Submarine pipeline vortex-induced vibration analysis method based on multiple Fourier characterization PINN

Abstract

The invention discloses a submarine pipeline vortex-induced vibration analysis method based on multiple Fourier characterization PINN, and the method comprises the steps: constructing a physical information neural network, determining a loss function of an initial condition, a boundary condition and a partial differential equation (PDEs) according to a submarine pipeline vortex-induced vibration model, and adding an anchor point to correct the network, so as to improve the fitting capability of the network; a Fourier transform technology is adopted, the key frequency of vibration and the frequency value of the key frequency are accurately identified, a Fourier mapping # imgabs0 # function is introduced, and a multiple Fourier mapping network is constructed to carry out refined mapping decomposition and feature extraction on vibration signals; according to the number and the value of the key vibration frequency, the mapping times and the hyper-parameter are dynamically adjusted to overcome the limitation of the PINN in processing complex vibration signals; integrating the spatial-temporal characteristics through point-by-point multiplication, and outputting a vibration learning result; according to the method, the analysis precision is improved, the capability of processing various high and low frequency superposition vibrations is enhanced, and the problem of spectrum deviation during high-frequency information learning is effectively solved.

Description

Technical Field

[0001] The invention relates to the field of vortex-induced vibration simulation of submarine pipelines, in particular to a vortex-induced vibration analysis method of submarine pipelines based on multi-Fourier characterization PINN. Background Art

[0002] Submarine pipelines play an irreplaceable role in deepwater flow transmission. While bearing the excitation of internal flow, they are also subjected to the dynamic load of ocean currents that changes with time and space, thus inducing vortex-induced vibration. When the vortex shedding frequency is close to the natural frequency of the pipeline, resonance is easily induced, which has always been the main factor in pipeline fatigue failure and one of the most challenging problems in marine engineering for a long time. Therefore, it is of practical significance to analyze the vortex-induced vibration during submarine pipeline flow transmission. However, due to the simultaneous action of internal and external flow on the submarine pipeline, its vibration characteristics show a multi-modal superposition effect on the spatiotemporal scale. The close interweaving and deep superposition of high-frequency and low-frequency components in the vibration spectrum cause the pipeline to show significant spatiotemporal multi-scale and multiple high- and low-frequency superposition vibration characteristics, which greatly increases the difficulty of accurately capturing and analyzing the vibration behavior of the pipeline. Therefore, it is necessary to develop a high-precision algorithm that can accurately strip and analyze the multi-frequency superposition characteristics from complex vibration signals to achieve accurate prediction of submarine vortex-induced vibration.

[0003] PINN encodes physical partial differential equations (PDEs) into the neural network architecture, and under the dual drive of data and physics, it shows good learning performance. However, when facing high-frequency vibration problems, PINN and similar deep neural networks generally face the problem of spectrum deviation, which becomes a bottleneck restricting their accuracy. When analyzing scenes with complex interweaving of multi-frequency components, they often converge too early in the low-frequency region and cannot achieve global high-precision prediction.

[0004] Therefore, in view of the multi-scale and multiple high- and low-frequency superimposed vibration problems of submarine pipeline vortex-induced vibration, how to effectively overcome the spectrum deviation problem when PINN processes high-frequency vibration, solve the problem of inability to achieve global accurate prediction when multiple frequencies are superimposed, and achieve more effective vibration prediction on this basis, is the current problem to be solved. Summary of the invention

[0005] The present invention proposes a submarine pipeline vortex-induced vibration analysis method based on multiple Fourier characterization PINN, which can realize the accurate and effective solution and analysis of the vortex-induced vibration of the submarine flow pipeline. By introducing Fourier characteristic mapping, multiple Fourier transforms are performed to deeply analyze and refine the spatial and temporal multi-scale and multi-frequency superposition information of the vibration signal, synchronously learn the low-frequency and high-frequency components, and specifically learn the vibration dynamic response under different high and low frequency superpositions, effectively improving the prediction accuracy of the model.

[0006] The present invention adopts the following technical solutions.

[0007] The vortex-induced vibration analysis method of submarine pipeline based on multi-Fourier characterization PINN includes the following steps:

[0008] Step S1, according to the submarine pipeline vortex-induced vibration model, determine the corresponding initial conditions, boundary conditions, and loss functions of the partial differential equation, and modify the network by adding a small number of anchor points to the network to improve the network's ability to fit the high-order nonlinear complex dynamic characteristics of the submarine pipeline vortex-induced vibration equation, so as to obtain the loss function of the model corresponding to the data fitting;

[0009] Step S2, using Fourier transform technology to identify the key frequencies of vibration and their frequency values, then introducing a multiple Fourier characteristic mapping γ function to construct a multiple Fourier mapping network, using the multiple Fourier characteristic mapping function to perform refined mapping decomposition and feature extraction on the vibration signal, and realizing multi-scale and multi-frequency mapping decomposition of the vortex-induced vibration signal;

