Method for analyzing vibration of fluid conveying pipe on basis of fourier featured pinn

Abstract

Provided in the present invention is a method for analyzing the vibration of a fluid conveying pipe on the basis of a Fourier featured PINN. The method comprises: constructing a physics-informed neural network (PINN), determining initial conditions and boundary conditions on the basis of a fluid conveying pipe model, and obtaining loss functions corresponding to the initial conditions and the boundary conditions; using a "soft" constraint to express a second-type boundary condition and initial excitation of a pipe into PINN loss functions, and using a "hard" constraint to code a first-type boundary condition and an initial amplitude into a DNN system structure; adding an observation anchor point to the network for correction when a peak value is insufficient during a network model training process; constructing a Fourier feature mapping function, adjusting a neural tangent kernel of the network, training two mapping results by means of a set network, and finally, in the network, connecting spatial and temporal hidden layers by means of point multiplication, so as to establish an FFNN; and by means of training and learning, finally outputting a prediction result by means of a linear layer connection.

Description

A Vibration Analysis Method for Fluid Conveyance Pipelines Based on Fourier Characterization PINN Technical Field

[0001] The present invention belongs to the technical field of pipeline vibration prediction, and in particular relates to a fluid transmission pipeline vibration analysis method based on Fourier characterization PINN. Background Art

[0002] Fluid pipelines are widely used in numerous fields. Fluid flowing through them can cause vibrations, ultimately leading to long-term fatigue failure. Therefore, studying the dynamic response of flow-induced vibrations in pipelines is of practical significance. However, due to the presence of fluid-solid interaction (FSI) in pipelines, pipeline vibrations exhibit multi-scale characteristics and a superposition of high and low frequencies, making analysis challenging. Therefore, it is necessary to develop an algorithm that can stably learn high-frequency information while accurately predicting the governing equations with high-order nonlinear terms at multiple time and space scales without increasing computational cost.

[0003] PINNs leverage the universal approximation capabilities of neural networks to train neural network parameters by simultaneously minimizing the sum of data fitting errors and partial differential equation residuals. This allows them to effectively solve partial differential equations (PDEs) with high-order nonlinear terms. However, due to the widespread spectral bias of the fully connected architecture used in PINNs, the network tends to learn low-frequency components and ignore high-frequency components during training, making it difficult to correctly learn high-frequency functions. When the solutions to partial differential equations exhibit high-frequency and multi-scale characteristics, PINNs often struggle to train stably and fail to produce accurate predictions. Summary of the Invention

[0004] In view of the spectral deviation problem and insufficient learning ability of high-frequency information in traditional deep fully connected networks, the present invention proposes a pipeline vibration analysis method based on Fourier characterization PINN as an effective solution. It slows down the rapid decline of NTK in the network caused by the increase in frequency by Fourier mapping the input data, reduces the degree of attenuation of eigenvalues ​​as the frequency increases, increases the eigenvalues ​​of high-frequency information, thereby improving the convergence speed of high-frequency information training and enhancing the ability of the fully connected network to learn high-frequency functions.

[0005] To this end, this paper provides a pipeline vibration analysis method based on Fourier-characterized PINN. The goal is to establish a numerical algorithm that can accurately and efficiently solve pipeline models, endow the algorithm with strong high-frequency learning capabilities, and apply it to predict pipeline vibration changes.

[0006] The solution includes constructing a physical information neural network (PINN), determining the initial and boundary conditions based on the flow pipeline model, and obtaining the corresponding loss function; expressing the second-class boundary conditions and the initial excitation of the pipeline into the PINN loss function with "soft" constraints, and encoding the first-class boundary conditions and initial amplitude into the DNN architecture with "hard" constraints. At the same time, observation anchor points are added to the network to correct for insufficient peak values ​​during network model training; and constructing Fourier feature maps. Function, decomposes the data input by the PINN feedforward neural network into spatial Fourier feature maps and temporal Fourier feature maps By choosing different The value is used to modify the Fourier basis frequency vector and adjust the neural tangent kernel of the network to control the frequency range of the neural network learning; after the above processing, the two mapping results are trained through the set network respectively, and finally the hidden layers of space and time are connected in the network through point multiplication to establish the FFNN network; after training and learning, the spatial output is obtained and time output , and use point-by-point multiplication to merge the two, and finally output the prediction result through a linear layer connection.

