A method and system for single pile horizontal seismic response analysis

Abstract

The application provides a single-pile horizontal seismic response analysis method and system, relates to the technical field of geotechnical engineering earthquake resistance and intelligent calculation, and comprises the following steps: obtaining pile body parameters, soil dynamics parameters and a target frequency set, determining free field displacement and soil horizontal dynamic complex impedance based on a pile-soil interaction model, and establishing a frequency domain complex fourth-order control equation; performing dimensionless and real and imaginary part coupling splitting to form a real value control equation set, constructing a Siren-PINN, and training based on a control equation residual and a boundary condition residual; obtaining a pile body horizontal displacement response by using the trained model, calculating a pile-soil motion interaction factor under each target frequency, and generating a frequency domain analysis result.

Description

Technical Field

[0001] This application belongs to the field of geotechnical engineering earthquake resistance and intelligent computing technology, specifically involving a method and system for analyzing the horizontal seismic response of a single pile. Background Technology

[0002] Pile foundations are widely used in infrastructure such as highway bridges, building construction, and port terminals. Under seismic loading, vertically incident shear waves propagate through the soil, causing dynamic interactions between the pile and the surrounding soil, resulting in a difference between the actual seismic response at the pile top and the free-field surface motion. Therefore, accurately obtaining the horizontal displacement response of a single pile and the characteristics of pile-soil motion interaction under different seismic frequencies is a crucial technical foundation for pile foundation seismic response analysis and seismic performance assessment.

[0003] Current methods for analyzing the horizontal seismic response of single piles typically employ numerical simulations such as the finite element method and the finite difference method, or analytical or semi-analytical methods based on elastic dynamics theory. Some techniques also utilize deep learning models to fit the relationship between pile-soil parameters and dynamic response. Numerical simulation methods usually require discretizing the pile-soil continuous medium and setting artificial boundaries for the semi-infinite soil mass; analytical or semi-analytical methods require derivation and solution based on the pile-soil dynamic control relationship and boundary conditions; deep learning methods mainly establish a mapping relationship between input parameters and response results through training data.

[0004] However, in wide-frequency, multi-frequency, and high-frequency oscillatory response analysis scenarios, numerical discretization methods are easily affected by grid density, artificial boundaries, and computational scale. Under high-frequency conditions, further grid refinement is usually required. Analytical or semi-analytical methods are cumbersome to solve when the governing equations and boundary conditions are complex. Purely data-driven methods rely heavily on training samples, and the consistency between their prediction results and the pile-soil dynamic control relationship is difficult to guarantee directly. For working conditions with significant frequency variations and large differences in pile-soil parameters, these problems further affect the consistency between response results at different frequencies and the efficiency of continuous frequency domain analysis. Summary of the Invention

[0005] This application provides a method and system for analyzing the horizontal seismic response of a single pile to solve the above-mentioned technical problems.

[0006] The technical solution adopted in this application is as follows: This application provides a method for analyzing the horizontal seismic response of a single pile, executed by a computing device, the method comprising: Obtain the pile body parameters, soil dynamic parameters, and target frequency set corresponding to the vertical incident shear wave of the target single pile; For each target frequency in the target frequency set, a plane strain pile-soil interaction model based on continuous medium theory is used to determine the free field displacement and the horizontal dynamic complex impedance of the soil according to the soil dynamic parameters and the target frequency. Based on the pile parameters, the free field displacement and the horizontal dynamic complex impedance of the soil, a fourth-order frequency domain complex value governing equation characterizing the horizontal displacement of the pile body, as well as the corresponding pile top boundary conditions and pile bottom boundary conditions are established. The frequency domain complex-valued fourth-order governing equations and their boundary conditions are dimensionless. The dimensionless pile horizontal displacement, free field displacement, and soil horizontal dynamic complex impedance are then split into real and imaginary parts. Based on the coupling relationship between the split real and imaginary parts, the frequency domain complex-valued fourth-order governing equations are transformed into a set of coupled real-valued governing equations consisting of real and imaginary governing equations, and the corresponding real-valued boundary conditions are obtained. A Siren-PINN model is constructed with dimensionless pile coordinates as input and the real and imaginary parts of the dimensionless horizontal displacement of the pile as output. The derivatives of the output with respect to the dimensionless pile coordinates are obtained based on automatic differentiation. The output and the derivatives are substituted into the coupled real-valued control equations and the real-valued boundary conditions to obtain the control equation residuals and boundary condition residuals, respectively. A comprehensive loss function is constructed based on the control equation residuals and boundary condition residuals. The Siren-PINN model corresponding to each target frequency is trained with the comprehensive loss function as the optimization objective to obtain the single pile horizontal displacement response model corresponding to each target frequency. The predicted coordinates of the pile body are input into the single pile horizontal displacement response model corresponding to the target frequency to obtain the real and imaginary parts of the dimensionless horizontal displacement of the pile body. The amplitude of the horizontal displacement at the pile top is determined based on the real and imaginary parts corresponding to the pile top position. The pile-soil motion interaction factor is determined by combining the amplitude of the horizontal displacement of the free ground surface at the corresponding target frequency. The frequency domain analysis results of the single pile horizontal seismic response are generated based on the pile-soil motion interaction factor corresponding to each target frequency.

[0007] According to one embodiment of this application, the step of determining the free-field displacement and the horizontal dynamic complex impedance of the soil based on the soil dynamic parameters and the target frequency, and establishing a frequency-domain complex-valued fourth-order governing equation characterizing the horizontal displacement of the pile body based on the pile parameters, the free-field displacement, and the horizontal dynamic complex impedance of the soil, includes: The horizontal displacement of the soil under seismic loading is decomposed into free field displacement and relative displacement of the soil with respect to the free field. Based on the attenuation condition of the scattering displacement at infinite radial distance of the soil and the displacement continuity condition of the pile-soil interface, the displacement distribution of the soil around the pile is determined. The stress components at the pile-soil interface are determined based on the displacement distribution and the stress-strain relationship of the soil. The stress components are integrated along the pile perimeter to obtain the horizontal resistance of the soil around the pile acting on the pile body. The horizontal dynamic complex impedance of the soil is determined based on the relationship between the horizontal resistance and the relative horizontal displacement between the pile and the soil. Substituting the horizontal resistance, characterized by the horizontal dynamic complex impedance of the soil, into the dynamic equilibrium relationship of the pile body, and performing frequency domain transformation, yields the frequency domain complex-valued fourth-order governing equation.

[0008] According to one embodiment of this application, the pile top boundary conditions and pile bottom boundary conditions include: The pile top is set as a free boundary, so that the bending moment and shear force at the pile top section are zero, and the free boundary is defined by the second and third derivatives of the horizontal displacement of the pile body with respect to the pile body coordinates, respectively. The pile bottom is set as a fixed boundary, so that the horizontal displacement of the pile bottom is consistent with the free field displacement at the corresponding position, and the pile bottom rotation angle is zero. The pile bottom rotation angle is limited by the first derivative of the horizontal displacement of the pile body with respect to the pile body coordinates.

[0009] According to one embodiment of this application, the dimensionless processing of the frequency domain complex-valued fourth-order governing equations and their boundary conditions includes: The pile length is used as the length characteristic scale to make the pile body coordinates dimensionless, and the bedrock displacement amplitude is used as the displacement characteristic scale to make the pile body horizontal displacement and free field displacement dimensionless. The corresponding dimensionless frequency is determined according to the target frequency, pile diameter and soil shear wave velocity. Based on the length characteristic scale and displacement characteristic scale, the coefficients of each term in the frequency domain complex-valued fourth-order governing equation, as well as the pile top boundary condition and pile bottom boundary condition, are transformed accordingly to obtain the dimensionless frequency domain complex-valued fourth-order governing equation and its boundary conditions.

[0010] According to one embodiment of this application, the dimensionless pile horizontal displacement, free field displacement, and soil horizontal dynamic complex impedance are decomposed into real and imaginary parts, respectively, and the coupled real-valued governing equations are formed based on the coupling relationship between the decomposed real and imaginary parts, including: The real and imaginary parts of the horizontal displacement of the pile body, the real and imaginary parts of the free field displacement, and the real and imaginary parts of the horizontal dynamic complex impedance of the soil are determined respectively. The horizontal resistance term, which is composed of the horizontal dynamic complex impedance of the soil and the relative horizontal displacement between the pile and the soil, is expanded into real and imaginary parts, so that the real and imaginary parts of the horizontal dynamic complex impedance of the soil form cross-coupled terms with the real and imaginary parts of the relative horizontal displacement between the pile and the soil, respectively. Based on the cross-coupling terms, the real part control equation and the imaginary part control equation are constructed respectively, so that the real part control equation and the imaginary part control equation jointly constrain the real part and the imaginary part of the horizontal displacement of the pile.

