure when the modal responses have significant cross-correlations:

[0196]

[0197] Where, l in Represents the correlation between the i-th and n-th modal responses.

[0198] The formula can be expanded as:

[0199]

[0200] In the CQC rules, the modal correlation coefficient l in It can be calculated by the following formula:

[0201]

[0202] In the above formula, u o is the maximum response (peak response) of the structure; N is the number of modes; u io The peak response of the i-th mode; ζ i The damping ratio of the i-th mode represents the ability of this mode to dissipate vibration energy; n The damping ratio of the nth mode; where, represents the natural frequency ratio of the i-th mode to the n-th mode, ω i and ω n are the natural frequencies of the i-th mode and the n-th mode, respectively.

[0203] This formula takes into account the relationship between different modal frequencies and damping ratios, which improves the accuracy of the CQC rule. The input result of the response spectrum method is recorded as RS in .

[0204] 2. TransTCN Prediction Model

[0205] TCN networks and Transformers are widely used in time series tasks. In order to overcome the forgetting defects and insufficient local feature extraction capabilities of RNNS-type networks in sequence data learning, this embodiment adopts a model fusion TCN and transformer framework to achieve effective capture of multi-sequence input correlations. Unlike the ordinary transformer framework, the TransTCN model described in this embodiment is an encoder-only framework; the nonlinear response of the structure is simultaneously affected by the structural features, time series features, and spatial features of the predicted position. In order to prevent the mutual influence between different features, it has two inputs, namely the time series matrix of the fused structural features and the calculation results of the response spectrum method. The TransTCN prediction model described in this embodiment, such as Figure 2 As shown, it includes the Transformer module and the TCN module.

[0206] Transformer modules, such as Figure 2 (c) is shown as a Transformer model built to capture seismic motion characteristics and response spectrum structural features. The Transformer model is a sequence-to-sequence model based on the self-attention mechanism. It is a typical encoder-decoder structure. The encoder usually uses the self-attention mechanism, which enables it to be highly parallelized during the calculation process. Compared with the recurrent neural network (RNN) that calculates recursive results sequentially, the encoder structure can perform calculations faster when processing long sequences, significantly improving the efficiency of training and inference. In some time series tasks (such as classification, regression, etc.), the goal of the model is to extract features from the input sequence and make predictions directly without generating a new sequence. In this case, only the encoder structure can extract the required features from the input data more directly, reducing the additional decoding steps, thereby simplifying the model and improving efficiency.

[0207] The Transformer module described in this embodiment includes a multi-head self-attention and a fully connected feedforward network layer. The input sequence first passes through an embedding layer, which encodes the position of the input sequence and converts it into a vector sequence, and then processes it through multiple stacked self-attention layers (multi-head self-attention). The core of the self-attention mechanism is that it can model the relationship between each element in the sequence and all other elements. This enables the encoder to effectively capture global contextual information when processing time series data, and is particularly effective in modeling long-term dependencies. This feature is crucial for capturing the seismic characteristics and structural features of the response spectrum method in this embodiment.

[0208] Input TransTCN model features XS = {xs1, xs2, ..., xs t} is converted into three trainable linear mapping feature matrices by the embedding layer, named query key Sum Scaled Dot-Product Attention and softmax function are used to consider its bidirectional temporal dependency on the historical sequence. The self-attention matrix is ​​expressed as:

[0209]

[0210] Among them, xs t is the characteristic of time series t, n model ,m modeldenote the number of queries and the length of the input sequence in the attention module, respectively, k and d v are the feature dimensions of the query, key, and value matrices respectively. It is the scaling factor of the optimized training to adjust the calculation of the inner product to prevent the inner product from being too large. In self-attention, d k and d v The Softmax function is used to convert the scaled dot product result into a probability distribution, and T is the transpose operator.

[0211] Since single-head attention may not extract sufficient local features, the focused location may not be completely accurate. This embodiment uses a multi-head attention structure to weighted sum the obtained results to form a multi-head attention mechanism, which enables the model to jointly focus on features at different locations in different representation subspaces.

[0212] The multi-head attention structure increases the accuracy of the model by processing different features in layers, splitting the input into fixed-size segments, and key Sum Perform different linear transformation projections, and then calculate the attention on each segment in parallel. Combine the h independent scaling dot product attention results through vector concatenation (connect the first vector) and linear transformation and obtain the output value (see Figure 2 (d)), satisfying:

[0213] MultiHead(Q,K,V)=Concatenate(head1,…,head h )W o (14)

[0214] Where h represents the number of heads, Represents the multi-head attention weight, head h Represents the attention head, and Concatenate represents the concatenation operation. The purpose of concatenation is to enable the model to focus on different parts or different features of the input, so as to learn richer information.