[0010] Step S3, dynamically adjust the mapping times M according to the number of key vibration frequencies, perform mapping decomposition of the network input spatiotemporal data for a corresponding number of times according to the mapping times M, decompose the input data to obtain the spatial Fourier characteristic map γ(ξ) and the temporal Fourier characteristic map γ(τ), so as to ensure the accurate decomposition of multi-frequency vibration spatiotemporal data, and effectively overcome the limitations of PINN in processing complex vibration signals;

[0011] Step S4, decomposing the neural tangent kernel of PINN in the frequency domain to obtain the corresponding relationship between different frequencies and the optimal hyperparameter σ, and selecting the corresponding optimal hyperparameter value σ according to the frequency corresponding to the key frequency of vibration to adjust the value of the Fourier base frequency vector, thereby improving the network's recognition and learning accuracy of specific frequency vibrations, and overcoming the problem that PINN converges too early in the low-frequency region and cannot achieve global high-precision prediction;

[0012] Step S5: Input the spatiotemporal mapping results into the same fully connected neural network for independent iterative learning to obtain spatial outputs and time output Finally, the spatial and temporal features are effectively integrated through point-by-point multiplication to output the vibration learning results.

[0013] In step S1, a vortex-induced vibration model of a submarine flow transmission pipeline with two ends simply supported is adopted, and the pipeline span length is L, the outer diameter of the pipe is D, and the inner diameter of the pipe is D. i , the x-axis is the central axis along the length of the pipe, the y-axis is the displacement of the pipeline in the downstream direction, the z-axis is the displacement of the pipeline in the transverse direction, and its deflection is w; without considering the influence of axial tension, gravity and buoyancy, the vortex-induced vibration equation of the submarine flow pipeline is expressed as:

[0014]

[0015] Where: E is the elastic modulus of the pipeline, I is the moment of inertia of the cross section of the pipeline, P is the internal pressure of the pipeline, A f is the cross-sectional area of ​​the fluid in the tube, m f , m p and m a are the mass of the fluid in the pipe per unit length, the mass of the pipe, and the additional mass of the ocean current, r s and r f are the pipeline structure damping and the additional ocean current damping, respectively; U is the internal flow velocity; V is the external ocean current velocity; C L0 is the vortex lift coefficient when the structure is stationary, ρ e is the ocean current density, q is the wake oscillator, ε is the wake oscillator coefficient, Ω f is the vortex shedding frequency, α is the dimensionless scaling factor;

[0016] The equation of formula 1 is made dimensionless for the convenience of calculation, and the dimensionless vortex-induced vibration equation of the submarine flow pipeline is obtained:

[0017]

[0018] The expression corresponding to the initial condition of the pipeline is:

[0019]

[0020] The boundary condition expression is:

[0021]

[0022] The residual network PDE(x, t; u, v, q) of the partial differential equation of vortex-induced vibration of submarine pipeline is introduced, that is,

[0023] Then the loss of initial conditions is expressed as:

[0024]

[0025] The loss of boundary conditions is expressed as:

[0026]

[0027] The loss of the partial differential equation is expressed as:

[0028]

[0029] A small number of anchor points are added to the network to modify the network in order to improve the network's ability to fit the high-order nonlinear complex dynamic characteristics of the submarine pipeline vortex-induced vibration equation. The corresponding anchor point data fitting loss function is expressed as:

[0030] The total loss of the network is expressed as the initial loss (L ic ), boundary loss (L bc ), PDE loss (L PDE ) and the loss of data fitting (L data ), that is

[0031] L total =L ic +L bc +L PDE +L data Formula 10;

[0032] Where N is the number of training data, x and t represent the spatial and temporal coordinates respectively, and η θ and η are the predicted and actual values ​​of each point in the network, respectively, θ represents the network learnable hyperparameters, which are weight W and bias b in this network. Wherein the subscripts bc1 and bc2 represent Dirichlet and Neumann boundary conditions, respectively, ic1 and ic2 represent the initial amplitude and initial excitation, respectively, PDE is the configuration point, which consists of the space-time coordinates randomly sampled in the computational domain, and data is the anchor point.

[0033] In step S2, the specific Fourier transform process is:

[0034] For a signal x[n] of length N, it is regarded as a time domain signal composed of N sampling points; by performing Fourier transform on the signal, it is converted into a representation X[k] in the frequency domain, where k = 0, 1, ..., N-1, where X[k] represents the complex value corresponding to the kth frequency of the signal in the frequency domain, and the frequency corresponding to X[k] is calculated by the following formula:

[0035]

[0036] Where: f s represents the sampling rate, f k It represents the frequency value corresponding to the kth frequency of the signal in the frequency domain. By calculating the frequency values ​​corresponding to all k frequencies, the frequency spectrum information of the signal in the frequency domain is jointly composed, thereby obtaining the specific frequency spectrum composition of the vibration;

[0037] Subsequently, the Fourier characteristic mapping function γ is introduced to construct a multiple Fourier mapping network to realize the multi-scale and multi-frequency mapping decomposition of the vortex-induced vibration signal. The mapping function is expressed as follows:

[0038]

[0039] In the formula, is from the Gaussian distribution N(0,σ 2) and keeps it fixed during the model training process, σ>0 is the hyperparameter specified by the user, o is half of the number of neurons in each layer of the network, and d is the input dimension of the network; the random Fourier feature map γ is used to map and decompose the input spatiotemporal vibration data to achieve multi-scale and multi-frequency mapping and decomposition of vortex-induced vibration.

