Multi-frequency seismic response analysis method based on Fourier enhanced physical information KAN network
By using the Fourier-enhanced KAN network (FPIKAN) model, the problems of slow high-frequency signal learning and parameter explosion in the traditional PINN and PIKAN multi-frequency seismic response analysis are solved. This achieves efficient and accurate multi-frequency seismic response analysis, improves the training accuracy and efficiency of the model, and meets the decision transparency requirements of engineering applications.
Patent Information
- Application Number
- CN202511158753.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-08-19
- Publication Date
- 2025-11-21
AI Technical Summary
Traditional PINN and PIKAN models suffer from slow or non-capturing high-frequency signals when handling multi-frequency seismic response analysis. The explosion in the number of parameters leads to high computational costs, low efficiency in model training and inference, and poor interpretability, making it difficult to meet the decision transparency requirements of engineering applications.
The Fourier Enhanced Physical Information KAN Network (FPIKAN) model is adopted. By using Fourier expansion and frequency domain sparsity to compress parameters, combined with Fourier series and lightweight neural networks, a linear combination activation function of Fourier basis is dynamically generated. A residual loss function is constructed to optimize model parameters, thereby improving the ability to capture high-frequency features and training efficiency.
It achieves accurate capture of structural dynamic response under multi-frequency loads, improves the training accuracy and efficiency of the model, and can simultaneously analyze low-order and high-order vibration data, thereby enhancing the interpretability and scalability of the model.
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Figure CN120995874A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of seismic response analysis, and particularly relates to a multi-frequency seismic response analysis method based on a Fourier-enhanced physics-informed Kolmogorov-Arnold network (KAN). BACKGROUND
[0002] As a core discipline for studying the response behavior of structures under dynamic loads, the key problem of structural dynamics is to solve the differential equation of structural dynamics under seismic action. In recent years, the physics-informed neural network (PINN) provides a new idea for solving differential equations by embedding physical equations into the neural network learning process. However, the traditional PINN is usually constructed on the basis of a multilayer perceptron (MLP) architecture, which has obvious shortcomings when dealing with multi-frequency target functions. During the training process, MLP tends to preferentially learn the low-frequency components of the target function, while the learning speed of high-frequency signals (such as rapid changes or detailed features) is slow, and may even completely fail.
[0003] In the PINN framework, spectral bias significantly hinders the convergence of high-frequency physical fields (such as shock waves, boundary layers, etc.), and an adaptive weighted loss function is proposed to balance the learning process of different frequency bands. In addition, the traditional PINN is based on the MLP architecture, and its data fitting and physical constraint mechanism has a "black box" feature, which leads to weak model interpretability and makes it difficult to meet the requirements of decision transparency in high-risk engineering applications.
[0004] To solve the problem of insufficient interpretability of the MLP architecture, a Kolmogorov-Arnold network (KAN) and a physics-informed deep learning framework based on KAN (PIKAN) are proposed. However, PIKAN faces two basic problems in practical application: the double-matrix structure of KAN (i.e., the activation function matrix and the B-spline coefficient matrix) leads to a rapid increase in the number of parameters, and the explosion of parameters not only increases the storage and computing cost of the model, but also significantly reduces the efficiency of training and inference; the smoothness of the B-spline basis function makes it difficult to accurately capture high-frequency components of signals, resulting in poor performance when dealing with data with rich spectral characteristics. The low efficiency and spectral distortion problems force the model to make a trade-off between interpretability and scalability. SUMMARY
[0005] The purpose of the present application is to provide a multi-frequency seismic response analysis method based on a Fourier-enhanced physics-informed KAN network to solve the above-mentioned deficiencies in the prior art.
[0006] In order to achieve the above object, the present application provides the following technical scheme: a multi-frequency seismic response analysis method based on Fourier enhanced physical information KAN network, an FPIKAN model is constructed, the FPIKAN model forwards and propagates, including the following steps:
[0007] In the input layer, Fourier expansion is calculated for each feature of the input spatial domain parameter, and Fourier basis combination is obtained;
[0008] In the hidden layer, Fourier spectrum is extracted from the sum result of all Fourier basis combinations, and Fourier spectrum is dynamically activated based on the linear combination activation function of Fourier basis;
[0009] In the output layer, all dynamically activated results are combined into the predicted displacement field output.