[0215] For the hi-th attention head, a single attention is calculated by different weight matrices:

[0216]

[0217] in, Represents the self-attention transformation weight, Attention is the self-attention calculation, d model is the characteristic dimension of the entire model.

[0218] However, when performing long time series prediction, the decoder will greatly increase the computing time. In addition, using different decoding methods during training and decoding may lead to problems such as low prediction accuracy or overfitting. Especially when the training dataset is small, problems such as low prediction accuracy or overfitting will be more prominent.

[0219] To avoid the aforementioned issues, extract temporal dependencies, and ensure accurate predictions, this embodiment uses only a Transformer combined with a TCN. The TCN module described in this embodiment is an encoder-only framework, a variant of a traditional convolutional neural network (CNN) designed specifically for efficient and effective processing of sequence data. It effectively captures complete temporal dependencies. Specifically, by combining the TCN module, point-by-point convolution is achieved for time series tasks.

[0220] The structure of the module described in this embodiment is as follows Figure 2 As shown in (a), it contains multiple dilated causal convolution (DCC) layers and a cross-layer path layer, using rectified linear units (ReLU) as the activation function and batch normalization as the convolution filter. When the length of the input and output data is different, a 1×1 convolution is used as the cross-path layer.

[0221] In this embodiment, dilated causal convolution is a key component of the TCN module. Unlike causal convolution, which can only expand its receptive field by increasing the kernel size or adding more layers, causal convolution only satisfies simple sequence mapping. The dilated convolution used in this embodiment enables the TransTCN prediction model to handle long sequence structures, preventing convolutional layers from becoming too deep and difficult to train.

[0222] For one-dimensional time series input With the filter f:{1,2,…,ζ-1}, the dilated convolution operation F on the elements θ in the one-dimensional sequence can be expressed as:

[0223]

[0224] Among them, θ represents the time step to be processed, ci is the time step currently being processed, cd is the expansion factor used to control the size of the interval, ζ is the size of the filter, and * represents the convolution operation. The input is a one-dimensional time series. As the number of convolutional layers increases, the dilation factor increases exponentially. The structure of the dilated causal convolution is as follows:

[0225] By adjusting the expansion factor, the amount of information received by the TCN can be controlled. The receptive field of the TCN can be calculated as:

[0226]

[0227] Among them, R field is the receptive field of TCN; N stack is the number of stacked TCN layers, that is, the number of stacked convolutional layers; is the size of the convolution kernel, and cd is the expansion factor of each convolution layer.

[0228] To shorten training time and avoid degradation issues caused by deep networks, residual modules are used to train the network. By stacking several TCN layers, a larger dilation factor can be used to obtain a larger receptive field. When the lengths of the input and output data are different, 1×1 convolution is used as the cross-path layer. The calculation process of the residual block is as follows:

[0229]

[0230] in, represents the feature representation learned by the DCC layer. It is the input and the output features obtained after the convolution operation of the DCC layer. These features can effectively represent the temporal information of the input sequence.

[0231] Based on the complete TransTCN prediction model above, Figure 3 As shown, TransTCN receives input PE in and RS in (They are the seismic structure fusion features and the response results based on the response spectrum method, respectively) and the attention scores of each sequence point are obtained through multi-head attention reconstruction, the attention tensor of the current sequence position is output, and tensor normalization and attention filtering are performed. The output features are enriched with temporal information through a two-layer DDC module, and the rectified linear unit (ReLU) is used as the activation function. The tensor features of the corresponding sequence position are further extracted through normalization as a convolution filter. The model uses a residual module to avoid model degradation.

[0232] 3. Output layer

[0233] Huber Loss is used as the loss function. By fine-tuning the parameter δ, an effective combination of mean square error (MSE) and mean absolute error (MAE) is achieved. This can effectively reduce the sensitivity to outliers while maintaining differentiability to reduce the impact of training. Huber Loss is defined as a piecewise function:

[0234]

[0235] in, is the predicted value, is the true value, and δ is the adjustment coefficient, which determines the transition threshold between the loss function from quadratic to linear. When , the loss function takes the form of a quadratic function, which imposes a greater penalty on the predicted value and has better smoothness and stability. Conversely, when the loss function becomes a linear function, the penalty on the predicted value is smaller when the error is large, reducing the impact of outliers on the model.

[0236] By continuously adjusting δ to make the predicted output consistent with the actual output, the adjusted network can be used to predict the peak displacement envelope curve of the sliding layer and track irregularity of the high-speed railway system considering the uncertainty of ground motion and pier height.