[0040] The spatial Fourier feature map γ(ξ) and the temporal Fourier feature map γ(τ) decomposed in step S3 are respectively:

[0041]

[0042] Where M is the number of Fourier feature mapping. Through the above Fourier feature embedding, the input spatiotemporal data are decomposed by Fourier feature mapping.

[0043] In step S3, if the vortex-induced vibration of the submarine pipeline under the excitation of internal and external flows is composed of many different frequency components, and it is difficult to achieve accurate decomposition through one or two mappings, then according to the specific frequency spectrum composition of the vibration, M is increased. ξ and M τ The number of M is increased to achieve a more detailed decomposition of the vibration. At the same time, considering the strong time-varying nature of vortex-induced vibration, a larger M is selected. τ value to ensure that PINN can learn more fine-grained time scales.

[0044] In step S4, when performing frequency domain decomposition, the neural tangent kernel function of PINN is expressed as follows:

[0045] When the number of K approaches infinity, the eigensystem of K approaches the kernel function K(x,x′), then

[0046] In the formula, g(x)=C1cos(bx)+C2sin(bx) is the characteristic function, C1, C2 are constants, is the non-zero eigenvalue eigenvalue corresponding to the characteristic function;

[0047] For the kernel function K(x,x′), the non-zero eigenvalues ​​corresponding to its eigenfunction g(x) satisfy the following equation:

[0048] In the formula, the value of the hyperparameter σ determines the composition of B. By changing the value of the hyperparameter σ, the value of B is changed, thereby changing the result of the Fourier initialization mapping decomposition. By calculating the frequency content of the eigenvector of the maximum eigenvalue corresponding to different σ, the distribution of the optimal frequency under different σ is determined, and then according to the critical vibration frequency obtained by the spectrum analysis in step S2, the corresponding hyperparameter value σ is selected to adjust the value of the Fourier fundamental frequency vector.

[0049] In step S4, the corresponding hyperparameter value σ is selected to adjust the value of the Fourier fundamental frequency vector. The specific method is: by increasing the value of the hyperparameter σ, the possibility of b obtaining a larger value is increased when the Fourier feature is initialized, so that the frequency of the dominant NTK feature vector increases, so that the NTK feature vector shows more high-frequency features, realizes the rapid convergence of high-frequency information, overcomes the problem of spectrum deviation, and selects the optimal σ to enable the network to better learn specific frequency vibrations, improve its recognition and learning accuracy of specific frequency vibrations, so as to overcome the problem that PINN converges too early in the low-frequency region and cannot achieve global high-precision prediction.

[0050] In step S5, the spatiotemporal multi-scale Fourier characterization is first performed and then input into a fully connected neural network. The feedforward network of the fully connected neural network is defined as:

[0051]

[0052] After iterative learning of the network, the spatial output and the temporal output are merged by point multiplication in the last layer of the network to achieve effective integration of spatiotemporal features. The expression is as follows:

[0053]

[0054] Finally, the network prediction results are output through the linear layer, and the formula is:

[0055]

[0056] In the formula, θ represents all learnable parameters of the network, which are weights W and bias b in this network, φ is the activation function, and η θ (ξ,τ) is the network output, where the superscript indicates the number of network layers.

[0057] The invention discloses a method for analyzing vortex-induced vibration of a submarine pipeline based on a multiple Fourier characterization PINN. The method comprises: in order to improve the fitting ability of the network to cope with the high-order nonlinear complex dynamic characteristics of the submarine pipeline vortex-induced vibration equation, a physical information neural network (PINN) is constructed, and the loss functions of the initial conditions, boundary conditions and partial differential equations (PDEs) are determined according to the submarine pipeline vortex-induced vibration model, and anchor points are added to correct the network to improve its fitting ability; in order to realize the multi-scale and multi-frequency mapping decomposition of the vortex-induced vibration, the Fourier transform technology is adopted to accurately identify the key frequencies of the vibration and their frequency values, and the multiple Fourier characteristic mapping γ function is introduced to construct the multiple Fourier mapping The network uses multiple Fourier feature mapping functions to perform refined mapping decomposition and feature extraction on the vibration signal; according to the number of key vibration frequencies, the mapping times M are dynamically adjusted to ensure the accurate decomposition of multi-frequency vibration spatiotemporal data, so as to effectively overcome the limitations of PINN in processing complex vibration signals; in order to improve the network's recognition and learning accuracy of specific frequency vibrations, according to the frequency size corresponding to the key vibration frequency, the corresponding optimal hyperparameter σ value is selected to adjust the Fourier fundamental frequency vector value to overcome the problem that PINN converges too early in the low-frequency region and cannot achieve global high-precision prediction; then the spatiotemporal mapping results are input into the same fully connected neural network for independent iterative learning, and the spatial outputs are obtained respectively. and time output Finally, the spatial and temporal features are effectively integrated through point-by-point multiplication to output comprehensive and accurate vibration learning results. The beneficial effects of the present invention are: the vibration is decomposed in multiple scales and frequencies through multiple Fourier mappings, which improves the accuracy of PINN in analyzing complex vibration signals, significantly enhances its ability to handle multiple high- and low-frequency superimposed vibrations, and effectively solves the problem of spectrum deviation when learning high-frequency information.