[0010] Further, the input layer of the FPIKAN model converts the parameters of the single variable function in the traditional PIKAN from the spatial domain to the Fourier domain, and represents the single variable function as a finite Fourier series.
[0011] Further, the Fourier coefficients in the finite Fourier series are optimized by back propagation, and the frequency domain gradient calculation is accelerated by FFT.
[0012] Further, the input spatial domain parameter is a time variable, and a second order ordinary differential equation of structural dynamics is a physical constraint, so that three features of displacement response, first order derivative of displacement response with respect to time and second order derivative of displacement response with respect to time are obtained.
[0013] Further, the Fourier expansion calculation for each feature of the input spatial domain parameter includes the following steps:
[0014] Based on the spectral characteristics of Fourier series, Fourier expansion is calculated for displacement response, first order derivative of displacement response with respect to time and second order derivative of displacement response with respect to time respectively, and corresponding Fourier basis is obtained.
[0015] The Fourier bases of displacement response, first order derivative of displacement response with respect to time and second order derivative of displacement response with respect to time are combined to obtain Fourier basis combination of input time variable.
[0016] Further, the hidden layer of the FPIKAN model replaces the static activation function in the PIKAN with the linear combination activation function of Fourier basis, and dynamically generates the coefficients of the linear combination activation function of Fourier basis by inputting the Fourier spectrum into a lightweight neural network.
[0017] Further, the dynamic activation is the input coefficient of the Fourier spectrum into the linear combination activation function of Fourier basis dynamically generated from the Fourier spectrum.
[0018] Further, the output layer of the FPIKAN model is respectively subjected to nonlinear mapping by a GELU activation function and random Fourier features on the normalized Fourier spectrum, and then summed after being scaled by scaling parameters, and then subjected to linear transformation and added with a bias term to obtain an output value of each neuron of the hidden layer.
[0019] Further, the FPIKAN model constructs a residual loss function based on a second-order ordinary differential equation of structural dynamics and total variation regularization, and determines parameters by minimizing the residual loss function.
[0020] Compared with the prior art, the Fourier-enhanced physical information KAN network-based multi-frequency seismic response analysis method provided by the application has the ability to capture complex frequency loads. When predicting the dynamic response of a structure under multi-frequency loads, the method can accurately capture multiple frequency loads.
[0021] Compared with the prior art, the Fourier-enhanced physical information KAN network-based multi-frequency seismic response analysis method provided by the application can simultaneously analyze low-order vibration and high-order vibration data in the dynamic response analysis of a single-degree-of-freedom system structure under multi-frequency harmonic load, while the traditional PINN model cannot be fitted with the true solution.
[0022] Compared with the prior art, the Fourier-enhanced physical information KAN network-based multi-frequency seismic response analysis method provided by the application has good accuracy when solving the structural dynamic response problem of a multi-degree-of-freedom system structure under input seismic load, while the traditional PINN model and the traditional PIKAN model cannot be fitted with the true solution.
[0023] Compared with the prior art, the Fourier-enhanced physical information KAN network-based multi-frequency seismic response analysis method provided by the application has better training accuracy, higher training efficiency and parameter efficiency.
[0024] Compared with the prior art, the Fourier-enhanced physical information KAN network-based multi-frequency seismic response analysis method provided by the application has better training accuracy, higher training efficiency and parameter efficiency. BRIEF DESCRIPTION OF DRAWINGS
[0025] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments will be briefly introduced as follows. Obviously, the drawings in the following description only represent some embodiments described in the present application, and other drawings can also be obtained by those skilled in the art based on these drawings.