[0237] 4. Model training data

[0238] The typical high-speed railway simply supported beam bridge described in this embodiment is an existing model, such as Figure 4 As shown in Figure 1, it is a finite element model of the bridge structure developed in OpenSees. The model has geometric and material nonlinearities and can accurately represent the potential nonlinear behavior of the bridge system.

[0239] In this example, the fiber cross-section of the simply supported high-speed railway bridge uses the Concrete01 material to simulate the concrete behavior, while the Steel02 material represents the force-displacement relationship of the steel bars. The Giuffre-Menegotto-Pinto model is used to account for cyclic loading and unloading, as well as the Bauschinger effect. The bridge piers are modeled using fiber cross-section nonlinear elements with displacement foundations. The track is consolidated at both ends to simulate the constraints of the roadbed. ZeroLength six-way springs are used to simulate the pile foundations to account for pile-soil interaction. The stiffnesses are calculated based on the site conditions of the bridge: 2.41e9N / m, 2.32e9N / m, 1.87e10N / m, 2.55e11N / m, 1.42e11N / m, and 1.42e11N / m. Elastic beam elements are used to model the main beam, base plate, track slab, and rails. TwoNodeLink elements are used to simulate the nonlinear behavior of the bearings, sliding layer, mortar layer, shear groove, shear reinforcement, side blocks, and fasteners.

[0240] 4.1 Ground motion records and uncertainty parameter sampling

[0241] In this embodiment, the ground motion records used for training, verification, and testing are obtained from the NGA-West2 strong motion database provided by the Pacific Earthquake Engineering Research Center (PAGE) NGA-West2 database. A total of 4,459 complete ground motion records of historical earthquake events are collected from this database. These records are distributed in 1,086 accelerometer sites of 78 events, covering different site conditions, different magnitudes, different stations, and different PGA values.

[0242] In order to ensure that the sampling can fully reflect the uncertainty and prove the universality of the proposed method, the cost of considering various types of uncertain parameters is high, and not considering or under-considering a certain type of earthquake will cause defects in the accuracy of the model. According to the provisions of China's "Railway Engineering Seismic Design Code", the seismic motion is selected, and the influence of seismic motion propagation path, site conditions and spectral characteristics is fully considered. The shear wave velocity of the soil layer is divided into four categories according to the site boundaries of the Chinese code. The design response spectra under the corresponding site conditions are generated respectively. The 50 groups of seismic motions with the highest matching degree are selected for each type of response spectrum. The response spectrum and average spectrum of the seismic motion set are selected as follows: Figure 5 shown.

[0243] Each horizontal acceleration record is processed as a separate record. The sampling frequency of all seismic motion records is uniformly adjusted to 0.02s, and the collected seismic motion duration is 30s to meet the calculation accuracy requirements of high-speed railway bridge structures. The seismic site parameter is used as the selection parameter e1, and PGA (with a value range of 0 to 0.8g) is used as the selection parameter e2. A random selection library of seismic motion parameters E = [e1, e2] is constructed. T , assuming it follows a uniform distribution.

[0244] Due to the constraints of terrain conditions, the height of bridge piers is usually subject to significant uncertainty. The change in pier height significantly affects the mass, stiffness, and damping matrix of the structure, and thus affects the post-earthquake damage state of the bridge. The variability of bridge structural response is usually at the same level as the variability of structural parameters. Fully considering the uncertainty of pier height in the dynamic analysis of bridge structures is of great significance for the seismic performance analysis of bridges. The pier height parameter is expressed as PH = [P1, P2, ..., P S-1 ] T .P S-1 is the height of pier S-1, S is the number of bridge spans, and it is assumed to obey the normal distribution.

[0245] In order to improve the representativeness of training samples and the coverage of parameter space, this embodiment obtains low-bias samples based on structured randomness based on Latin hypercube stratified sampling method to construct a training data sample library. T and the uncertain parameters of pier height PH=[P1,P2,…,P S-1 ] T Construct S+1 dimensional probability space R U , represents the total space containing the uncertainty of all random variables (seismic motion and pier height parameters). Let ξ=S+1, which represents the probability space dimension of the combination of bridge structure and seismic motion. For the ξ-dimensional random variable set The random variables include all the bridge pier height parameters and ground motion parameters, which together define R uEach sample point in . Decompose it into P mutually non-intersecting orthogonal subspaces where a single subspace ξ j The dimensions satisfy:

[0246]

[0247] It means that the sum of the dimensions of each subspace is equal to the dimension ξ of the entire probability space, and the dimension of each subspace does not exceed ξ.

[0248] At the same time, the orthogonality condition should be met:

[0249]

[0250]

[0251] in, is an empty set, ∩ and are the intersection operation and the union operation of all subspaces, respectively. Equations (21) and (22) indicate that different subspaces are independent of each other, that is, there is no overlap between them, and the union of all subspaces is equal to the original probability space R U .