[0011] According to one embodiment of this application, the construction of the Siren-PINN, which takes dimensionless pile coordinates as input and the real and imaginary parts of the dimensionless horizontal displacement of the pile as output, includes: For each target frequency, the corresponding horizontal dynamic complex impedance of the soil and the equation coefficients of the coupled real-valued control equations are determined based on the target frequency, and the corresponding Siren-PINN is trained for the target frequency, wherein the target frequency is used as a pre-parameter for model training, and the dimensionless pile coordinates are used as the network input for Siren-PINN. A sinusoidal activation function is used in the hidden layer of the Siren-PINN, and the real and imaginary parts of the dimensionless horizontal displacement of the pile body are output through the output layer.

[0012] According to one embodiment of this application, constructing a comprehensive loss function based on the residuals of the governing equations and the residuals of the boundary conditions includes: Automatic differentiation is used to determine the first to fourth derivatives of the real and imaginary parts of the dimensionless horizontal displacement of the pile body relative to the dimensionless pile body coordinates. The real part, imaginary part and their corresponding derivatives are substituted into the real part control equation and the imaginary part control equation to obtain the real part control equation residuals and the imaginary part control equation residuals at the corresponding configuration points. The higher-order derivative relationship of the free field displacement is determined based on the frequency domain wave equation satisfied by the free field displacement, and the higher-order derivative relationship is used to determine the higher-order derivative term of the free field displacement in the residual of the control equation. The boundary conditions at the top and bottom of the pile are decomposed into real and imaginary parts, respectively. The boundary condition residuals are determined based on the decomposed boundary conditions and the corresponding derivatives of the dimensionless horizontal displacement of the pile. The physical residual loss of the control equation is determined based on the real part residual and the imaginary part residual of the control equation at each configuration point, and the boundary condition loss is determined based on the boundary condition residual. The comprehensive loss function is composed of the physical residual loss of the control equation and the boundary condition loss.

[0013] According to one embodiment of this application, training the Siren-PINN corresponding to each target frequency includes: According to the Chebyshev node distribution, configuration points are set in the dimensionless solution domain of the pile body to determine the physical residual loss of the governing equation, and the configuration point density in the boundary areas of the corresponding pile top and pile bottom is increased. During model training, the weights of the boundary condition loss and the physical residual loss of the control equation in the comprehensive loss function are adjusted in stages. Specifically, as the model training progresses, the weight of the boundary condition loss is gradually reduced and the weight of the physical residual loss of the control equation is increased.

[0014] According to one embodiment of this application, training the Siren-PINN corresponding to each target frequency in the target frequency set further includes: The dimensionless frequency corresponding to each target frequency is compared with a preset frequency threshold. A first training round is configured for target frequencies that are lower than the preset frequency threshold, and a second training round is configured for target frequencies that reach or exceed the preset frequency threshold, wherein the second training round is greater than the first training round. Train each Siren-PINN sequentially according to the frequency order in the target frequency set. After completing the training of the Siren-PINN corresponding to the current target frequency, use the hidden layer parameters of the trained Siren-PINN that are closer to the input side as the initial parameters of the Siren-PINN corresponding to the next target frequency, and re-initialize the remaining hidden layer parameters and output layer parameters. The frequency domain analysis results of generating the horizontal seismic response of a single pile based on the pile-soil motion interaction factors corresponding to each target frequency include: Predicted coordinate points are set within the dimensionless solution domain of the pile body. The predicted coordinate points are then input into the single pile horizontal displacement response model corresponding to the target frequency to obtain the real and imaginary parts of the dimensionless horizontal displacement of the pile body at each predicted coordinate point. The dimensionless horizontal displacement of the pile body is converted into the corresponding horizontal displacement of the pile body according to the displacement characteristic scale, and the amplitude of the horizontal displacement at the pile top is determined according to the real and imaginary parts of the horizontal displacement corresponding to the pile top position. The amplitude of the horizontal displacement at the surface of the free field is determined based on the propagation relationship of the free field shear wave at the corresponding target frequency. The ratio of the amplitude of the horizontal displacement at the pile top to the amplitude of the horizontal displacement at the surface of the free field is determined as the pile-soil motion interaction factor corresponding to the target frequency. The pile-soil motion interaction factors are combined according to the target frequency set to obtain the frequency domain analysis results of the pile-soil motion interaction factors as a function of frequency.

[0015] This application also provides a single-pile horizontal seismic response analysis system, including: The parameter acquisition module is used to acquire the pile body parameters, soil dynamic parameters, and target frequency set corresponding to the vertical incident shear wave of the target single pile. The physical model construction module is used to determine the free field displacement and the horizontal dynamic complex impedance of the soil based on the plane strain pile-soil interaction model of the continuous medium theory for each target frequency in the target frequency set. It also establishes the frequency domain complex value fourth-order governing equations characterizing the horizontal displacement of the pile body, as well as the corresponding pile top boundary conditions and pile bottom boundary conditions, based on the pile body parameters, the free field displacement and the horizontal dynamic complex impedance of the soil. The control equation transformation module is used to perform dimensionless processing on the frequency domain complex-valued fourth-order control equation and its boundary conditions, and to decompose the dimensionless pile horizontal displacement, free field displacement and soil horizontal dynamic complex impedance into real and imaginary parts respectively. Based on the coupling relationship between the decomposed real and imaginary parts, the frequency domain complex-valued fourth-order control equation is transformed into a coupled real-valued control equation set composed of real and imaginary control equations, and the corresponding real-valued boundary conditions are obtained. The response model training module is used to construct a Siren-PINN with dimensionless pile coordinates as input and the real and imaginary parts of the dimensionless horizontal displacement of the pile as output. Based on automatic differentiation, the derivatives of the output with respect to the dimensionless pile coordinates are obtained. The output and the derivatives are substituted into the coupled real-valued control equations and the real-valued boundary conditions to obtain the control equation residuals and boundary condition residuals, respectively. A comprehensive loss function is constructed based on the control equation residuals and boundary condition residuals. The Siren-PINN corresponding to each target frequency is trained with the comprehensive loss function as the optimization objective to obtain the single pile horizontal displacement response model corresponding to each target frequency. The response result generation module is used to input the predicted coordinates of the pile body into the single pile horizontal displacement response model corresponding to the target frequency, obtain the real and imaginary parts of the dimensionless horizontal displacement of the pile body, determine the amplitude of the horizontal displacement at the pile top based on the real and imaginary parts corresponding to the pile top position, and determine the pile-soil motion interaction factor by combining the amplitude of the free-field surface horizontal displacement at the corresponding target frequency. Based on the pile-soil motion interaction factor corresponding to each target frequency, the frequency domain analysis results of the single pile horizontal seismic response are generated.

[0016] Due to the adoption of the above technical solution, the beneficial effects achieved by this application are as follows: This application obtains the pile body parameters, soil dynamic parameters, and target frequency set of the target single pile, determines the free field displacement and horizontal dynamic complex impedance of the soil for each target frequency, and combines the horizontal dynamic complex impedance of the soil with the dynamic equilibrium relationship of the pile body to form a fourth-order complex-valued governing equation in the frequency domain. Then, the governing equation and boundary conditions are converted into physical constraints that can be used for Siren-PINN training, so that the network solution process is simultaneously constrained by the dynamic characteristics of the pile body, the dynamic response of the soil, and the boundary conditions of the pile foundation. This ensures that the obtained horizontal seismic response of the single pile is consistent with the physical relationship of pile-soil dynamic interaction, and can further generate frequency domain analysis results of pile-soil motion interaction factors corresponding to different target frequencies.

[0017] This application decomposes the horizontal displacement of soil under seismic loading into free-field displacement and displacement components relative to the free field. Based on the displacement continuity of the pile-soil interface, it determines the soil displacement and interface stress around the pile. Then, it obtains the horizontal resistance around the pile and determines the horizontal dynamic complex impedance of the soil by integrating the interface stress. This allows the horizontal action of the soil on the pile to vary with the target frequency and soil dynamic parameters, and further acts on the fourth-order governing equation in the frequency domain. Compared with the method of describing the soil action using fixed foundation parameters, this application can more completely reflect the correlation between the relative motion of the pile and soil and the dynamic response of the soil.