[0026] Figure 1 The FPIKAN model training step provided for the embodiments of the present application;
[0027] Figure 2 The structural dynamic response training architecture diagram of the FPIKAN model provided for the embodiments of the present application;
[0028] Figure 3 The excitation force provided for the embodiments of the present application The displacement training result of the FPIKAN model under the excitation force The iteration diagram;
[0029] Figure 4 The excitation force provided for the embodiments of the present application The displacement training result of the FPIKAN under the excitation force The iteration diagram;
[0030] Figure 5 The loss function comparison diagram of the traditional PINN model, the traditional PIKAN model (the grid size is 10 and 50 respectively) and the FPIKAN model under the excitation force provided for the embodiments of the present application The iteration diagram;
[0031] The loss function comparison diagram of the traditional PINN model, the traditional PIKAN model (the grid size is 10 and 50 respectively) and the FPIKAN model under the excitation force provided for the embodiments of the present application Figure 6 The iteration diagram; The NTK matrix eigenvalue curve diagram of the FPIKAN model (1 layer of hidden layer, 60 neurons per layer, the grid size is 10) under the excitation force provided for the embodiments of the present application
[0032] Figure 7 The displacement training result of the first layer under the seismic motion, the traditional PINN model, the traditional PIKAN model and the FPIKAN model provided for the embodiments of the present application The iteration diagram;
[0033] Figure 8 The displacement training result of the first layer under the seismic motion, the traditional PINN model, the traditional PIKAN model and the FPIKAN model provided for the embodiments of the present application The iteration diagram;
[0034] Figure 9 FIG. 9 is a diagram of displacement training results of a real value, a traditional PINN model, a traditional PIKAN model, and an FPIKAN model in the ninth layer under seismic motion according to an embodiment of the present application; the second iteration;
[0035] Figure 10 FIG. 9 is a diagram of displacement training results of a real value, a traditional PINN model, a traditional PIKAN model, and an FPIKAN model in the ninth layer under seismic motion according to an embodiment of the present application; the second iteration;
[0036] Figure 11 FIG. 9 is a diagram of displacement training results of a real value, a traditional PINN model, a traditional PIKAN model, and an FPIKAN model in the ninth layer under seismic motion according to an embodiment of the present application; the second iteration. DETAILED DESCRIPTION
[0037] In order to make the technical solution of the present application better understood by those skilled in the art, the present application will be further described in detail below with reference to the accompanying drawings.
[0038] In the following, example embodiments will be described more fully with reference to the accompanying drawings, in which example embodiments can be embodied in different forms and should not be construed as being limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.
[0039] In the case of no conflict, each embodiment of the present disclosure and each feature in the embodiments can be combined with each other.
[0040] As used herein, the term "and / or" includes any and all combinations of one or more of the associated listed items.
[0041] The terms used in the present application are only used to describe specific embodiments, and are not intended to limit the present disclosure. As used herein, the singular forms "a" and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will also be understood that the terms "comprise" and / or "consist of," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.
[0042] Embodiments described herein can be described with reference to plan views and / or cross-sectional views by virtue of the fact that the idealized illustrations shown in the drawings provide only a plan view and / or cross-sectional view of the illustrated structures and devices. Thus, the example illustrations can be modified, consistent with the scope of the present disclosure, to include modifications of the illustrated devices and / or configurations. Accordingly, embodiments are not limited to the illustrative examples shown in the drawings, but rather the embodiments are intended to include configurations that are within the scope of the appended claims. Thus, the regions illustrated in the figures are schematic only and their shapes are not intended to be limiting.
[0043] Kolmogorov-Arnold (KAN) neural network is based on Kolmogorov-Arnold representation theorem, any multivariate continuous function can be expressed as a combination of a finite number of univariate functions:
[0044]
[0045] where, is a univariate function, called internal function; is an external function. KAN builds network layers based on this theorem, replacing traditional linear weighting with spline functions. Each "weight" connection is a learnable function , such as splines and orthogonal polynomials, using additive structure between layers (such as internal and external function combination in Kolmogorov decomposition layer), KAN's structure is more sparse and interpretable, and has a natural advantage for high-dimensional problems. Introduce physical information into KAN architecture, that is, introduce physical residual loss term in training, get traditional PIKAN model, aiming to meet the second-order ordinary differential equation of structural dynamics and the prescribed boundary conditions, the second-order ordinary differential equation of structural dynamics:
[0046]
[0047] where , , are mass, damping and stiffness matrices respectively, is displacement field, is external excitation.