[0252] Since the subspaces are mutually orthogonal, The sample points extracted from the space can represent the uncertainty of the random variables in the space. The stratified sampling set without replacement for each orthogonal subspace becomes the sample library. The final sample set is the Latin hypercube uniformly distributed sampling points in the probability space. The probability density satisfies the linear relationship. For different probability distributions, the sample points of the random parameters in the specific probability distribution space can be obtained by using the equal probability density transformation, and finally the required seismic motion-structure random parameter sample set is obtained.

[0253] Taking the Latin hypercube sampling of seismic motion as an example, for different seismic motion zoning, Latin hypercube sampling is used to obtain the corresponding seismic motion area, seismic motion number and PGA. For all seismic motions in the selected area, one seismic motion is randomly sampled without replacement and the seismic motion is amplitude modulated to the corresponding PGA size. The processed seismic motion is then used as the seismic motion input of the finite element model.

[0254] The nonlinear time history analysis is performed on the finite element model established by inputting the constructed random parameter sample set to obtain the corresponding structural seismic response, such as Figure 6 In addition, in order to provide a more accurate prediction of the structural seismic performance, this embodiment also establishes a three-dimensional model of bidirectional input motion and response, and adopts a bidirectional earthquake motion input direction with a horizontal to vertical ratio of 1:0.65.

[0255] Example 2

[0256] This embodiment uses the prediction method described in Example 1 to predict the lateral displacement peak response envelope curve of the sliding layer with a significant probability of damage in the track structure under earthquake and the track irregularity peak envelope curve that seriously threatens driving safety. The data preparation process is as follows: Figure 7 At the same time, the effect of the prediction method described in the embodiment is quantitatively analyzed, ablated and qualitatively evaluated in a comprehensive test scenario.

[0257] 1. Method Validation

[0258] Track structures are susceptible to damage during transverse earthquakes due to the low yield strength of their supports, manifesting as extensive damage to the sliding layer and significant track irregularities. Therefore, to demonstrate the practicality and potential value of this model, this example uses two datasets, each with 2,000 data points, for training. The model focuses on predicting the peak displacement envelope curves of the sliding layer and track irregularities for a CRTSII high-speed railway track-simply supported beam bridge with nine spans and varying pier heights. This data captures the potential seismic risks under various site conditions.

[0259] For the finite element model running the complete time history record, select the load step where the peak displacement appears in the nonlinear time history analysis and extract the lateral peak displacement envelope curve of the sliding layer node. For nodes The lateral peak displacements of the left and right rails in the sliding layer are recorded as Left and right rail nodes respectively The peak displacement of the node, the peak track irregularity is recorded as TI = (R l +R r ) / 2.

[0260] To evaluate the performance of the prediction method described in Example 1, the dataset was divided into training (80%), validation (10%), and test (10%) sets. The data was randomly shuffled before training to ensure fair comparison of prediction results. The network was implemented using MATLAB, and the prediction model was trained on an NVIDIA GTX3090 GPU. The TransTCN model was trained for 200 epochs, with a batch size of 128 and an Adam optimizer with an initial learning rate of 0.01. To enhance convergence, the learning rate was gradient descent, with a 25% decrease every 10 epochs. Weights were initialized with a standard Gaussian distribution, with a base filterSize of 3, numFilters of 128, and a dropoutFactor of 0.2. The data was shuffled at each epoch, and the network was validated every 10 epochs. An early stopping strategy and L2 regularization were used to prevent overfitting. Early stopping was set to stop training after 20 epochs if there was no significant improvement in accuracy, and the L2 regularization value was set to 0.001.

[0261] 2. Evaluation Method

[0262] In order to achieve quantitative evaluation, this embodiment uses the mean absolute error (MAE), the root mean square error (RMSE) and the coefficient of determination (R 2 ) is used as the evaluation metric to measure the overall model performance. The formula is as follows:

[0263]

[0264] Where Ω is the sample size, is the predicted value, is the true value, It is the true value mean, and the indicator is calculated in m units.

[0265] 3. Accuracy comparison

[0266] To demonstrate the superiority of the prediction method described in Example 1, this example uses a deep learning benchmark method that has been proven to be effective in structural earthquake response prediction and damage prediction in recent years for comparison. The benchmark method used is as follows.

[0267] (1) Bayesian Bi-LSTM: A fast prediction method using a Bayesian self-optimizing bidirectional long short-term memory (Bi-LSTM) network is used to develop a fast prediction framework for the seismic response of HSR track-bridge systems.