[0018] This application uses pile length and bedrock displacement amplitude as characteristic scales to make the governing equations and boundary conditions dimensionless, which can reduce the impact of the order-of-magnitude differences between different geometric parameters, material parameters and frequency parameters on the network training process. On this basis, the horizontal displacement of the pile body, the free field displacement and the horizontal dynamic complex impedance of the soil are decomposed into real and imaginary parts respectively, and the real and imaginary cross-coupling relationship generated by the multiplication of complex impedance and pile-soil relative displacement is retained. This allows the frequency domain complex-valued dynamics problem to be transformed into coupled real-valued constraints that can be directly processed by the real-domain neural network, thereby avoiding the separation of the real and imaginary parts of the complex response for solution.

[0019] This application constructs a Siren-PINN network with dimensionless pile coordinates as input and the real and imaginary parts of the dimensionless horizontal displacement of the pile as output. It utilizes automatic differentiation to obtain the first to fourth derivatives of the network output with respect to the pile coordinates. These derivatives are then substituted into the real and imaginary control equations, as well as the pile top and bottom boundary conditions, to form control equation residuals and boundary condition residuals. This allows the network training process to be directly constrained by the fourth-order dynamic control equations and higher-order boundary conditions. Simultaneously, the sinusoidal activation function can adapt to the spatially varying oscillating characteristics of the seismic response, thus helping to maintain the stability of the higher-order derivative calculations and the frequency domain response solution process.

[0020] This application employs Chebyshev placement points and increases the density of placement points in the boundary regions at the pile top and bottom, thereby distributing more training placement points in the regions affected by higher-order boundary conditions. Simultaneously, by adjusting the weights of boundary condition loss and physical residual loss of the control equation in stages during the training process, the model prioritizes satisfying boundary constraints in the early stages of training, and then gradually strengthens the constraints of the dynamic control equation. This reduces the impact of asynchronous convergence between boundary residuals and control equation residuals on the solution process, enabling the boundary conditions and internal dynamic control relationships to converge in a coordinated manner during the same training process.

[0021] This application improves the continuity and computational efficiency of multi-frequency model training by configuring different training rounds according to different dimensionless frequencies and transferring some hidden layer parameters close to the input side in the network corresponding to the previous target frequency to the network corresponding to the next target frequency during the solution of continuous frequency points. This allows some pile-soil dynamic response features learned under adjacent frequency conditions to be used for the initialization of subsequent frequency models, while preserving space for the subsequent network layers to readjust for the current frequency. This avoids using the same training rounds for all target frequencies or retraining entirely from random parameters, thereby improving the continuity of multi-frequency model training and computational efficiency during the solution of continuous frequency domain.

[0022] This application uses a trained single-pile horizontal displacement response model to obtain the real and imaginary parts of the dimensionless horizontal displacement of the pile body. Based on the reset displacement amplitude at the pile top position and the horizontal displacement amplitude of the free surface at the corresponding target frequency, the pile-soil motion interaction factor is determined. This allows the pile displacement results obtained by the neural network to be further converted into frequency domain parameters that characterize the degree of correction of the pile top seismic input. By combining the pile-soil motion interaction factors corresponding to each target frequency in frequency order, continuous frequency domain response results can be obtained, providing direct frequency domain data for subsequent horizontal seismic response analysis of pile foundations. Attached Figure Description

[0023] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a schematic diagram of a pile-soil system model provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of the overall architecture for solving the seismic response of a single pile using PINNs, as provided in an embodiment of the present invention. Figure 3 This is a comparison chart of the PINNs solution results and analytical results provided in the embodiments of the present invention; Figure 4 This is a comparison chart of the average training time per frequency under different technologies provided in the embodiments of the present invention; Figure 5This is a comparison chart of various errors under different technologies provided in the embodiments of the present invention; Figure 6 A comparison chart of single-frequency average training time with and without transfer learning provided in an embodiment of the present invention; Figure 7 The pile-soil motion interaction factor-frequency curves under different soil damping ratios β are provided in the embodiments of the present invention. Figure 8 A flowchart provided for an embodiment of the present invention. Detailed Implementation

[0024] The technical solution of this application will be described in detail below with reference to embodiments thereof. However, it should be understood that these descriptions are only for illustrating specific implementations of this application and are not intended to limit the scope of protection of this application. In the following detailed description, to facilitate understanding of the technical solution of this application by those skilled in the art, the pile-soil dynamic interaction model, frequency domain complex-valued control equation, dimensionless transformation and real-imaginary part coupling processing, Siren-PINN network structure, physical constraint loss function, model training strategy, and response result output process are described in detail. One or more embodiments can be implemented without using some specific parameters.

[0025] The terms "comprising," "including," etc., as used herein, indicate the presence of the stated features, steps, operations, and / or components, but do not exclude the presence or addition of one or more other features, steps, operations, and / or components. Technical and scientific terms used herein, unless otherwise defined, have the meanings commonly understood by those skilled in the art and should be interpreted in the context of this specification. The following embodiments are presented in a hierarchical manner, following the structure of "overall scheme—physical model—data and parameter processing—model construction—loss function—model training—result output—engineering application—optional implementation," consistent with the organizational form of the specific embodiments provided in the full text of this second edition.

[0026] This application provides a method for analyzing the horizontal seismic response of a single pile based on the frequency-domain complex-valued physical constraint Siren-PINN. It belongs to the interdisciplinary field of geotechnical engineering seismic resistance and intelligent computing. It is mainly aimed at the frequency domain solution of pile-soil dynamic interaction under vertical incident shear wave action. It can be used for pile foundation horizontal seismic response analysis, pile foundation seismic performance evaluation, seismic design parameter analysis and dynamic response prediction in highway bridges, building engineering, port docks and other projects.

[0027] Under seismic loading, vertically incident shear waves propagate through the soil layer, and the pile interacts dynamically with the surrounding soil, causing a difference between the actual seismic motion at the pile top and the free-field surface motion under pileless conditions. To characterize this difference, this application first establishes a pile-soil dynamic interaction model under continuous medium conditions. Based on the soil dynamic parameters and target frequency, the horizontal dynamic complex impedance of the soil is calculated, and this horizontal dynamic complex impedance is embedded into the single-pile dynamic equilibrium equation to form a fourth-order frequency-domain complex-valued governing equation. Furthermore, through dimensionless transformation and coupling decomposition of the real and imaginary parts, the frequency-domain complex-valued governing relationship is transformed into a physical constraint that can be processed by a real-domain neural network. Then, a Siren-PINN network is constructed and trained for each target frequency. The first to fourth derivatives of the pile's horizontal displacement are calculated through automatic differentiation, and the residuals of the governing equations and boundary conditions are used together to train the constraint network. Finally, the horizontal displacement of the pile is obtained based on the trained network, and the pile-soil motion interaction factor and the corresponding frequency-domain curve are calculated.

[0028] In one embodiment, the method of this application can be executed by a computing device with data processing capabilities. The computing device may include a processor, memory, and a computing program for performing neural network training and numerical computation. Specifically, Siren-PINN can be built based on the PyTorch framework, and an automatic differentiation module can be used to calculate the first to fourth derivatives of the network output with respect to the input coordinates.

[0029] In this embodiment, the overall processing can sequentially include establishing a pile-soil motion interaction model, dimensionless transformation and complex-valued coupling decomposition of the governing equations, constructing a Siren-PINN, constructing a comprehensive loss function that integrates physical constraints, training the model using a multi-strategy collaborative approach, solving for the horizontal seismic response of a single pile and outputting the results, and using the obtained frequency domain response results for pile foundation seismic design and performance evaluation. The core of the relevant technical solution is to transform the horizontal dynamic complex impedance of the soil, the fourth-order complex-valued governing equation of the single pile in the frequency domain, and the pile foundation boundary conditions into physical constraints during the neural network training process, rather than simply obtaining the single pile seismic response through statistical mapping between input data and output labels.

[0030] like Figure 1 As shown, a pile-soil system consisting of a single pile, a viscoelastic soil layer, and rigid bedrock can be established. Figure 1 In the diagram, H represents the pile length, and D represents the pile diameter. This represents simple harmonic horizontal motion at the bedrock. Indicates the amplitude of bedrock displacement. The angular frequency of the seismic wave is represented by t, and time is represented by t. G represents the Lamé parameter of the elastic medium. Indicates the hysteretic damping ratio of the soil. Indicates soil density. Indicates the Poisson's ratio of the soil. The vector represents the soil shear wave velocity; z is the vertical coordinate of the pile, and r is the radial coordinate. Here are the corner coordinates, and O is the origin.