[0058] The beneficial effects of the present invention are: accurately stripping and analyzing multi-frequency superposition characteristics from complex vibration signals, overcoming the defects of PINN in dealing with complex multi-frequency superposition vortex-induced vibration problems, enabling the neural network to specifically learn the vibration dynamic response under different high and low frequency superpositions, and realize accurate prediction of multi-scale and multi-frequency superposition vortex-induced vibrations, effectively improving the prediction accuracy of the model, and providing an effective solution for the modeling and simulation of multiple high, medium and low frequency superposition problems in science and engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments:

[0060] Attached Figure 1 It is a schematic diagram of the implementation process of the method of the present invention;

[0061] Attached Figure 2 It is a schematic diagram of the overall structural framework of the present invention;

[0062] Attached Figure 3 Schematic diagram for comparison of PINN and MFF-PINN submarine pipeline vortex-induced vibration simulation results;

[0063] Attached Figure 4 Schematic diagram of the dimensionless amplitude of vortex-induced vibration changing with time under different internal and external flow velocities. DETAILED DESCRIPTION

[0064] As shown in the figure, the vortex-induced vibration analysis method of submarine pipeline based on multi-Fourier characterization PINN includes the following steps:

[0065] Step S1, according to the submarine pipeline vortex-induced vibration model, determine the corresponding initial conditions, boundary conditions, and loss functions of the partial differential equation, and modify the network by adding a small number of anchor points to the network to improve the network's ability to fit the high-order nonlinear complex dynamic characteristics of the submarine pipeline vortex-induced vibration equation, so as to obtain the loss function of the model corresponding to the data fitting;

[0066] Step S2, using Fourier transform technology to identify the key frequencies of vibration and their frequency values, then introducing a multiple Fourier characteristic mapping γ function to construct a multiple Fourier mapping network, using the multiple Fourier characteristic mapping function to perform refined mapping decomposition and feature extraction on the vibration signal, and realizing multi-scale and multi-frequency mapping decomposition of the vortex-induced vibration signal;

[0067] Step S3, dynamically adjust the mapping times M according to the number of key vibration frequencies, perform mapping decomposition of the network input spatiotemporal data for a corresponding number of times according to the mapping times M, decompose the input data to obtain the spatial Fourier characteristic map γ(ξ) and the temporal Fourier characteristic map γ(τ), so as to ensure the accurate decomposition of multi-frequency vibration spatiotemporal data, and effectively overcome the limitations of PINN in processing complex vibration signals;

[0068] Step S4, decomposing the neural tangent kernel of PINN in the frequency domain to obtain the corresponding relationship between different frequencies and the optimal hyperparameter σ, and selecting the corresponding optimal hyperparameter value σ according to the frequency corresponding to the key frequency of vibration to adjust the value of the Fourier base frequency vector, thereby improving the network's recognition and learning accuracy of specific frequency vibrations, and overcoming the problem that PINN converges too early in the low-frequency region and cannot achieve global high-precision prediction;

[0069] Step S5: Input the spatiotemporal mapping results into the same fully connected neural network for independent iterative learning to obtain spatial outputs and time output Finally, the spatial and temporal features are effectively integrated through point-by-point multiplication to output the vibration learning results.

[0070] In step S1, a vortex-induced vibration model of a submarine flow transmission pipeline with two ends simply supported is adopted, and the pipeline span length is L, the outer diameter of the pipe is D, and the inner diameter of the pipe is D. i , the x-axis is the central axis along the length of the pipe, the y-axis is the displacement of the pipeline in the downstream direction, the z-axis is the displacement of the pipeline in the transverse direction, and its deflection is w; without considering the influence of axial tension, gravity and buoyancy, the vortex-induced vibration equation of the submarine flow pipeline is expressed as:

[0071]

[0072] Where: E is the elastic modulus of the pipeline, I is the moment of inertia of the cross section of the pipeline, P is the internal pressure of the pipeline, A f is the cross-sectional area of ​​the fluid in the tube, m f , m p and m a are the mass of the fluid in the pipe per unit length, the mass of the pipe, and the additional mass of the ocean current, r s and r f are the pipeline structure damping and the additional ocean current damping, respectively; U is the internal flow velocity; V is the external ocean current velocity; C L0 is the vortex lift coefficient when the structure is stationary, ρ e is the ocean current density, q is the wake oscillator, ε is the wake oscillator coefficient, Ω f is the vortex shedding frequency, α is the dimensionless scaling factor;

[0073] The equation of formula 1 is made dimensionless for the convenience of calculation, and the dimensionless vortex-induced vibration equation of the submarine flow pipeline is obtained:

[0074]

[0075] The expression corresponding to the initial condition of the pipeline is:

[0076]

[0077] The boundary condition expression is:

[0078]

[0079] The residual network PDE(x, t; u, v, q) of the partial differential equation of vortex-induced vibration of submarine pipeline is introduced, that is,

[0080] Then the loss of initial conditions is expressed as:

[0081]

[0082] The loss of boundary conditions is expressed as:

[0083]

[0084] The loss of the partial differential equation is expressed as:

[0085]

[0086] A small number of anchor points are added to the network to modify the network in order to improve the network's ability to fit the high-order nonlinear complex dynamic characteristics of the submarine pipeline vortex-induced vibration equation. The corresponding anchor point data fitting loss function is expressed as:

[0087] The total loss of the network is expressed as the initial loss (L ic ), boundary loss (L bc ), PDE loss (L PDE ) and the loss of data fitting (L data ), that is

[0088] L total =L ic +L bc +L PDE +L data Formula 10;

[0089] Where N is the number of training data, x and t represent the spatial and temporal coordinates respectively, and η θ and η are the predicted and actual values ​​of each point in the network, respectively, θ represents the network learnable hyperparameters, which are weight W and bias b in this network. Wherein the subscripts bc1 and bc2 represent Dirichlet and Neumann boundary conditions, respectively, ic1 and ic2 represent the initial amplitude and initial excitation, respectively, PDE is the configuration point, which consists of the space-time coordinates randomly sampled in the computational domain, and data is the anchor point.

[0090] In step S2, the specific Fourier transform process is:

[0091] For a signal x[n] of length N, it is regarded as a time domain signal composed of N sampling points; by performing Fourier transform on the signal, it is converted into a representation X[k] in the frequency domain, where k = 0, 1, ..., N-1, where X[k] represents the complex value corresponding to the kth frequency of the signal in the frequency domain, and the frequency corresponding to X[k] is calculated by the following formula:

[0092]

[0093] Where: f s represents the sampling rate, f k It represents the frequency value corresponding to the kth frequency of the signal in the frequency domain. By calculating the frequency values ​​corresponding to all k frequencies, the frequency spectrum information of the signal in the frequency domain is jointly composed, thereby obtaining the specific frequency spectrum composition of the vibration;

[0094] Subsequently, the Fourier characteristic mapping function γ is introduced to construct a multiple Fourier mapping network to realize the multi-scale and multi-frequency mapping decomposition of the vortex-induced vibration signal. The mapping function is expressed as follows:

[0095]

[0096] In the formula, is from the Gaussian distribution N(0,σ 2 ) and keeps it fixed during the model training process, σ>0 is the hyperparameter specified by the user, o is half of the number of neurons in each layer of the network, and d is the input dimension of the network; the random Fourier feature map γ is used to map and decompose the input spatiotemporal vibration data to achieve multi-scale and multi-frequency mapping and decomposition of vortex-induced vibration.

[0097] The spatial Fourier feature map γ(ξ) and the temporal Fourier feature map γ(τ) decomposed in step S3 are respectively:

[0098]

[0099] Where M is the number of Fourier feature mapping. Through the above Fourier feature embedding, the input spatiotemporal data are decomposed by Fourier feature mapping.

[0100] In step S3, if the vortex-induced vibration of the submarine pipeline under the excitation of internal and external flows is composed of many different frequency components, and it is difficult to achieve accurate decomposition through one or two mappings, then according to the specific frequency spectrum composition of the vibration, M is increased. ξ and M τ The number of M is increased to achieve a more detailed decomposition of the vibration. At the same time, considering the strong time-varying nature of vortex-induced vibration, a larger M is selected. τ value to ensure that PINN can learn more fine-grained time scales.

[0101] In step S4, when performing frequency domain decomposition, the neural tangent kernel function of PINN is expressed as follows:

[0102] When the number of K approaches infinity, the eigensystem of K approaches the kernel function K(x,x′), then

[0103] In the formula, g(x)=C1cos(bx)+C2sin(bx) is the characteristic function, C1, C2 are constants, is the non-zero eigenvalue eigenvalue corresponding to the characteristic function;

[0104] For the kernel function K(x,x′), the non-zero eigenvalues ​​corresponding to its eigenfunction g(x) satisfy the following equation:

[0105] In the formula, the value of the hyperparameter σ determines the composition of B. By changing the value of the hyperparameter σ, the value of B is changed, thereby changing the result of the Fourier initialization mapping decomposition. By calculating the frequency content of the eigenvector of the maximum eigenvalue corresponding to different σ, the distribution of the optimal frequency under different σ is determined, and then according to the critical vibration frequency obtained by the spectrum analysis in step S2, the corresponding hyperparameter value σ is selected to adjust the value of the Fourier fundamental frequency vector.

[0106] In step S4, the corresponding hyperparameter value σ is selected to adjust the value of the Fourier fundamental frequency vector. The specific method is: by increasing the value of the hyperparameter σ, the possibility of b obtaining a larger value is increased when the Fourier feature is initialized, so that the frequency of the dominant NTK feature vector increases, so that the NTK feature vector shows more high-frequency features, realizes the rapid convergence of high-frequency information, overcomes the problem of spectrum deviation, and selects the optimal σ to enable the network to better learn specific frequency vibrations, improve its recognition and learning accuracy of specific frequency vibrations, so as to overcome the problem that PINN converges too early in the low-frequency region and cannot achieve global high-precision prediction.