[0268] (2) AttLSTM: An attention-based long short-term memory neural network (AttLSTM) is used to learn dynamics from limited training data and predict bridge responses under unseen earthquakes. The proposed method with attention mechanism outperforms the state-of-the-art LSTM in terms of accuracy and reliability.

[0269] (3) 2D CNN: LSTM, WaveNet, and 2D CNN were studied and compared in predicting the earthquake time-history responses of three types of buildings and bridge structures. Among them, CNN showed better performance in predicting displacement responses compared with the other two methods.

[0270] (4) ConvLSTM: A deep learning method ConvLSTM for predicting the nonlinear response of multi-component structures, which can achieve indirect prediction of the nonlinear response of building structures and the structural load curve.

[0271] It's worth noting that while the aforementioned benchmark methods have been widely demonstrated to demonstrate superior performance in aspects such as node response prediction and damage prediction, they differ from the approach proposed in this paper. Furthermore, these benchmark methods are all targeted at deterministic structures. When the structure type changes, the network needs to be retrained to obtain valid prediction results. Therefore, to ensure reasonable model comparison, during training, the seismic motion is combined with structural measures using the multi-input fusion feature method of this invention to achieve displacement prediction for different pier heights. The test results for different models are shown in Table 1 and Figures 8 and 9.

[0272] Table 1. Peak displacement envelope or track irregularity prediction accuracy under different network models

[0273]

[0274] From the results in Table 1, it can be seen that the TransTCN network proposed in the present invention shows excellent performance in both sliding layer peak envelope prediction and track peak irregularity prediction. The sliding layer prediction performance is slightly improved compared to the performance of several other networks. Compared with the best-performing 2D CNN network, the R2 is improved by about 0.6%, the root mean square error (RMSE) is reduced by 66.3%, and the mean absolute error (MAE) is reduced by 0.5%. The improvement in track peak irregularity prediction is more significant. Among them, compared with the worst-performing model Bi-LSTM, the R2 of track peak irregularity prediction is improved by about 87.5%, the root mean square error (RMSE) is reduced by 51.3%, and the mean absolute error (MAE) is reduced by 92.2%. Compared with the best-performing model 2D CNN, the R2 of track peak irregularity prediction is improved by about 9.0%, the root mean square error (RMSE) is reduced by 9.0%, and the mean absolute error (MAE) is reduced by 78.2%.

[0275] The improvement of R2 indicates the model's ability to explain the data and reflects the superiority of the model of the present invention in fitting the data. The lower RMSE value indicates that the deviation between the predicted value and the true value becomes smaller, which enhances the accuracy and reliability of the model. The lower MAE value indicates that the average deviation between the predicted value and the actual value is smaller, indicating that the model provides more accurate predictions as a whole, which fully demonstrates the superiority of the proposed model.

[0276] Figure 7 The performance comparison of several selected models for peak track irregularity at nine spans with varying pier heights shows that the TransTCN model's prediction curves closely match the finite element method results in terms of track irregularity prediction accuracy under varying terrain and pier heights. ConvLSTM and 2DCNN networks also effectively predict peak track irregularity under certain terrain conditions, while other networks perform significantly worse than the numerical results. Compared to other models, the proposed model exhibits significant advantages in both prediction accuracy and stability.

[0277] Figure 8 shows the track irregularity prediction R under different data input conditions. 2 , MAE, and MSE changes. From the results in the figure, it can be seen that when the number of training data is small (less than 500), the R 2 The MAE and MSE of TransTCN decrease rapidly, showing its efficient learning ability and fast convergence ability under small amounts of data. Compared with other models (AttLSTM, BiLSTM, ConvLSTM, 2DCNN), TransTCN performs better in the initial stage with smaller prediction errors. After the number of training data reaches 1000, TransTCN's R 2 The value tends to stabilize and remains close to 1.0, indicating high prediction accuracy with large amounts of data. MAE and MSE values ​​continue to remain low, with the MAE value being the lowest compared to other models. While the MAE values ​​of other models fluctuate to varying degrees and exhibit high errors within the stable region, TransTCN consistently maintains the lowest prediction error, demonstrating its significant advantage in prediction accuracy.

[0278] From the above analysis, it can be seen that the prediction method of the present invention performs well under different training data sizes. With a small amount of data, TransTCN can quickly learn and achieve high-precision prediction; with a large amount of data, TransTCN can maintain stable high performance, and R 2 The , MAE, and MSE values ​​are all superior to those of the other comparison models. These results fully demonstrate the superiority of the TransTCN model in prediction accuracy and robustness, as well as the superiority of data volume!