[0031] like Figure 2 , Figure 3 , Figure 4 , Figure 5 , Figure 6 , Figure 7 and Figure 8 As shown, in one specific embodiment, the pile-soil system can be described by the following physical assumptions: the soil around the pile is a homogeneous, isotropic viscoelastic semi-infinite space medium; the pile is a linear elastic circular pile with uniform cross-section, and the shear deformation and rotational inertia of the pile body are ignored; the bedrock is a rigid medium, and the ground motion propagates upward in the form of vertically incident simple harmonic shear waves; the pile-soil interface is in continuous contact, without relative slippage or separation, and the displacement and stress at the interface are continuous; at infinite radial distance of the soil, the scattered displacement generated by the pile body decays to zero.

[0032] The pile parameters of the target single pile may include pile length H, pile diameter D, and pile elastic modulus. Pile density Pile cross-sectional area and the moment of inertia of the pile section Soil dynamic parameters may include soil shear modulus G and soil density. Poisson's ratio Hysteresis damping ratio Shear wave velocity And parameters related to the Lamé parameters. These parameters can be obtained from geotechnical investigation data, pile design data, or pile testing data in engineering applications.

[0033] The target frequency set is used to define the range of seismic frequencies to be analyzed. For each target frequency in the target frequency set, the dynamic complex impedance of the soil is calculated, the coefficients of the governing equations are determined, Siren-PINN training is performed, and the response is solved, thereby obtaining the horizontal seismic response of a single pile corresponding to multiple frequency points.

[0034] In some embodiments, a dynamic response model of soil is established based on the theory of continuous medium elastic dynamics and the Novak plane strain assumption.

[0035] For the radial displacement of the soil and tangential displacement The following simplified governing equations can be established: (1) (2) In the formula, , Here are the soil composite parameters, where:

[0036]

[0037] The volumetric strain e is expressed as:

[0038] The Laplace operator is represented as:

[0039] Under the action of seismic waves, the absolute displacement of the soil can be decomposed into free-field displacement and the scattered displacement of the soil relative to the free field. The radial and tangential displacements are written as follows: (3) (4) in, , These represent the radial and tangential displacements of the soil relative to the free-field soil under seismic loading, respectively. The free field displacement can be written as:

[0040] in:

[0041] Substituting equations (3) and (4) into the soil governing equations and performing differential transformations, we can obtain the soil relative to volumetric strain. The governing equations for the core variables are: (5) in:

[0042] Furthermore, let:

[0043] Solving equation (5) using the method of separation of variables, after introducing the Bessel function, the general solutions for the radial and angular functions are respectively: (6) (7) in:

[0044] and Let m represent the corresponding Bessel functions of order m. , , , These are coefficients to be determined.

[0045] Since the scattering displacement at radial infinity of the soil approaches zero, the following condition is satisfied: (8) Considering You can get .

[0046] For the pile-soil interface, let the pile radius be:

[0047] The displacement continuity condition at the pile-soil interface can be expressed as: (9) in, This indicates the horizontal displacement of the pile under earthquake action.

[0048] Further consideration of volumetric strain e regarding The even function relation can be obtained. , And by solving and filtering the undetermined coefficients, the analytical forms of the radial and tangential displacements of the soil are obtained: (10) (11) in:

[0049]

[0050] These are coefficients to be determined.

[0051] The continuity condition at the junction of the pile and the soil is obtained as follows: (12) in:

[0052] Based on the stress-strain constitutive relationship of a linear elastic body, substituting the above soil displacement results, the radial normal stress and tangential shear stress of the soil around the pile can be obtained. The radial normal stress is: (13) The tangential shear stress is: (14) Integrating the stress at the pile-soil interface along the circumferential direction yields the horizontal resistance f exerted by the surrounding soil on the pile shaft. (15) Based on the definition of the horizontal dynamic complex impedance of soil, which is the ratio between the horizontal resistance of the soil around the pile and the relative horizontal displacement between the pile and the soil, the horizontal dynamic complex impedance of soil K can be obtained: (16) The above process provides a specific calculation path for determining the horizontal dynamic complex impedance of the soil from parameters such as soil dynamics, target frequency, and pile diameter. This allows the soil reaction force in the subsequent governing equations to no longer use a fixed subgrade coefficient, but rather to be expressed in the form of a complex impedance that varies with frequency. The relevant physical models, boundary conditions, and complex impedance derivation form the basis for the subsequent frequency domain physical constraints in this application.

[0053] In some embodiments, a single pile is considered as a one-dimensional linear elastic Euler beam, and the shear deformation and rotational inertia of the pile body are neglected. The dynamic equilibrium equation of the pile body micro-element can be expressed as: (17) The first term on the left corresponds to the restoring force provided by the bending stiffness of the pile, and the second term corresponds to the inertial force of the pile. This represents the horizontal resistance per unit length of the soil surrounding the pile acting on the pile shaft.

[0054] Based on the aforementioned horizontal dynamic complex impedance of the soil, the horizontal resistance of the soil around the pile can be expressed as: (18) in, It is a complex number, with its real part representing the dynamic stiffness component of the soil and its imaginary part representing the radiation damping and material damping components of the soil; the negative sign indicates that the direction of the horizontal resistance of the soil is opposite to the direction of the pile-soil relative motion.

[0055] For a vertically incident harmonic shear wave, a time factor is introduced. The pile displacement and free field displacement can be expressed as: (19) Taking the second derivative with respect to time, we get: (20) After eliminating the common time factor, the time-domain control equation is transformed into a frequency-domain complex-valued ordinary differential equation: (twenty one) in, The elastic modulus of the pile body. The moment of inertia of the pile section, For pile density, This represents the cross-sectional area of ​​the pile.

[0056] In this embodiment, the engineering boundary condition of "free pile top and fixed pile bottom" can be adopted.

[0057] When there is no external load acting on the pile top, the bending moment and shear force at the pile top section are both zero, and the corresponding boundary conditions are: (twenty two) For a pile bottom embedded in rigid bedrock, in an implementation method using bedrock displacement as the amplitude reference, it can be expressed as: (twenty three) Pile-soil movement interaction factor Defined as the ratio of the horizontal displacement amplitude at the pile top to the horizontal displacement amplitude at the free surface of the site: (twenty four) This parameter is used to quantify the degree to which the pile body corrects for free-field seismic motion. When When the value is greater than 1, it corresponds to the amplification of the pile top response relative to the free field within a certain frequency range; when When the value is less than 1, it can reflect the filtering effect of the pile body on the seismic motion of the corresponding frequency.

[0058] In some embodiments, to reduce the impact of the order-of-magnitude differences in pile length, stiffness, density, frequency, and soil dynamic complex impedance on the network gradient descent process, the governing equations and boundary conditions can be made dimensionless.

[0059] The pile length H can be used as a length characteristic scale, and the bedrock displacement amplitude can be used as the characteristic scale. As a displacement characteristic scale, dimensionless pile coordinates are defined as follows:

[0060] At the same time, dimensionless frequencies can be defined for engineering applications:

[0061] in:

[0062] In some embodiments, a scattering displacement W is introduced to address the additional response of the pile relative to the free field: (25) Substituting the scattering displacement relationship into the governing equation, we can obtain: (26) The free-field equivalent load on the right side of equation (26) includes the bending equivalent load caused by the change in free-field displacement curvature and the inertial equivalent load caused by the inertial effect of free-field motion. For the mid-to-high frequency band of interest in this embodiment, the order of the bending equivalent load with increasing frequency is higher than that of the inertial equivalent load. Therefore, in one specific implementation, the bending equivalent load term can be retained as a physical reinforcement constraint, and the total displacement U can be used as the network output to obtain: (27) Using the total displacement as the network output allows for more direct application of the engineering boundary conditions at the pile top and bottom, while the scattered displacement is mainly used for the transformation of the governing equations.

[0063] Since commonly used real-domain neural networks cannot be directly calculated in complex number form, the pile displacement, free-field displacement, and soil complex impedance can be decomposed into real and imaginary parts respectively:

[0064]

[0065]

[0066] Substituting the above complex-valued variables into the frequency domain control equation: (28) The soil resistance coupling term can be expanded as follows: (29) As can be seen from equation (29), the product of the soil complex impedance and the displacement produces a real-imaginary cross-coupling term. Therefore, the real part equation contains the imaginary part of the displacement, and the imaginary part equation contains the real part of the displacement. Both real-valued equations need to be satisfied simultaneously.

[0067] After simplification, the real part governing equations are: (30) The imaginary part governing equation is: (31) based on The coordinate transformation relationship for derivatives of different orders is as follows: (32) Substituting equation (32) into the real part of the governing equation: (33) The imaginary part governing equation is: (34) bedrock displacement amplitude As a displacement scale, it can be defined as follows: (35) Further definition:

[0068]

[0069]

[0070] Using max normalization, we can define:

[0071] in:

[0072] Substituting the dimensionless coefficients, we obtain the dimensionless governing equations for the real part: (36) The dimensionless governing equation for the imaginary part is: (37) The aforementioned dimensionless transformation and the coupling relationship between real and imaginary parts correspond to the core processing procedure in this application for transforming the frequency domain complex-valued control equations into physical constraints of the real-domain neural network.