[0007] The technical solution specifically adopted by the present invention to solve the technical problem is:

[0008] A method for analyzing vibration of a fluid delivery pipeline based on Fourier-characterized PINN, characterized by comprising the following steps:

[0009] Step 1: Introduce the residual network PDE of the flow pipeline equation and construct the corresponding PINN network;

[0010] Step 2: Based on the flow pipeline model, determine the initial conditions and boundary conditions and obtain the corresponding loss function in the PINN network. The initial conditions include the initial amplitude and initial excitation of the pipeline, and the boundary conditions include the first-type Dirichlet condition and the second-type Neumann condition.

[0011] Step 3: Separately process the first and second boundary conditions and initial conditions in the pipeline problem, express the Neumann boundary conditions and pipeline initial excitation into the PINN loss function with "soft" constraints, constrain the boundary conditions by minimizing the loss function, and encode the Dirichlet boundary conditions and initial amplitude into the DNN architecture with "hard" constraints. Construct a function that satisfies the initial boundary value conditions and impose hard constraints on the Dirichlet boundary conditions and initial amplitude.

[0012] Step 4: In view of the high-order terms and strong nonlinear properties of the flow pipeline equation, internal observation anchor points are added to the network to correct the problem of insufficient peak values ​​generated by the neural network model during prediction;

[0013] Step 5: Through Fourier feature mapping Function, decomposes the input data of the PINN network into spatial Fourier feature maps and temporal Fourier feature maps ;

[0014] Step 6: By selecting different The value modifies the Fourier basis frequency vector and adjusts the neural tangent kernel of the network, thereby controlling the frequency range of neural network learning;

[0015] Step 7: Substitute the results of spatial and temporal Fourier feature mapping into the same PINN network for training, and connect the spatial and temporal hidden layers in the network through point multiplication to establish a FFNN network;

[0016] Step 8: Get spatial output after training and learning and time output , and use point-by-point multiplication to merge the two, and finally output the prediction result through a linear layer connection.

[0017] Furthermore, in step 1, the residual network of the equation is introduced for the flow pipeline model The specific expression is as follows:

[0018]

[0019]

[0020] Where a is the input quantity, Indicates that the network can learn hyperparameters, including weights W and bias b, is the output vector, and the superscript indicates the number of network layers;

[0021] It is an L-layer neural network whose input vector Including spatial coordinates in piping vibration models and time coordinates , the output vector is the pipeline amplitude , the network is a feed-forward neural network.

[0022] Furthermore, in Step 2, the loss function of the second type of Neumann boundary condition under soft constraints is:

[0023]

[0024] Its initial condition, that is, the pipeline initial excitation loss function is:

[0025]

[0026] Pipeline differential equations The mean square error of the network is:

[0027]

[0028] Therefore, the pipeline loss function under soft constraints is:

[0029]

[0030] Where, The number of training data for the Neumann boundary condition of the model when the support condition is fixed at both ends, is the number of initial incentive training, The number of internal configuration points. is the neural network output, and are the boundary condition points and initial condition points as network input, 、 is the actual value of the corresponding point; is passed to Configuration points of the network, parameters Corresponding to weight coefficients in the loss function, effectively assigning different learning rates to each individual loss term; these weights can be specified by the user or automatically adjusted during network training.

[0031] Furthermore, in step 3, the loss function of the first type of Dirichlet boundary condition under hard constraints is:

[0032]

[0033] Its initial condition, i.e. the initial amplitude loss function of the pipeline, is:

[0034]

[0035] Pipeline differential equations The mean square error of the network is:

[0036]

[0037] Therefore, the pipeline loss function under hard constraints is:

[0038]

[0039] Where, the function of the special solution that satisfies the boundary conditions constructed by the “hard” constraint is: The number of training data for the Dirichlet boundary conditions of the model when the two ends are fixed support conditions, is the number of initial amplitude training, are the weight coefficients corresponding to each loss function term, and the subscripts ic and bc represent the initial condition and boundary condition, respectively.

[0040] Furthermore, in Step 4, for the high-order terms and strong nonlinear properties of the flow pipeline equation, internal observation anchor points are added to the network, corresponding to the subscript domain. The loss function of the anchor point term is:

[0041]

[0042] The pipeline loss function with anchor point correction term added under hard constraints is:

[0043]

[0044] In the formula, the weight Set to 0 or 1, depending on the governing equation itself.