[0279] In summary, the long-sequence prediction framework TransTCN, constructed by combining the Transformer and TCN networks, combines the results of response spectrum analysis to predict the seismic peak response of track structures and track irregularities under uncertain pier heights in high-speed rail systems. Specifically, the proposed method combines the advantages of TCN temporal convolution with the Transformer multi-head self-attention mechanism to accurately capture the complex spatiotemporal dependencies between features across multiple time steps. When evaluated on a dataset of two high-speed railway track-bridge systems with random piers of varying spans, the proposed method achieved superior results compared to existing methods.

[0280] The multimodal hybrid neural network model proposed in this paper can effectively predict the seismic peak response and track irregularity of track structures under uncertain pier heights in high-speed rail systems, demonstrating significant computational accuracy advantages over other similar technologies. The spatial feature input of the superposition response spectrum method described in this paper significantly improves model performance, offering potential advantages in assessing regional earthquake losses under structural randomness. Furthermore, the model proposed in this paper requires significantly less data than other models, exhibiting better convergence and faster convergence speed, with an R² greater than 0.9 when the number of ground motions exceeds 600.

[0281] In summary, the model and method of the present invention provide an effective calculation tool for real-time identification of regional high-speed rail system damage and regional structural seismic risk assessment, which helps to overcome the limitations of traditional finite element methods in large-sample earthquake damage calculations.

[0282] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the present invention can be implemented in other specific forms without departing from the spirit or essential features of the present invention. Therefore, the embodiments should be viewed as exemplary and non-restrictive in all respects. Furthermore, it should be understood that although this specification is described in terms of implementation methods, it does not encompass only one technical solution. This narrative is provided for clarity only, and those skilled in the art should consider the specification as a whole. The technical solutions in the embodiments may also be appropriately combined to form other implementation methods that are understandable to those skilled in the art.

Claims

1. A method for predicting peak displacement envelope curves and track irregularities of a high-speed railway system, taking into account the uncertainty of ground motion and pier height, characterized by: The steps include: 1) Constructing a multimodal hybrid neural network model TransTCN that embeds domain prior knowledge; the multimodal hybrid neural network model TransTCN is a long sequence prediction framework that combines Transformer and TCN networks, which includes a Transformer module and a TCN module; The Transformer module is used to capture the temporal characteristics of earthquake motion and the spatial structure characteristics of the response spectrum method. It includes an embedding layer, a multi-head self-attention layer, and a fully connected feedforward network layer. The TCN module is used to capture the complete temporal dependency, which includes multiple dilated causal convolutional layers and a cross-path layer, and uses rectified linear units as activation functions and batch normalization as convolution filters; 2) Extract multimodal input features from the time series of pier height characteristics fused with earthquake motion characteristics and the spatial series of peak displacement output based on the elastic response spectrum method; The seismic motion characteristics and pier height characteristics are dimensionally fused as time series characteristics to handle temporal dependencies. The elastic response spectrum analysis results are used as spatial characteristics to handle spatial dependencies. The time series based on seismic motion fusion structural characteristics is the sequence obtained by fusing seismic data processing and pier height characteristics. The specific fusion process is as follows: The earthquake motion sequence is expressed as E = {e1, e2, ..., e t }∈R t , where e t is the seismic acceleration at each moment, t is the time sequence number; The pier height sequence is expressed as P = {p1, p2, ..., p s-1 }∈R s-1 , p s-1 is the height of the pier, s is the number of bridge spans; The predicted output of the peak displacement envelope or track irregularity of the structure after the earthquake is O = {O1, O2, …, oα}∈Rα, where O k is the peak displacement of the output point, α is the number of nodes at the predicted location of the structure; Using multiplication fusion, the structural characteristic time series of a single fused ground motion is expressed as: ON in =P T E (1); Where: P is the pier height sequence; E is the earthquake motion sequence; PE in Represents input features; T represents transposition operation; 3) The multimodal input features extracted in step 2) are input into the multimodal hybrid neural network model TransTCN constructed in step 1) for processing, so as to realize the prediction of the peak displacement envelope curve of the sliding layer and the track irregularity of the high-speed railway system considering the uncertainty of the earthquake motion and the pier height.