[0107] In step S5, the spatiotemporal multi-scale Fourier characterization is first performed and then input into a fully connected neural network. The feedforward network of the fully connected neural network is defined as:

[0108]

[0109] After iterative learning of the network, the spatial output and the temporal output are merged by point multiplication in the last layer of the network to achieve effective integration of spatiotemporal features. The expression is as follows:

[0110]

[0111] Finally, the network prediction results are output through the linear layer, and the formula is:

[0112]

[0113] In the formula, θ represents all learnable parameters of the network, which are weights W and bias b in this network, φ is the activation function, and η θ (ξ,τ) is the network output, where the superscript indicates the number of network layers.

[0114] Example:

[0115] In order to better explain the specific process of the present invention, the method flow is as follows ( Figure 1 ), the implementation of the present invention is described in detail in conjunction with specific embodiments:

[0116] This case considers the dimensionless vortex-induced vibration equations (II), initial condition equation (III) and boundary condition equation (IV) of a two-end simply-supported submarine flow pipeline, and solves the spatial domain ξ∈(0,1) and the time domain τ∈(0,15). Consider the dimensionless flow velocity u=1 in the pipe and the ocean current velocity v=1. By using the proposed spatiotemporal multiscale multi-Fourier characterization neural network η θ (ξ,τ) approximates the potential solution η(ξ,τ). Specifically, before the input network is trained, the input vector a (0) =(ξ,τ) Fourier feature map decomposition is applied with different σ initializations, and then input into PINN for iterative learning. A fully connected neural network is used in all training, with 3 hidden layers, each with 50 neurons, and the network batch size is set to N f =5000, N bc =100, N ic =100, N domain = 10000. In order to better evaluate the performance of the multi-Fourier characterization PINN, the generalized finite difference method (GFDM) was used to create a reference data set.

[0117] Firstly, the PINN algorithm and the multiple Fourier characterization PINN (MFF-PINN) algorithm with different mapping times are used to learn and predict the vortex-induced vibration of submarine pipelines. The learning prediction results of different algorithms are compared. Figure 3 As shown in the figure, it can be seen that as the number of iterations increases, MFF-PINN shows more accurate learning ability ( Figure 3 b and c parts).

[0118] After 30,000 epochs of training iterations, MFF-PINN accurately simulates the vortex-induced vibration of the pipeline. In sharp contrast, ( Figure 3 a), PINN has a spectrum deviation problem when facing the spatiotemporal multi-scale and high- and low-frequency superposition characteristics of vortex-induced vibration. Even after 30,000 epochs of iterative learning, PINN still fails to effectively simulate the vortex-induced vibration of the pipeline, and its calculated error is close to the true value of the GFDM reference solution.

[0119] Further comparison of the simulation results of MFF-PINN after Fourier mapping decomposition with different mapping times ( Figure 3It can be seen from the figure that after 20,000 epochs of training iterations, compared with the MFF-PINN using two-times Fourier mapping (σ=[1,30]), the MFF-PINN using three-times Fourier mapping (σ=[1,10,30]) can decompose the pipeline vibration more finely and achieve more effective vortex-induced vibration simulation. Figure 3 The relative error analysis in (d) further confirms the overall superiority of MFF-PINN, and the cubic mapping scheme is particularly outstanding in accelerating convergence, with its error rapidly decreasing from 1.0115 to 0.2228, indicating that increasing the number of mappings can effectively refine the vibration feature analysis and promote rapid convergence of high-frequency components, thereby achieving a more stable and accurate learning effect overall.

[0120] In actual engineering, the environment of submarine pipelines is complex and changeable, and the internal and external flow velocities are often not constant. As the internal and external flow velocities change, the vibration characteristics of the pipeline change accordingly. In order to verify the universality of MFF-PINN, different internal and external flow velocity combinations (u=1.0, v=0.5; u=2.0, v=1.0; u=3.0, v=1.5) are selected for vortex-induced vibration simulation. The obtained cloud diagram of the dimensionless amplitude of the submarine pipeline at each position along the pipeline axis with time is shown in Figure 4 .

[0121] Under the excitation of different internal and external flow velocity combinations, the vortex-induced vibration of the pipeline exhibits vibration characteristics of various forms. As the flow velocity increases, the vortex-induced vibration becomes more complex. MFF-PINN applies separate Fourier feature embedding to the input coordinates of space and time for the frequency composition of vibration under different internal and external flow excitations, and independently iterates and learns through neural networks one by one, successfully solving the problem of spectrum deviation. For the vortex-induced vibration of the pipeline under different internal and external flow combinations, MFF-PINN can accurately learn the high- and low-frequency superposition vibration of the target function, and the simulation error is controlled within 10 -4 The magnitude shows good learning performance.