[0048] Please refer to Figures 1-2 , based on Fourier enhanced physical information KAN network multi-frequency seismic response analysis method, construct FPIKAN model, FPIKAN model forward propagation includes the following steps:
[0049] S1, calculate the Fourier expansion of each feature of the input spatial domain parameter in the input layer to obtain the Fourier basis combination.
[0050] In the traditional PIKAN, the univariate function The parameters are converted from the spatial domain (e.g., spline coefficients) to the Fourier domain, the parameter size is compressed by utilizing the frequency domain sparsity, and the ability to capture high-frequency characteristics is enhanced. For each single-variable function , it is expressed as a finite Fourier series:
[0051]
[0052] where the complex parameter is directly learned by frequency domain optimization. To reduce parameter redundancy, low-frequency truncation and symmetry constraints are introduced: the low-frequency dominance is to retain only low-frequency components of , and the high-frequency components are suppressed by regularization; the conjugate symmetry is to force to ensure is a real function. The Fourier coefficients are optimized by backpropagation, and the frequency domain gradient calculation can be accelerated by using the fast Fourier transform (FFT):
[0053] .
[0054] The input spatial domain parameters are time-varying, and the second-order ordinary differential equation of structural dynamics is the physical constraint, obtaining three features of displacement response, first-order derivative of displacement response with respect to time, and second-order derivative of displacement response with respect to time. For each feature of the input spatial domain parameters, the Fourier expansion is calculated, including the following steps: based on the spectral characteristics of the Fourier series, the Fourier expansion of the displacement response, the first-order derivative of the displacement response with respect to time, and the second-order derivative of the displacement response with respect to time are calculated respectively, and the corresponding Fourier basis is obtained; the Fourier bases of the displacement response, the first-order derivative of the displacement response with respect to time, and the second-order derivative of the displacement response with respect to time are combined to obtain the Fourier basis combination of the input time variable.
[0055] Neural Tangent Kernel (NTK), let be the output of the neural network, be the network parameters, then the NTK is defined as: Based on the NTK formula, the Fourier basis-based Kolmogorov-Arnold neural network (FKAN) is proposed, which maps the input to the Fourier basis combination: Each Fourier basis corresponds to a frequency component, which greatly enhances the KAN model's ability to analyze the dynamic response of various frequency components of ground motion.
[0056] S2, the Fourier spectrum of the sum of all Fourier basis combinations in the hidden layer is extracted, and the Fourier spectrum is dynamically activated based on the linear combination activation function of the Fourier basis.
[0057] Replace the static activation function in PIKAN with a frequency-adaptive dynamic activation function that can adjust its shape according to the frequency-domain characteristics of the input signal. Define the activation function as a linear combination of Fourier bases, with parameters dynamically generated from the frequency spectrum of the input signal:
[0058]
[0059] where is the Fourier spectrum of the input ; coefficients are dynamically generated by a lightweight neural network :
[0060]
[0061] The activation function dynamically adjusts according to the frequency-domain characteristics of the input signal, enhancing the model's adaptability to multi-band signals. The nonlinear combination of Fourier bases can surpass the expressive power of traditional ReLU / SiLU.
[0062] S3, combine all the results of dynamic activation in the output layer to form the predicted displacement field output.
[0063] The Fourier basis non-linearly maps the input in each layer, then uses a linear transformation to realize the combination of "outer function" and "inner function", and uses adaptive Fourier coefficients to adjust the spectral bias. Specifically, the learnable activation function modeled using Fourier series enables the network to flexibly capture a wide range of frequency information. By utilizing the spectral characteristics of Fourier series, they effectively represent low and high frequency elements of the input signal. Specifically, the core calculation process of each FPIKAN layer can be formalized as:
[0064]
[0065] where normalized input at layer denotes a nonlinear mapping based on random Fourier features (RFF); is a learnable scaling parameter to adjust the contribution of RFF features; is a linear transformation weight; is a bias term. By stacking multiple layers of the above transformation, the KAF network builds an effective multi-layer approximation structure. Due to the excellent parallelism of RFF on GPU, this structure avoids the high computational burden of B-spline basis functions, significantly improving training and inference efficiency while maintaining strong function approximation ability, demonstrating high computational efficiency in solving structural dynamics problems.