[0073] In some embodiments, when a vertically incident plane shear wave propagates in a homogeneous viscoelastic soil layer, the free field displacement... Satisfies the one-dimensional frequency domain Helmholtz equation: (38) in:

[0074]

[0075] Represents the complex shear wavenumber. This represents the complex shear wave velocity, including soil hysteretic damping.

[0076] Take the second derivative of both sides of equation (38) with respect to z: (39) Simplifying, we get: (40) According to equation (38), the second derivative term can be substituted into equation (40) to obtain: (41) Further results were obtained: (42) In dimensionless coordinates: (43) Based on the above relationship, when constructing the residual of the governing equation, it is no longer necessary to calculate the fourth derivative of the free field displacement using numerical methods, but the corresponding higher-order derivative terms can be directly determined using equation (43).

[0077] like Figure 2 As shown, in some embodiments, Siren-PINN can employ a single-frequency-point successive training architecture. For each target dimensionless frequency... First, the corresponding horizontal dynamic complex impedance of the soil and the coefficients of the governing equation are calculated based on the frequency. Then, the corresponding Siren-PINN is trained for the frequency.

[0078] In this training architecture, the target frequency is used as a pre-training parameter to determine the complex impedance and governing equation coefficients, but it is not used as input neurons to the network. The network input is only the dimensionless pile body coordinates. The network outputs are the real and imaginary parts of the dimensionless horizontal displacement of the pile, respectively.

[0079] In one specific embodiment, a fully connected feedforward neural network can be constructed. The input layer includes one neuron for receiving dimensionless pile coordinates; the hidden layer can be set to 6 layers, each including 192 neurons; the output layer has 2 neurons, which output the real and imaginary parts of the pile's horizontal displacement, respectively.

[0080] All hidden layers can use the sine function Sin as the activation function. The sine activation function has periodicity and continuous high-order differentiability, which is suitable for describing the spatial oscillation behavior in seismic response, and is also suitable for calculating the first to fourth derivatives using automatic differentiation.

[0081] In a specific implementation of weight initialization, the first-layer weights follow the following rules:

[0082] Subsequent weights in each layer follow the following order:

[0083] During network training, global float64 double-precision computation can be used to reduce the impact of numerical precision loss on residual calculation during high-order automatic differentiation.

[0084] For any input coordinate The network outputs respectively and And the network output is calculated using the autograd automatic differentiation module in the PyTorch framework. The first, second, third, and fourth derivatives are obtained. The obtained derivatives are further substituted into equations (36), (37), and the boundary conditions to construct the physical constraint loss.

[0085] In some embodiments, the mean square error can be used to characterize the degree to which the network output satisfies the dynamic control equations.

[0086] Substituting the network output and the corresponding fourth derivative into equations (36) and (37), respectively, we obtain the real residuals. and imaginary residuals : (44) (45) For N placement points, the physical residual loss of the governing equations can be expressed as: (46) Therefore, the adjustment of network parameters does not depend on a large amount of measured label data, but is directly constrained by the frequency domain pile-soil dynamics control equation.

[0087] For the free boundary at the pile top, equation (22), after being transformed to dimensionless coordinates, can be expressed as: (47) Constraints are applied to the real and imaginary parts respectively, and the second derivative constraint is: (48) The third derivative constraint is: (49) The pile top boundary loss can be constructed as follows: (50) For the fixed boundary at the pile bottom, we can obtain: (51) Since the bedrock displacement in this embodiment takes a real constant as the phase reference, the imaginary displacement is 0, and the displacement constraint can be written as: (52) The corner constraint is: (53) The pile bottom boundary loss can be constructed as follows: (54) The total loss due to boundary conditions is: (55) The residuals of the relevant control equations, the residuals of the higher-order boundaries at the pile top, and the residuals of the boundary at the pile bottom together form the physical constraints required for network training.

[0088] In some embodiments, the physical residual loss of the governing equations and the boundary condition loss can be weighted to obtain a comprehensive loss function: (56) in, This represents the weight corresponding to the physical residual loss of the governing equation. This represents the weight corresponding to the boundary condition loss.

[0089] Since the residuals of the fourth-order complex-valued governing equations and the residuals of higher-order boundary conditions may have different orders of magnitude in the early stages of training, a three-stage dynamic weighting mechanism is adopted in one implementation.

[0090] In the initial training phase, specifically within 0-20% of the training rounds, the following settings can be configured:

[0091] Prioritize satisfying boundary conditions in the network.

[0092] During the middle of training, specifically within 20% to 50% of the training rounds, the following settings can be configured:

[0093] Gradually increase the weight of the physical residuals in the governing equations.

[0094] In the later stages of training, i.e., within more than 50% of the training rounds, the following settings can be configured:

[0095] This shifts the training focus further towards compressing the physical residuals of the governing equations, while maintaining boundary constraints.

[0096] Through the above methods, the focus of optimizing the comprehensive loss function gradually shifts from the boundary feasibility in the early stage of training to the physical consistency of the pile-soil dynamic control equation.

[0097] In one specific embodiment, the dimensionless solution domain of the pile body can be:

[0098] Generate 2000 Chebyshev configuration points internally.

[0099] Furthermore, it can be done near the top of the pile. and near the bottom of the pile 200 placement points are added to each boundary region. Since the fourth-order governing equations involve second- and third-order derivative constraints corresponding to bending moment and shear force at the boundaries, increasing the density of placement points near the boundaries can enhance the network's learning of higher-order boundary conditions.

[0100] The configuration points are used to calculate the control equation residuals corresponding to equations (44) and (45), and participate in the calculation of physical residual loss in equation (46); the configuration points at the top and bottom of the piles are further used to calculate the boundary losses corresponding to equations (50) and (54).

[0101] It should be noted that the 2000 configuration points and the addition of 200 points each at the boundary represent one specific implementation method. In other implementation methods, the number of configuration points can be adjusted based on pile length, frequency range, computing resources, and error requirements.

[0102] In some embodiments, the AdamW optimizer can be used to update network parameters and combined with a pure cosine annealing learning rate scheduling strategy for training.

[0103] Specifically, the initial learning rate can be set as follows:

[0104] The weight decay coefficient can be set as follows:

[0105] As training rounds progress, the learning rate gradually decreases according to a cosine variation until:

[0106] In AdamW, decoupled weight decay is used to constrain model parameters, while cosine annealing is used to smoothly adjust the learning rate during model iteration.

[0107] Because the complexity of the pile-soil dynamic response varies at different target frequencies, it can be determined based on the dimensionless frequency. Adaptively set the number of training rounds.

[0108] In one specific implementation, when:

[0109] At that time, the training rounds were set to 3500.

[0110] when:

[0111] At that time, the training rounds were set to 8500.

[0112] Among them, the pile-soil interaction response curve in the low-frequency band is relatively smooth, so the first training round is used; under higher frequency conditions, the response oscillation is enhanced, so more training rounds are used.

[0113] In scenarios involving continuous multi-frequency solutions, Siren-PINN can be trained sequentially according to the frequency order in the target frequency set.

[0114] For example, after completing the training of the Siren-PINN corresponding to the j-th target frequency, the weights of the first four hidden layers closest to the input side in the trained network are transferred to the Siren-PINN corresponding to the (j+1)-th target frequency as the initial weights of the next model, and the weights of the last two hidden layers and the output layer are reinitialized.

[0115] The pile-soil dynamic response corresponding to adjacent frequency points has a certain continuity. Therefore, the partial response characterization formed in the front hidden layer can be used for the next frequency point. Reinitializing the network layer near the output side preserves the space for adaptive optimization of the next target frequency based on its own control equation coefficients.

[0116] Under the condition of solving for continuous frequency points, the above-mentioned hierarchical transfer learning method can be combined with the single-frequency successive training structure without having to use all target frequencies as neural network inputs at the same time.

[0117] In some embodiments, a dual-index early stopping mechanism can be used to monitor the model convergence status.

[0118] When the overall loss decreases less than 50 consecutive training rounds during a period of time:

[0119] Alternatively, if the physical residual loss of the control equation does not decrease significantly for 100 consecutive rounds, the training of the model corresponding to the current target frequency can be terminated early.

[0120] Furthermore, a gradient clipping threshold can be set to 0.3. When a gradient exceeds 0.3 during backpropagation, the gradient exceeding the threshold is truncated, thereby limiting the occurrence of large gradients during fourth-order automatic differentiation.