[0122] For those skilled in the art, the above examples are only used to illustrate the technical solutions and effects of the present invention, and do not limit the present invention. The present invention can be implemented in other specific forms without departing from the spirit or basic features of the present invention. Any changes made within the spirit and principles of the present invention should be included in the present invention, and any figure mark in the claims should not be regarded as limiting the claims involved.

Claims

1. A method for analyzing vortex-induced vibration of submarine pipelines based on multi-Fourier characterization PINN, characterized by: The following steps are involved: Step S1, according to the submarine pipeline vortex-induced vibration model, determine the corresponding initial conditions, boundary conditions, and loss functions of the partial differential equation, and modify the network by adding a small number of anchor points to the network to improve the network's ability to fit the high-order nonlinear complex dynamic characteristics of the submarine pipeline vortex-induced vibration equation, so as to obtain the loss function of the model corresponding to the data fitting; Step S2, using Fourier transform technology to identify the key frequencies of vibration and their frequency values, then introducing a multiple Fourier characteristic mapping γ function to construct a multiple Fourier mapping network, using the multiple Fourier characteristic mapping function to perform refined mapping decomposition and feature extraction on the vibration signal, and realizing multi-scale and multi-frequency mapping decomposition of the vortex-induced vibration signal; Step S3, dynamically adjust the mapping times M according to the number of key vibration frequencies, perform mapping decomposition of the network input spatiotemporal data for a corresponding number of times according to the mapping times M, decompose the input data to obtain the spatial Fourier characteristic map γ(ξ) and the temporal Fourier characteristic map γ(τ), so as to ensure the accurate decomposition of multi-frequency vibration spatiotemporal data, and effectively overcome the limitations of PINN in processing complex vibration signals; Step S4, decomposing the neural tangent kernel of PINN in the frequency domain to obtain the corresponding relationship between different frequencies and the optimal hyperparameter σ, and selecting the corresponding optimal hyperparameter value σ according to the frequency corresponding to the key frequency of vibration to adjust the value of the Fourier base frequency vector, thereby improving the network's recognition and learning accuracy of specific frequency vibrations, and overcoming the problem that PINN converges too early in the low-frequency region and cannot achieve global high-precision prediction; Step S5: Input the spatiotemporal mapping results into the same fully connected neural network for independent iterative learning to obtain spatial outputs and time output Finally, the spatial and temporal features are effectively integrated through point-by-point multiplication to output the vibration learning results.

2. The method for analyzing vortex-induced vibration of submarine pipelines based on multi-Fourier characterization PINN according to claim 1 is characterized in that: In step S1, a vortex-induced vibration model of a submarine flow transmission pipeline with two ends simply supported is adopted, and the pipeline span length is L, the outer diameter of the pipe is D, and the inner diameter of the pipe is D. i , the x-axis is the central axis along the length of the pipe, the y-axis is the displacement of the pipeline in the downstream direction, the z-axis is the displacement of the pipeline in the transverse direction, and its deflection is w; without considering the influence of axial tension, gravity and buoyancy, the vortex-induced vibration equation of the submarine flow pipeline is expressed as: Where: E is the elastic modulus of the pipeline, I is the moment of inertia of the cross section of the pipeline, P is the internal pressure of the pipeline, A f is the cross-sectional area of ​​the fluid in the tube, m f , m p and m a are the mass of the fluid in the pipe per unit length, the mass of the pipe, and the additional mass of the ocean current, r s and r f are the pipeline structure damping and the additional ocean current damping, respectively; U is the internal flow velocity; V is the external ocean current velocity; C L0 is the vortex lift coefficient when the structure is stationary, ρ e is the ocean current density, q is the wake oscillator, ε is the wake oscillator coefficient, Ω f is the vortex shedding frequency, α is the dimensionless scaling factor; The equation of formula 1 is made dimensionless for the convenience of calculation, and the dimensionless vortex-induced vibration equation of the submarine flow pipeline is obtained: The expression corresponding to the initial condition of the pipeline is: The boundary condition expression is: The residual network PDE(x, t; u, v, q) of the partial differential equation of vortex-induced vibration of submarine pipeline is introduced, that is, Then the loss of initial conditions is expressed as: The loss of boundary conditions is expressed as: The loss of the partial differential equation is expressed as: A small number of anchor points are added to the network to modify the network in order to improve the network's ability to fit the high-order nonlinear complex dynamic characteristics of the submarine pipeline vortex-induced vibration equation. The corresponding anchor point data fitting loss function is expressed as: The total loss of the network is expressed as the initial loss (L ic ), boundary loss (L bc ), PDE loss (L PDE ) and the loss of data fitting (L data ), that is L total =L ic +L bc +L PDE +L data Formula 10; Where N is the number of training data, x and t represent the spatial and temporal coordinates respectively, and η θ and η are the predicted and actual values ​​of each point in the network, respectively, θ represents the network learnable hyperparameters, which are weight W and bias b in this network. Wherein the subscripts bc1 and bc2 represent Dirichlet and Neumann boundary conditions, respectively, ic1 and ic2 represent the initial amplitude and initial excitation, respectively, PDE is the configuration point, which consists of the space-time coordinates randomly sampled in the computational domain, and data is the anchor point.