2. The method for predicting peak displacement envelope curves and track irregularities of a high-speed railway system considering ground motion and pier height uncertainty according to claim 1 is characterized by: In step 2), The spatial sequence of peak displacements output based on the elastic response spectrum method is the analysis result of integrating domain prior knowledge into the neural network using the response spectrum method. The specific analysis process is as follows: For a multi-span high-speed railway track bridge system composed of arbitrary structures, its motion equation is expressed as A spring-mass system with damping in degrees of freedom: Where: M is the mass matrix of the multi-span high-speed railway track bridge system; C is the damping matrix of the multi-span high-speed railway track bridge system; K is the stiffness matrix of the multi-span high-speed railway track bridge system; is the acceleration vector of the multi-span high-speed railway track bridge system; is the velocity vector of the multi-span high-speed railway track bridge system; x(t) is the displacement vector of the multi-span high-speed railway track bridge system; P(t) is the equivalent load acting on the system; Where C is the classic Rayleigh damping, and the equation can be naturally decoupled; Under a given earthquake motion sequence, P(t) can be expressed as: Where: ι represents the influence vector, represents the ground acceleration; For a linear elastic system with multiple degrees of freedom, the system response is expressed as an effective combination of multiple modal components: Where: u(t) is the total response of the system; u n (t) is the nth-order modal response; is the static response of the nth mode; A n (t) is the pseudo acceleration response of the nth mode, N is the number of modes, n represents the modal order; Among them A n (t) is: Where: A n (t) is the pseudo acceleration response of the nth mode; represents the square of the natural frequency of the nth mode; D n (t) is the displacement response of the system; nth-order modal static response for Where: Γ n represents the modal participation factor, φ n nth-order mode shape vector; Since the modal static response is related to the modal participation factor, modal frequency and mode shape vector; Combining equations (4)-(6), the total response u(t) equation of the system is transformed into: Where Γ n and φ n represent the modal participation factor and the nth-order mode shape vector respectively; Obtaining the maximum structural response during earthquake excitation is a key step in the seismic design or seismic assessment of structures, that is, solving, Where: you no =C n f n D n (t), represents the peak response of the nth mode, obtained from the response spectrum calculation; Taking into account the correlation between the peak modal responses to prevent significant errors, the perfect square combination method is used to predict the maximum response of the structure when the modal responses have significant cross-correlation: Where, ρ in represents the correlation between the i-th and n-th modal responses; the formula can be expanded as follows: In the CQC rule, the modal correlation coefficient ρ in Calculated by the following formula: Where: u o is the maximum response of the structure, N is the number of modes, u io is the peak response of the ith mode, ζ i is the damping ratio of the i-th mode, which indicates the ability of this mode to dissipate vibration energy. ζ n is the damping ratio of the nth mode, represents the natural frequency ratio of the i-th mode to the n-th mode, ω i and ω n are the natural frequencies of the i-th mode and the n-th mode, respectively.

3. The method for predicting peak displacement envelope curves and track irregularities of a high-speed railway system considering ground motion and pier height uncertainty according to claim 1 is characterized by: In step 3), the features extracted in step 2) are input into the multimodal hybrid neural network model TransTCN constructed in step 1), and the specific processing is as follows: After the time series of earthquake motion fusion structural features and the spatial series of peak displacement output based on the elastic response spectrum method are input into the multimodal hybrid neural network model TransTCN, they first pass through the embedding layer of the Transformer module to encode the position of the input sequence and convert it into a vector sequence. Then, they are processed by the multi-head self-attention layer to obtain the attention score of each sequence point and output the attention tensor of the current sequence position. The TCN module then captures the complete temporal dependency. The attention tensor obtained by the Transformer module is normalized and filtered. The output features are enriched with temporal information through multiple dilated causal convolutional layers. The rectified linear unit is used as the activation function, and normalization is used as a convolution filter to further extract tensor features at the corresponding sequence position. Huber Loss is used as the loss function. By fine-tuning the parameter δ, an effective combination of mean square error and mean absolute error is achieved. The output result is adjusted. The adjusted network can be used to realize the prediction of the peak displacement envelope curve of the sliding layer and track irregularity of the high-speed railway system considering the uncertainty of ground motion and pier height.

4. The method for predicting peak displacement envelope curves and track irregularities of a high-speed railway system considering ground motion and pier height uncertainty according to claim 3 is characterized by: The processing process of inputting the time series of structural characteristics based on earthquake motion fusion and the spatial series of peak displacement output based on elastic response spectrum method into the Transformer module is as follows: Input TransTCN model features XS = {xs1, xs2, ..., xs t } is converted into three trainable linear mapping feature matrices by the embedding layer, named as: query key Sum Among them, xs t is the characteristic of time series t, n model ,m model denote the number of queries and the length of the input sequence in the attention module, respectively, k and d v are the feature dimensions of the query, key, and value matrices respectively; The scaled point product attention and softmax function are used to consider its bidirectional temporal dependency on the historical sequence. The self-attention matrix is ​​expressed as in, It is the scaling factor of the optimized training to adjust the calculation of the inner product to prevent the inner product from being too large. In self-attention, d k and d v Equal; T is the transpose symbol; Multiple queries key Sum Perform different linear transformation projections, and then calculate the attention on each segment in parallel. Combine h independent scaled dot product attention results through vector concatenation and linear transformation and obtain the output value through linear transformation, satisfying: MultiHead(Q,K,V)=Concatenate(head1,…,head h )W o (14); Where: h represents the number of heads, represents the multi-head attention weight, head h represents the attention head, Concatenate represents the concatenation operation; For the hi-th attention head, a single attention is calculated by different weight matrices: Where: represents the self-attention weight, Represents the multi-head attention weight, Attention is the self-attention calculation, d model is the characteristic dimension of the entire model.