[0121] The Chebyshev placement points, three-stage dynamic loss weights, AdamW and cosine annealing, frequency-adaptive training rounds, hierarchical partial transfer learning, dual-index early stopping, and gradient clipping together constitute a specific collaborative training method.

[0122] After training, the Siren-PINN, which has converged at the corresponding target frequency, is used for forward prediction of the horizontal seismic response of the pile.

[0123] The pile-soil motion interaction factor is defined as: (57) The real part of the pile top reset displacement output by the neural network and the virtual part The horizontal displacement amplitude at the pile top is: (58) For a simple harmonic shear wave in a homogeneous viscoelastic soil layer, the pile bottom is the bedrock surface. The displacement amplitude at that point is Surface The complex amplitude of the free field at that point is: (59) in:

[0124]

[0125] The amplitude of horizontal displacement of the free surface is: (60) Under dimensionless conditions, when:

[0126] At that time, the pile-soil motion interaction factor can be expressed as: (61) The above calculation relationship allows the pile displacement results output by the neural network to be further converted into engineering parameters that characterize the degree of correction of the effective seismic input of the pile foundation.

[0127] After training the network at a specific target frequency, a predicted coordinate point matrix can be generated within the dimensionless solution domain of the pile body. The density of the predicted coordinate points can be set independently of the training configuration point density.

[0128] The predicted coordinates are input into the trained network corresponding to the target frequency to obtain the dimensionless real and imaginary parts of the pile displacement at each predicted location within the solution domain. Then, based on the bedrock displacement amplitude... Perform reverse scaling to convert dimensionless displacement into horizontal displacement of the pile body in actual physical dimensions.

[0129] The horizontal displacement amplitude of the pile top is obtained according to the real and imaginary parts corresponding to the pile top position according to formula (58), and the horizontal displacement amplitude of the free field surface is determined by combining formula (59) and formula (60). Then, the pile-soil motion interaction factor corresponding to the current target frequency is calculated according to formula (57) or formula (61).

[0130] In one specific embodiment, multiple frequency points can be continuously set within the dimensionless frequency range of 0.01 to 1.0, and the corresponding frequencies can be calculated sequentially according to the single-frequency successive training method. This allows us to plot the complete frequency domain curve of the pile-soil motion interaction factor as a function of dimensionless frequency.

[0131] This output process constitutes a complete data chain of "target frequency - complex impedance and control equation - Siren-PINN training - pile body repositioning - pile top displacement amplitude - free site surface displacement amplitude - pile-soil motion interaction factor - frequency domain curve".

[0132] In a specific simulation embodiment, the following parameters can be used: The pile length H is 20m, the pile diameter D is 1m, and the elastic modulus of the pile body is... for MPa, pile density The soil density is 2500 kg / m³; the soil shear modulus G is 30 MPa; and the soil density is... The soil hysteretic damping ratio is 1800 kg / m³. The Poisson's ratio of the soil is 0.05. It is 0.4.

[0133] Based on the above parameters, the complex impedance calculation, control equation coefficient determination, dimensionless transformation, Siren-PINN training, and response output are performed sequentially at the target frequency point, and the results are compared with the corresponding Novak plane strain analytical results.

[0134] Under this specific working condition, within the dimensionless frequency range of 0.1 to 0.6 corresponding to the main seismic frequency, the average relative error between the calculated and analytical results is 8.7%, with a maximum relative error not exceeding 10%. As the frequency further increases, the error tends to increase.

[0135] From the frequency domain variation characteristics, in the lower frequency range, the horizontal movement of the pile is close to the free field movement; as the frequency increases, the pile-soil motion interaction factor can show a peak amplification region; subsequently, as the dimensionless frequency continues to increase, the pile top displacement relative to the free field surface displacement gradually decreases, showing the corresponding frequency filtering characteristics.

[0136] In some embodiments, to analyze the effect of each training mechanism, the engineering physical parameters, the total number of configuration points, and the stopping conditions can be kept basically consistent, and one training condition can be changed for comparison.

[0137] The first comparison method is to replace the sinusoidal activation function in the Siren network with the ordinary Tanh activation function.

[0138] The second comparison method is to replace the Chebyshev locus with the uniformly sampled locus.

[0139] The third comparison method is to replace the frequency-adaptive training rounds with fixed training rounds.

[0140] The fourth comparison method is to cancel cross-frequency parameter transfer, so that each target frequency starts training from random initialization.

[0141] The fifth comparison method is to replace the three-stage dynamic loss weights with fixed loss weights.

[0142] In the comparative tests used, the average training time per frequency under the normal training strategy was approximately 374.4 s; under uniform sampling conditions, it was approximately 391.9 s; under no transfer learning conditions, it was approximately 568.8 s; under fixed weight conditions, it was approximately 392.9 s; and under the Tanh activation function, it was approximately 489.6 s.

[0143] In the corresponding error comparison, the average error for the normal training strategy is approximately 13.78%, the maximum error is approximately 35.24%, and the error for the main frequency band is approximately 5.25%; under Tanh activation conditions, the errors are approximately 19.58%, 37.67%, and 20.25%, respectively; under uniform sampling conditions, the errors are approximately 27.30%, 55.66%, and 28.36%, respectively; under no transfer learning conditions, the errors are approximately 22.09%, 36.58%, and 24.25%, respectively; and under fixed weight conditions, the errors are approximately 24.87%, 48.09%, and 28.35%, respectively.

[0144] For a separate comparison of whether or not transfer learning was used, the average training time per frequency was approximately 374.4 s with hierarchical partial transfer learning and approximately 568.8 s without transfer learning, representing a reduction of approximately 35% in training time. These results correspond to specific experimental results under the same test conditions and are used to illustrate the synergistic effect of various training strategies in this embodiment.

[0145] In some embodiments, the pile length, pile diameter, pile material parameters, and other soil parameters can be fixed, and only the soil hysteretic damping ratio can be changed. The complex shear wave velocity, horizontal dynamic complex impedance of the soil, and coefficients of the governing equations at the corresponding target frequency are recalculated, and the corresponding Siren-PINN is trained.

[0146] For example, they can be used respectively:

[0147] Obtain the pile-soil motion interaction factor-frequency curves under different damping conditions.

[0148] By changing By re-executing the complete solution process, the influence of soil material damping variations on the pile top seismic response and the pile-soil motion interaction factor can be analyzed. This process does not require changes to the overall network and physical constraint architecture; it only requires updating the physical coefficients corresponding to the current soil parameters and target frequency.

[0149] In actual engineering projects, the geometric parameters of the pile, the material parameters of the pile body, and the dynamic parameters of the soil can be obtained by relying on the geotechnical investigation report and the pile test data.

[0150] Subsequently, a target frequency set is established based on the range of earthquake frequencies to be analyzed, and the corresponding horizontal dynamic complex impedance of the soil and the dimensionless governing equation coefficients are calculated based on each target frequency.

[0151] For each target frequency, Siren-PINN is generated using the aforementioned single-frequency successive training method, and the network is trained using the physical residuals of the governing equations and higher-order boundary residuals. During the solution process for consecutive frequencies, partial transfer learning between adjacent frequencies can be employed.

[0152] After completing the solution, the frequency domain curve of the pile-soil motion interaction factor as a function of frequency can be obtained.

[0153] Furthermore, the free-field surface seismic motion can be multiplied by the pile-soil motion interaction factor at the corresponding frequency to obtain the actual effective seismic input at the pile top, which can then be used for seismic response analysis of the superstructure.

[0154] For different analysis conditions such as frequent earthquakes, design earthquakes, and rare earthquakes, the corresponding frequency domain response results can be obtained according to the corresponding earthquake input frequency components and pile-soil parameters, thereby conducting pile foundation dynamic response analysis and seismic performance evaluation.

[0155] For different pile diameters, pile lengths, or soil dynamic parameters, the corresponding engineering input parameters can be modified and the solution process can be re-executed, thereby enabling response comparison between multiple schemes.

[0156] Therefore, at the engineering application level, a closed-loop processing mechanism can be formed, which includes "acquiring on-site parameters, transforming physical parameters into dimensionless quantities, solving the target frequency point by point, outputting the horizontal response of the pile body, determining the pile-soil motion interaction factor, correcting the effective seismic input, and analyzing the seismic performance."

[0157] It should be noted that the Siren network with 6 hidden layers and 192 neurons per layer described above is one specific implementation. In other embodiments, while maintaining the frequency domain complex-valued physical constraints, the number of hidden layers can be set to 4 to 10 layers, and each layer can include 128 to 256 neurons.