[0066] S4, parameters of the structural dynamic response of the FPIKAN model are obtained by minimizing the residual loss function:
[0067]
[0068] where the first term is the dynamic equation residual, the second term is the total variation regularization term, is a hyperparameter. If L is less than the set error threshold ε, update the FPIKAN model parameters, return to S1, if not, output the parameters.
[0069] In one example, a structural model of single degree of freedom of structural dynamics is constructed and the displacement dynamic response of two structures is trained, and the training ability of the FPIKAN model under single frequency harmonic load and multiple frequency harmonic loads is compared. Training is carried out on a single NVIDIA RTX5080 GPU with 16GB of graphics memory, and the calculation equation is: where , , are the mass, damping and stiffness coefficients respectively, is the displacement field, is the external excitation. A single degree of freedom structural model is established, and the dynamic characteristic parameters are designed as: =1.0kg, =49.0N / m, where the damping ratio =0.1.
[0070] The external excitation is respectively a single harmonic load and a combination of two harmonic loads N, and the training is iterated times.
[0071] The hidden layer of the FPIKAN model is set to 1 layer, the grid size is one time of the input time (10s), and the correlation analysis is carried out.
[0072] Referring to Figures 3-4 , the FPIKAN model has good fitting performance when solving single harmonic load and multiple harmonic loads. The Newmark represents the true solution and the FPIKAN model prediction solution curve overlaps, and the four evaluation indexes show better accuracy compared with the traditional PINN model and the traditional PIKAN model (grid size is 10, 50 respectively).
[0073] Table 1, respectively under and excitation force FPIKAN model displacement training accuracy comparison ( times of iteration)
[0074]
[0075] Table 2, respectively in and Comparison of training time of displacement of traditional PINN model, traditional PIKAN model (grid size is 10, 50 respectively) and FPIKAN model under excitation force Sub-iteration
[0076]
[0077] Reference Figure 5 , FPIKAN model and traditional PIKAN model under the same neural network parameters (grid size is 10) compared with FPIKAN model has faster training speed and training accuracy, after changing the grid size of traditional PIKAN model to 50, it is found that the loss function of traditional PIKAN model decreases faster with the increase of iteration times, but the accuracy is not as good as that of FPIKAN model, so FPIKAN model shows better parameter efficiency; From table 1 and table 2, it can be seen that under the evaluation of four evaluation indexes, the accuracy of FPIKAN model is good, the L2 norm error is kept at about 1%, FPIKAN model has shorter training time than traditional PIKAN model (grid size is 10, 50 respectively), and shows better parameter efficiency and training accuracy under single frequency harmonic load.
[0078] From Figure 6 , FPIKAN model and traditional PIKAN model under the same neural network parameters (grid size is 10) compared with FPIKAN model has faster training speed and training accuracy, after changing the grid size of traditional PIKAN model to 50, it is found that the loss function of traditional PIKAN model decreases with the increase of iteration times similar to FPIKAN model, but the accuracy is not as good as that of FPIKAN model, FPIKAN model shows better parameter efficiency; From table 1 and table 2, it can be seen that under the evaluation of four evaluation indexes, the accuracy of FPIKAN model is still good, the training time is also lower than that of traditional PIKAN model, FPIKAN model shows better parameter efficiency and training accuracy under multi-frequency load.