5. The method for predicting peak displacement envelope curves and track irregularities of a high-speed railway system considering ground motion and pier height uncertainty according to claim 3 is characterized by: The complete temporal dependency capture process using the TCN module is as follows: One-dimensional time series input and filter f:{1,2,…,ζ-1}, the dilated convolution operation F on the element θ in the one-dimensional sequence is expressed as: Where: θ represents the time step to be processed; ci is the time step currently being processed; cd is the dilation factor used to control the size of the interval; ζ is the size of the filter; * represents the convolution operation; It is a one-dimensional time series input; The receptive field of TCN is calculated as: Where: R field is the receptive field of TCN; N stack is the number of stacked TCN layers, that is, the number of stacked convolutional layers; is the size of the convolution kernel, cd is the expansion factor of each convolution layer; The multimodal hybrid neural network model TransTCN uses a residual module to avoid model degradation. The calculation process of the residual module is as follows: Where: Represents the feature representation learned by the DCC layer; It is the input and the output features obtained after the convolution operation of the DCC layer. These features can effectively represent the temporal information of the input sequence.

6. The method for predicting peak displacement envelope curves and track irregularities of a high-speed railway system considering ground motion and pier height uncertainty according to claim 5, characterized in that: The output layer of the multimodal hybrid neural network model TransTCN uses Huber Loss as the loss function, which is defined as a piecewise function: Where: is the predicted value, is the true value, δ is the adjustment coefficient, which determines the transition threshold between the loss function from quadratic to linear; By continuously adjusting δ to make the predicted output consistent with the actual output, the adjusted network can be used to predict the peak displacement envelope curve of the sliding layer and track irregularity of the high-speed railway system considering the uncertainty of ground motion and pier height.

7. The method for predicting peak displacement envelope curves and track irregularities of a high-speed railway system considering ground motion and pier height uncertainty according to claim 1, characterized in that: The method further includes the step of training the multimodal hybrid neural network model TransTCN constructed in step 1), wherein the ground motion records and uncertainty parameter sampling process used in the training are as follows: The ground motion records used for training, validation, and testing are matched with the design response spectra that fully consider the influence of ground motion propagation path, site conditions, spectral characteristics, and soil shear wave velocity. Then, a training data sample library is obtained based on the Latin hypercube stratified sampling method. The uncertainty parameter of the earthquake motion E=[e1,e2] T and the uncertain parameters of pier height PH=[P1,P2,…,P S-1 ] T Construct S+1 dimensional probability space R U , represents the total space of uncertainty of all random variables including ground motion and pier height parameters; let ξ = S + 1, which represents the probability space dimension of the combination of bridge structure and ground motion; For a ξ-dimensional random variable The random variables include all the bridge pier height parameters and ground motion parameters, which together define R u Each sample point in ; Decompose it into P mutually disjoint orthogonal subspaces where a single subspace ξ j The dimensions satisfy: The sum of the dimensions of each subspace is equal to the dimension ξ of the entire probability space, and the dimension of each subspace does not exceed ξ; at the same time, the orthogonal condition should be satisfied: Where: is an empty set, ∩ and are the intersection operation and the union operation of all subspaces, respectively. Equations (21) and (22) indicate that different subspaces are independent of each other, that is, there is no overlap between them, and the union of all subspaces is equal to the original probability space R U ; Since the subspaces are mutually orthogonal, The sample points extracted from the space can represent the uncertainty of the random variables in the space. The stratified sampling set without replacement for each orthogonal subspace becomes the sample library. The final sample set is the Latin hypercube uniformly distributed sampling points in the probability space. The probability density satisfies the linear relationship. For different probability distributions, the sample points of the random parameters in the specific probability distribution space can be obtained by using the equal probability density transformation, and finally the required seismic motion-structure random parameter sample set is obtained.

8. Application of the method for predicting track irregularity and peak displacement envelope curve of the sliding layer of a high-speed railway system considering uncertainty of ground motion and pier height according to any one of claims 1 to 7, characterized in that: It is used to predict the peak displacement envelope curve of the sliding layer and track irregularity of the high-speed railway system with a typical high-speed railway simply supported beam bridge as the benchmark structure.

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