3. The method for analyzing vortex-induced vibration of submarine pipelines based on multi-Fourier characterization PINN according to claim 1 is characterized in that: In step S2, the specific Fourier transform process is: For a signal x[n] of length N, it is regarded as a time domain signal composed of N sampling points; by performing Fourier transform on the signal, it is converted into a representation X[k] in the frequency domain, where k = 0, 1, ..., N-1, where X[k] represents the complex value corresponding to the kth frequency of the signal in the frequency domain, and the frequency corresponding to X[k] is calculated by the following formula: Where: f s represents the sampling rate, f k It represents the frequency value corresponding to the kth frequency of the signal in the frequency domain. By calculating the frequency values ​​corresponding to all k frequencies, the frequency spectrum information of the signal in the frequency domain is jointly composed, thereby obtaining the specific frequency spectrum composition of the vibration; Subsequently, the Fourier characteristic mapping function γ is introduced to construct a multiple Fourier mapping network to realize the multi-scale and multi-frequency mapping decomposition of the vortex-induced vibration signal. The mapping function is expressed as follows: In the formula, is from the Gaussian distribution N(0,σ 2 ) and keeps it fixed during the model training process, σ>0 is the hyperparameter specified by the user, o is half of the number of neurons in each layer of the network, and d is the input dimension of the network; the random Fourier feature map γ is used to map and decompose the input spatiotemporal vibration data to achieve multi-scale and multi-frequency mapping and decomposition of vortex-induced vibration.

4. The method for analyzing vortex-induced vibration of submarine pipelines based on multi-Fourier characterization PINN according to claim 1 is characterized in that: The spatial Fourier feature map γ(ξ) and the temporal Fourier feature map γ(τ) decomposed in step S3 are respectively: Where M is the number of Fourier feature mapping. Through the above Fourier feature embedding, the input spatiotemporal data are decomposed by Fourier feature mapping.

5. The method for analyzing vortex-induced vibration of submarine pipelines based on multi-Fourier characterization PINN according to claim 1 is characterized in that: In step S3, if the vortex-induced vibration of the submarine pipeline under the excitation of internal and external flows is composed of many different frequency components, and it is difficult to achieve accurate decomposition through one or two mappings, then according to the specific frequency spectrum composition of the vibration, M is increased. ξ and M τ The number of M is increased to achieve a more detailed decomposition of the vibration. At the same time, considering the strong time-varying nature of vortex-induced vibration, a larger M is selected. τ value to ensure that PINN can learn more fine-grained time scales.

6. The method for analyzing vortex-induced vibration of submarine pipelines based on multi-Fourier characterization PINN according to claim 1 is characterized in that: In step S4, when performing frequency domain decomposition, the neural tangent kernel function of PINN is expressed as follows: When the number of K approaches infinity, the eigensystem of K approaches the kernel function K(x,x′), then In the formula, g(x)=C1cos(bx)+C2sin(bx) is the characteristic function, C1, C2 are constants, is the non-zero eigenvalue eigenvalue corresponding to the characteristic function; For the kernel function K(x,x′), The non-zero eigenvalues ​​corresponding to its characteristic function g(x) satisfy the following equation: In the formula, the value of the hyperparameter σ determines the composition of B. By changing the value of the hyperparameter σ, the value of B is changed, thereby changing the result of the Fourier initialization mapping decomposition. By calculating the frequency content of the eigenvector of the maximum eigenvalue corresponding to different σ, the distribution of the optimal frequency under different σ is determined, and then according to the critical vibration frequency obtained by the spectrum analysis in step S2, the corresponding hyperparameter value σ is selected to adjust the value of the Fourier fundamental frequency vector.

7. The method for analyzing vortex-induced vibration of submarine pipelines based on multi-Fourier characterization PINN according to claim 1 is characterized in that: In step S4, the corresponding hyperparameter value σ is selected to adjust the value of the Fourier fundamental frequency vector. The specific method is: by increasing the value of the hyperparameter σ, the possibility of b obtaining a larger value is increased when the Fourier feature is initialized, so that the frequency of the dominant NTK feature vector increases, so that the NTK feature vector shows more high-frequency features, realizes the rapid convergence of high-frequency information, overcomes the problem of spectrum deviation, and selects the optimal σ to enable the network to better learn specific frequency vibrations, improve its recognition and learning accuracy of specific frequency vibrations, so as to overcome the problem that PINN converges too early in the low-frequency region and cannot achieve global high-precision prediction.

8. The method for analyzing vortex-induced vibration of submarine pipelines based on multi-Fourier characterization PINN according to claim 1 is characterized in that: In step S5, the spatiotemporal multi-scale Fourier characterization is first performed and then input into a fully connected neural network. The feedforward network of the fully connected neural network is defined as: After iterative learning of the network, the spatial output and the temporal output are merged by point multiplication in the last layer of the network to achieve effective integration of spatiotemporal features. The expression is as follows: Finally, the network prediction results are output through the linear layer, and the formula is: In the formula, θ represents all learnable parameters of the network, which are weights W and bias b in this network, φ is the activation function, and η θ (ξ,τ) is the network output, where the superscript indicates the number of network layers.

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