[0158] In other embodiments, in addition to the sinusoidal activation function, an adaptive periodic activation function or a multi-scale wavelet activation function can also be used to process the dynamic response of the pile body with different spatial frequency characteristics.

[0159] In other embodiments, the configuration point sampling method may also employ Latin hypercube sampling or adaptive encrypted sampling based on the current physical residual.

[0160] In other embodiments, the optimizer may also employ the L-BFGS quasi-Newton method or the Adamax optimizer.

[0161] The above alternative methods mainly involve the network internal structure, configuration point settings and optimization methods. When adopting the above alternative methods, physical constraints can still be constructed according to the aforementioned pile-soil frequency domain dynamic control relationship.

[0162] The above embodiment adopts a boundary form of "free pile top and fixed pile bottom". In actual engineering, if there is an elastic connection at the pile top, or if the pile bottom is not completely fixed but has elastic support, the boundary residual can be redefined according to the corresponding engineering boundary relationship.

[0163] For example, for elastic constraints at the pile top, the corresponding elastic support relationship can be converted into boundary equations between displacement, rotation, bending moment or shear force at the pile top, and the residual of the boundary equation can be used as the new pile top boundary loss.

[0164] For elastic supports at the pile bottom, the corresponding boundary residuals can be constructed based on the relationship between the actual pile bottom displacement and the pile bottom reaction force.

[0165] When the boundary conditions described above change, the frequency domain complex-valued control equations, the coupling relationship between real and imaginary parts, the Siren-PINN input-output form, and the construction method of the physical residuals of the control equations can remain unchanged; the main adjustment is to the boundary condition loss terms. This extension method can cover non-standard engineering boundary conditions.

[0166] In the aforementioned main implementation, each target frequency corresponds to an independently trained Siren-PINN. The target frequency is used as a training pre-parameter to determine the horizontal dynamic complex impedance of the soil and the coefficients of the governing equation, rather than being directly used as the network input.

[0167] In other embodiments, if it is necessary to achieve a single training batch solution across multiple pile-soil parameter combinations, one or more dimensionless pile-soil parameters can be set as additional network inputs in addition to the dimensionless pile body coordinates, thereby forming a parameterized PINN.

[0168] For example, at least one of the dimensionless frequency, pile-soil stiffness ratio, or other dimensionless pile-soil physical parameters can be used as input along with dimensionless coordinates. The network still outputs the real and imaginary parts of the pile's horizontal displacement and is trained using the residuals of the governing equations and boundary residuals under the corresponding parameter conditions.

[0169] This parameterized implementation is used to extend the ability to solve problems across different operating conditions without changing the overall technical concept of using physical control equations as constraints for network training.

[0170] In summary, in a complete implementation, the pile parameters, soil dynamic parameters, and target frequency set of the target single pile are first obtained. For each target frequency, the free field displacement and the horizontal dynamic complex impedance of the soil are determined based on the continuous medium theory and the Novak plane strain assumption. The horizontal dynamic complex impedance of the soil is substituted into the dynamic equilibrium relationship of the pile body to obtain the fourth-order complex-valued governing equation in the frequency domain and the boundary conditions of the pile top and bottom. Subsequently, dimensionless transformation is performed using the pile length and bedrock displacement amplitude as characteristic scales, and the horizontal displacement of the pile body, the free field displacement, and the horizontal dynamic complex impedance of the soil body are decomposed into real and imaginary parts. The complex product of the complex impedance and the pile-soil relative displacement forms a cross-coupling relationship between the real and imaginary parts. Then, a Siren-PINN is established for each target frequency, using the dimensionless pile body coordinates as input and the horizontal displacement of the pile body as the input. The real and imaginary parts are used as outputs, and the first to fourth derivatives with respect to the coordinates are output through an automatic differential calculation network. The obtained derivatives are substituted into the real and imaginary part control equations and boundary conditions to form the physical residuals of the control equations, the pile top boundary residuals, and the pile bottom boundary residuals, and a comprehensive loss function is constructed from them. During training, Chebyshev placement points, three-stage dynamic loss weights, optimizer and learning rate scheduling, frequency adaptive training rounds, hierarchical transfer learning of adjacent frequencies, dual-index early stopping, and gradient clipping are combined. After the model converges, forward prediction is performed in the pile solution domain to obtain the real and imaginary parts of the pile horizontal displacement. The amplitude of the pile top horizontal displacement and the amplitude of the free-field surface displacement are further calculated, and the pile-soil motion interaction factor corresponding to different target frequencies is determined. Finally, the frequency domain curve of the single pile horizontal seismic response is generated.

Claims

1. A method for analyzing the horizontal seismic response of a single pile, characterized in that, Performed by a computing device, the method includes: Obtain the pile body parameters, soil dynamic parameters, and target frequency set corresponding to the vertical incident shear wave of the target single pile; For each target frequency in the target frequency set, a plane strain pile-soil interaction model based on continuous medium theory is used to determine the free field displacement and the horizontal dynamic complex impedance of the soil according to the soil dynamic parameters and the target frequency. Based on the pile parameters, the free field displacement and the horizontal dynamic complex impedance of the soil, a fourth-order frequency domain complex value governing equation characterizing the horizontal displacement of the pile body, as well as the corresponding pile top boundary conditions and pile bottom boundary conditions are established. The frequency domain complex-valued fourth-order governing equations and their boundary conditions are dimensionless. The dimensionless pile horizontal displacement, free field displacement, and soil horizontal dynamic complex impedance are then split into real and imaginary parts. Based on the coupling relationship between the split real and imaginary parts, the frequency domain complex-valued fourth-order governing equations are transformed into a set of coupled real-valued governing equations consisting of real and imaginary governing equations, and the corresponding real-valued boundary conditions are obtained. A Siren-PINN model is constructed with dimensionless pile coordinates as input and the real and imaginary parts of the dimensionless horizontal displacement of the pile as output. The derivatives of the output with respect to the dimensionless pile coordinates are obtained based on automatic differentiation. The output and the derivatives are substituted into the coupled real-valued control equations and the real-valued boundary conditions to obtain the control equation residuals and boundary condition residuals, respectively. A comprehensive loss function is constructed based on the control equation residuals and boundary condition residuals. The Siren-PINN model corresponding to each target frequency is trained with the comprehensive loss function as the optimization objective to obtain the single pile horizontal displacement response model corresponding to each target frequency. The predicted coordinates of the pile body are input into the single pile horizontal displacement response model corresponding to the target frequency to obtain the real and imaginary parts of the dimensionless horizontal displacement of the pile body. The amplitude of the horizontal displacement at the pile top is determined based on the real and imaginary parts corresponding to the pile top position. The pile-soil motion interaction factor is determined by combining the amplitude of the horizontal displacement of the free ground surface at the corresponding target frequency. The frequency domain analysis results of the single pile horizontal seismic response are generated based on the pile-soil motion interaction factor corresponding to each target frequency.

2. The method according to claim 1, characterized in that, The process of determining the free-field displacement and the horizontal dynamic complex impedance of the soil based on the soil dynamic parameters and the target frequency, and establishing a fourth-order frequency-domain complex-valued governing equation characterizing the horizontal displacement of the pile body based on the pile parameters, the free-field displacement, and the horizontal dynamic complex impedance of the soil, includes: The horizontal displacement of the soil under seismic loading is decomposed into free field displacement and relative displacement of the soil with respect to the free field. Based on the attenuation condition of the scattering displacement at infinite radial distance of the soil and the displacement continuity condition of the pile-soil interface, the displacement distribution of the soil around the pile is determined. The stress components at the pile-soil interface are determined based on the displacement distribution and the stress-strain relationship of the soil. The stress components are integrated along the pile perimeter to obtain the horizontal resistance of the soil around the pile acting on the pile body. The horizontal dynamic complex impedance of the soil is determined based on the relationship between the horizontal resistance and the relative horizontal displacement between the pile and the soil. Substituting the horizontal resistance, characterized by the horizontal dynamic complex impedance of the soil, into the dynamic equilibrium relationship of the pile body, and performing frequency domain transformation, yields the frequency domain complex-valued fourth-order governing equation.

3. The method according to claim 2, characterized in that, The pile top boundary conditions and pile bottom boundary conditions include: The pile top is set as a free boundary, so that the bending moment and shear force at the pile top section are zero, and the free boundary is defined by the second and third derivatives of the horizontal displacement of the pile body with respect to the pile body coordinates, respectively. The pile bottom is set as a fixed boundary, so that the horizontal displacement of the pile bottom is consistent with the free field displacement at the corresponding position, and the pile bottom rotation angle is zero. The pile bottom rotation angle is limited by the first derivative of the horizontal displacement of the pile body with respect to the pile body coordinates.