[0079] Figure 5 , Figure 6The convergence behavior of the three models during training is clearly shown. It can be observed that the loss of the traditional PINN model hardly decreases significantly throughout the training process, showing poor convergence. This phenomenon is consistent with the NTK analysis results described earlier, that is, the traditional PINN model is limited by its network structure's insufficient ability to express high-frequency modalities when dealing with target functions containing high-frequency components, leading to NTK feature spectrum degradation and optimization process difficulties. In contrast, the traditional PIKAN model, due to its parameterized additive network structure, can capture some high-frequency information, and its loss value decreases rapidly and remains stable in the initial stage, indicating that it has certain advantages in low-frequency fitting. The ability to capture high-frequency signals is further improved after the grid size is increased from 10 to 50, however, its convergence stagnation indicates that there is still a limitation of insufficient response to higher frequency modalities. The FPIKAN model is significantly better than the first two, not only achieving rapid loss reduction in the initial stage, but also entering the second stage of rapid convergence in the middle and later stages of training (after about the 20000th step), with the loss value continuously decreasing to the order of 10−7. This "two-stage convergence behavior" can be seen as an embodiment of the improvement of its frequency domain adaptability.
[0080] Combining the NTK feature spectrum analysis results, Figure 7 The NTK matrix eigenvalue curve of the FPIKAN model (1 layer of hidden layer, 60 neurons) is shown under the excitation force It is found that the FPIKAN model quickly captures the dominant frequency components in the early training stage, and then gradually optimizes the high-frequency details. The NTK matrix of the FPIKAN model has better results than the traditional PINN model. Experiments show that as the training step increases, the NTK eigenvalue presents a relatively uniform decay rate, which means that the model can learn the multi-frequency load well within a limited training time, and the FPIKAN model can capture the multi-frequency load signal well.
[0081] The NTK spectral analysis shows that the NTK eigenvalue of the FPIKAN model exhibits higher dominant spectral value in the early training stage, indicating that the model has stronger fitting ability for the main frequency components of the target function. At the same time, the convergence trend of the feature spectrum is also smoother, which means that the gradient direction is more stable, which is conducive to the optimizer quickly finding the optimal solution in the parameter space. In addition, compared to the phenomenon that the NTK eigenvalue of the traditional PINN model tends to degenerate in the later training stage, the feature spectrum of the FPIKAN model maintains good diversity, reflecting its sustained ability to maintain different modal information throughout the training process.
[0082] In summary, the FPIKAN model can effectively capture high-frequency information, thereby improving the practicality of the model. For structural dynamics problems, it provides a new way to solve the structural response under the excitation of multi-frequency superimposed loads such as earthquakes.
[0083] In another embodiment, the ground motion acceleration time history parameters are shown in Table 3, the mass of each floor of the nine-story frame shear structure is 81570 kg, and the inter-story stiffness is 8,748,000 N / m. The conventional PIKAN model, the FPIKAN model neural network hyperparameter settings are shown in Table 4. The L2 norm error, the coefficient of determination , the root mean square error and the mean absolute error indicators are used to evaluate the fitting accuracy of the model.
[0084] Table 3, Northridge earthquake ground motion record information
[0085]
[0086] Table 4, Hyperparameters of the conventional PIKAN model and the FPIKAN model of the nine-story frame shear structure
[0087]
[0088] Table 5, Comparison of displacement training accuracy of the conventional PINN model, the conventional PIKAN model and the FPIKAN model of the first, fifth and ninth floors of the structure under ground motion times of iteration
[0089] (a) First floor
[0090]
[0091] (b) Fifth floor
[0092]
[0093] (c) Ninth floor
[0094]
[0095] Table 6, Comparison of displacement training time of the conventional PINN model, the conventional PIKAN model and the FPIKAN model of the structure under ground motion times of iteration
[0096]
[0097] Figures 8-10 are the comparison diagrams of the training results of the conventional PINN model, the conventional PIKAN model and the FPIKAN model and the Newmark numerical solution. As can be seen from the diagram, the conventional PINN and the conventional PIKAN model cannot solve the corresponding problem of the structure under ground motion, while the training results of the FPIKAN model are well fitted with the true solution. ThroughFigure 11 It can be found that the FPIKAN model exhibits good spectrum capturing ability and optimization stability. By comparing the L2 norm error, the coefficient of determination , the root mean square error and the mean absolute error index evaluation, it is found that the nine-story shear frame structure using the FPIKAN model under the excitation of ground motion shows good training accuracy, as shown in Table 5, and from Table 6 it can be seen that the training time is shortened by 50% compared with the traditional PIKAN model. Therefore, the FPIKAN model can well solve the problems that the traditional PINN model and the traditional PIKAN model are difficult to handle, and has better accuracy and shorter training time.
[0098] The above only describes some exemplary embodiments of the present application in a descriptive manner, and it is needless to say that those skilled in the art can modify the described embodiments in various ways without departing from the spirit and scope of the present application. Therefore, the above drawings and descriptions are illustrative in nature and should not be understood as limiting the scope of protection of the claims of the present application.
Claims
1. A multi-frequency seismic response analysis method based on Fourier-enhanced physical information KAN network, characterized in that: Constructing the FPIKAN model, the forward propagation of the FPIKAN model includes the following steps: In the input layer, Fourier expansion is calculated for each feature of the input spatial domain parameters to obtain a combination of Fourier bases; Fourier spectra are extracted from the summation of all Fourier basis combinations in the hidden layer, and the Fourier spectra are dynamically activated based on the linear combination activation function of the Fourier basis. The output layer combines the results of all dynamic activations into a predicted displacement field output.
2. The multi-frequency seismic response analysis method based on Fourier-enhanced physical information KAN network according to claim 1, characterized in that: The input layer of the FPIKAN model transforms the parameters of the univariate function in the traditional PIKAN from the spatial domain to the Fourier domain, and represents the univariate function as a finite Fourier series.
3. The multi-frequency seismic response analysis method based on Fourier-enhanced physical information KAN network according to claim 2, characterized in that: The Fourier coefficients in the finite Fourier series are optimized through backpropagation and the frequency domain gradient calculation is accelerated by FFT.
4. The multi-frequency seismic response analysis method based on Fourier-enhanced physical information KAN network according to claim 1, characterized in that: The input spatial domain parameters are time-dependent, and the second-order ordinary differential equations of structural dynamics are physical constraints, resulting in three characteristics: displacement response, the first derivative of displacement response with respect to time, and the second derivative of displacement response with respect to time.
5. The multi-frequency seismic response analysis method based on Fourier-enhanced physical information KAN network according to claim 4, characterized in that: The calculation of the Fourier expansion for each feature of the input spatial domain parameters includes the following steps: Based on the spectral characteristics of Fourier series, Fourier expansions are calculated for the displacement response, the first derivative of the displacement response with respect to time, and the second derivative of the displacement response with respect to time, respectively, to obtain the corresponding Fourier basis. The Fourier basis of the displacement response, the first derivative of the displacement response with respect to time, and the second derivative of the displacement response with respect to time are combined to obtain the Fourier basis combination of the input time variable.
6. The multi-frequency seismic response analysis method based on Fourier-enhanced physical information KAN network according to claim 1, characterized in that: The hidden layer of the FPIKAN model replaces the static activation function in PIKAN with a linear combination activation function of Fourier basis; and dynamically generates the coefficients of the linear combination activation function of Fourier basis by inputting the Fourier spectrum into a lightweight neural network.
7. The multi-frequency seismic response analysis method based on Fourier-enhanced physical information KAN network according to claim 6, characterized in that: The dynamic activation is a linear combination activation function that takes the Fourier spectrum input coefficients as inputs to the Fourier basis dynamically generated from the Fourier spectrum.
8. The multi-frequency seismic response analysis method based on Fourier-enhanced physical information KAN network according to claim 1, characterized in that: The output layer of the FPIKAN model performs nonlinear mapping on the normalized Fourier spectrum using the GELU activation function and random Fourier features, then scales the spectrum using scaling parameters and sums the results. Finally, a bias term is added to the summation result after a linear transformation to obtain the output value of each neuron in the hidden layer.
9. The multi-frequency seismic response analysis method based on Fourier-enhanced physical information KAN network according to claim 1, characterized in that: The FPIKAN model is based on the second-order ordinary differential equation of structural dynamics and the total variation regularization to construct a residual loss function, and the parameters are determined by minimizing the residual loss function.