4. The method according to claim 1, characterized in that, The dimensionless processing of the frequency domain complex-valued fourth-order governing equations and their boundary conditions includes: The pile length is used as the length characteristic scale to make the pile body coordinates dimensionless, and the bedrock displacement amplitude is used as the displacement characteristic scale to make the pile body horizontal displacement and free field displacement dimensionless. The corresponding dimensionless frequency is determined according to the target frequency, pile diameter and soil shear wave velocity. Based on the length characteristic scale and displacement characteristic scale, the coefficients of each term in the frequency domain complex-valued fourth-order governing equation, as well as the pile top boundary condition and pile bottom boundary condition, are transformed accordingly to obtain the dimensionless frequency domain complex-valued fourth-order governing equation and its boundary conditions.

5. The method according to claim 4, characterized in that, The dimensionless pile horizontal displacement, free field displacement, and soil horizontal dynamic complex impedance are decomposed into real and imaginary parts, respectively. Based on the coupling relationship between the decomposed real and imaginary parts, the coupled real-valued governing equations are formed, including: The real and imaginary parts of the horizontal displacement of the pile body, the real and imaginary parts of the free field displacement, and the real and imaginary parts of the horizontal dynamic complex impedance of the soil are determined respectively. The horizontal resistance term, which is composed of the horizontal dynamic complex impedance of the soil and the relative horizontal displacement between the pile and the soil, is expanded into real and imaginary parts, so that the real and imaginary parts of the horizontal dynamic complex impedance of the soil form cross-coupled terms with the real and imaginary parts of the relative horizontal displacement between the pile and the soil, respectively. Based on the cross-coupling terms, the real part control equation and the imaginary part control equation are constructed respectively, so that the real part control equation and the imaginary part control equation jointly constrain the real part and the imaginary part of the horizontal displacement of the pile.

6. The method according to claim 5, characterized in that, The construction of the Siren-PINN, which takes dimensionless pile coordinates as input and the real and imaginary parts of the dimensionless horizontal displacement of the pile as output, includes: For each target frequency, the corresponding horizontal dynamic complex impedance of the soil and the equation coefficients of the coupled real-valued control equations are determined based on the target frequency, and the corresponding Siren-PINN is trained for the target frequency, wherein the target frequency is used as a pre-parameter for model training, and the dimensionless pile coordinates are used as the network input for Siren-PINN. A sinusoidal activation function is used in the hidden layer of the Siren-PINN, and the real and imaginary parts of the dimensionless horizontal displacement of the pile body are output through the output layer.

7. The method according to claim 6, characterized in that, The construction of the comprehensive loss function based on the residuals of the governing equations and the residuals of the boundary conditions includes: Automatic differentiation is used to determine the first to fourth derivatives of the real and imaginary parts of the dimensionless horizontal displacement of the pile body relative to the dimensionless pile body coordinates. The real part, imaginary part and their corresponding derivatives are substituted into the real part control equation and the imaginary part control equation to obtain the real part control equation residuals and the imaginary part control equation residuals at the corresponding configuration points. The higher-order derivative relationship of the free field displacement is determined based on the frequency domain wave equation satisfied by the free field displacement, and the higher-order derivative relationship is used to determine the higher-order derivative term of the free field displacement in the residual of the control equation. The boundary conditions at the top and bottom of the pile are decomposed into real and imaginary parts, respectively. The boundary condition residuals are determined based on the decomposed boundary conditions and the corresponding derivatives of the dimensionless horizontal displacement of the pile. The physical residual loss of the control equation is determined based on the real part residual and the imaginary part residual of the control equation at each configuration point, and the boundary condition loss is determined based on the boundary condition residual. The comprehensive loss function is composed of the physical residual loss of the control equation and the boundary condition loss.

8. The method according to claim 7, characterized in that, Training the Siren-PINN for each target frequency includes: According to the Chebyshev node distribution, configuration points are set in the dimensionless solution domain of the pile body to determine the physical residual loss of the governing equation, and the configuration point density in the boundary areas of the corresponding pile top and pile bottom is increased. During model training, the weights of the boundary condition loss and the physical residual loss of the control equation in the comprehensive loss function are adjusted in stages. Specifically, as the model training progresses, the weight of the boundary condition loss is gradually reduced and the weight of the physical residual loss of the control equation is increased.

9. The method according to claim 8, characterized in that, Training the Siren-PINN corresponding to each target frequency in the target frequency set also includes: The dimensionless frequency corresponding to each target frequency is compared with a preset frequency threshold. A first training round is configured for target frequencies that are lower than the preset frequency threshold, and a second training round is configured for target frequencies that reach or exceed the preset frequency threshold, wherein the second training round is greater than the first training round. Train each Siren-PINN sequentially according to the frequency order in the target frequency set. After completing the training of the Siren-PINN corresponding to the current target frequency, use the hidden layer parameters of the trained Siren-PINN that are closer to the input side as the initial parameters of the Siren-PINN corresponding to the next target frequency, and re-initialize the remaining hidden layer parameters and output layer parameters. The frequency domain analysis results of generating the horizontal seismic response of a single pile based on the pile-soil motion interaction factors corresponding to each target frequency include: Predicted coordinate points are set within the dimensionless solution domain of the pile body. The predicted coordinate points are then input into the single pile horizontal displacement response model corresponding to the target frequency to obtain the real and imaginary parts of the dimensionless horizontal displacement of the pile body at each predicted coordinate point. The dimensionless horizontal displacement of the pile body is converted into the corresponding horizontal displacement of the pile body according to the displacement characteristic scale, and the amplitude of the horizontal displacement at the pile top is determined according to the real and imaginary parts of the horizontal displacement corresponding to the pile top position. The amplitude of the horizontal displacement at the surface of the free field is determined based on the propagation relationship of the free field shear wave at the corresponding target frequency. The ratio of the amplitude of the horizontal displacement at the pile top to the amplitude of the horizontal displacement at the surface of the free field is determined as the pile-soil motion interaction factor corresponding to the target frequency. The pile-soil motion interaction factors are combined according to the target frequency set to obtain the frequency domain analysis results of the pile-soil motion interaction factors as a function of frequency.

10. A single-pile horizontal seismic response analysis system, characterized in that, include: The parameter acquisition module is used to acquire the pile body parameters, soil dynamic parameters, and target frequency set corresponding to the vertical incident shear wave of the target single pile. The physical model construction module is used to determine the free field displacement and the horizontal dynamic complex impedance of the soil based on the plane strain pile-soil interaction model of the continuous medium theory for each target frequency in the target frequency set. It also establishes the frequency domain complex value fourth-order governing equations characterizing the horizontal displacement of the pile body, as well as the corresponding pile top boundary conditions and pile bottom boundary conditions, based on the pile body parameters, the free field displacement and the horizontal dynamic complex impedance of the soil. The control equation transformation module is used to perform dimensionless processing on the frequency domain complex-valued fourth-order control equation and its boundary conditions, and to decompose the dimensionless pile horizontal displacement, free field displacement and soil horizontal dynamic complex impedance into real and imaginary parts respectively. Based on the coupling relationship between the decomposed real and imaginary parts, the frequency domain complex-valued fourth-order control equation is transformed into a coupled real-valued control equation set composed of real and imaginary control equations, and the corresponding real-valued boundary conditions are obtained. The response model training module is used to construct a Siren-PINN with dimensionless pile coordinates as input and the real and imaginary parts of the dimensionless horizontal displacement of the pile as output. Based on automatic differentiation, the derivatives of the output with respect to the dimensionless pile coordinates are obtained. The output and the derivatives are substituted into the coupled real-valued control equations and the real-valued boundary conditions to obtain the control equation residuals and boundary condition residuals, respectively. A comprehensive loss function is constructed based on the control equation residuals and boundary condition residuals. The Siren-PINN corresponding to each target frequency is trained with the comprehensive loss function as the optimization objective to obtain the single pile horizontal displacement response model corresponding to each target frequency. The response result generation module is used to input the predicted coordinates of the pile body into the single pile horizontal displacement response model corresponding to the target frequency, obtain the real and imaginary parts of the dimensionless horizontal displacement of the pile body, determine the amplitude of the horizontal displacement at the pile top based on the real and imaginary parts corresponding to the pile top position, and determine the pile-soil motion interaction factor by combining the amplitude of the free-field surface horizontal displacement at the corresponding target frequency. Based on the pile-soil motion interaction factor corresponding to each target frequency, the frequency domain analysis results of the single pile horizontal seismic response are generated.

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