[0045] Furthermore, in step 5, the spatial Fourier feature map and temporal Fourier feature maps It is expressed as follows:

[0046]

[0047]

[0048] Where, Represents a Gaussian distribution The mapping matrix obtained by sampling in m represents the Fourier basis frequency vector, is a hyperparameter that specifies how often the network tends to learn, is the number of neurons in each layer of the PINN network, and d is the input dimension of the network.

[0049] Furthermore, the neural tangent kernel in step 6 is expressed as follows:

[0050]

[0051] Where, , is the input point data.

[0052] Furthermore, in step 7, for the spatial Fourier feature map , the first layer and hidden layer of the network are:

[0053]

[0054]

[0055] For the temporal Fourier feature map , the first layer and hidden layer of the network are:

[0056]

[0057]

[0058] The spatial and temporal hidden layers are combined as follows:

[0059]

[0060] Where, is the activation function, and the superscript in the formula represents the number of network layers.

[0061] Compared with the existing technology, the present invention and its preferred solution overcome the shortcomings of fully connected neural networks in learning and predicting high-frequency functions. Combined with the use of physical information by PINN, it can accurately capture high-frequency oscillations for the flow pipeline model and better learn and predict pipeline vibrations. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments:

[0063] FIG1 is a schematic diagram of a method flow diagram according to an embodiment of the present invention;

[0064] FIG2 is a schematic diagram of the structural framework of an embodiment of the present invention;

[0065] FIG3 is a cloud diagram comparing the GFDM numerical solution and the PINN and FF-PINN prediction solutions according to an embodiment of the present invention;

[0066] FIG4 is a numerical solution of GFDM and the predicted amplitudes of PINN and FF-PINN according to an embodiment of the present invention. Comparison picture. DETAILED DESCRIPTION

[0067] To make the features and advantages of this patent more clearly understood, the following embodiments are specifically described in detail as follows:

[0068] It should be noted that the following detailed description is illustrative and is intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used in this specification have the same meaning as commonly understood by those skilled in the art to which this application belongs.

[0069] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0070] As shown in FIG1 to FIG4 , an embodiment of the present invention provides a method for analyzing vibration of a fluid delivery pipeline based on Fourier-characterized PINN, comprising the following steps:

[0071] Step 1: Introduce the residual network PDE of the flow pipeline equation and construct the corresponding PINN network;

[0072] The fluid delivery pipeline model selected by the present invention is an improved equation of the classic Euler-Bernoulli beam model.

[0073] Consider a straight pipe with fixed branches at both ends, the length of the pipe is L, and the direction along the central axis of the pipe is Axis, along the transverse direction of the pipeline Axis. Ignoring the effects of gravity, internal damping, and fluid pressure, and also ignoring the effects of temperature loads and the resulting axial tension, the differential equation for lateral motion can be expressed as:

[0074] (1)

[0075] Where E is the elastic modulus of the pipe, I is the moment of inertia of the cross section of the pipe, and are the mass of the fluid in the pipe per unit length and the mass of the pipe, respectively, and U is the fluid flow rate.

[0076] The expression of the pipeline boundary condition is:

[0077] (2)

[0078] By non-dimensionalizing the equation, we can obtain the differential equation of lateral motion of a straight pipe with fixed branches at both ends:

[0079] (3)

[0080] Similarly, the corresponding dimensionless boundary conditions and initial conditions at both ends can be written as:

[0081] (4)

[0082] (5)

[0083] It should be noted that for the lateral vibration problem of the fluid transmission pipeline, the network in the PINN established in Step 1 is It is an L-layer neural network whose input vector Including spatial coordinates in piping vibration models and time coordinates , the output vector is the pipeline amplitude , since the network is a feedforward neural network, each layer creates data for the next layer through the following nested transformations:

[0084] (6)

[0085] (7)

[0086] The expression of the output vector is:

[0087] (8)

[0088] Where a is the input quantity, Indicates that the network can learn hyperparameters, including weights W and bias b, is the activation function, the superscript indicates the number of network layers, is the output vector.

[0089] Then introduce the residual network of equation ,Right now

[0090] (9)

[0091] Step 2: Based on the flow pipeline model, determine the initial conditions and boundary conditions and obtain their corresponding loss function in the PINN network. The initial conditions include the initial amplitude and initial excitation of the pipeline, and the boundary conditions include the first-type Dirichlet condition and the second-type Neumann condition.

[0092] It should be noted that the loss function corresponding to the initial condition in step 2 is:

[0093] (10)

[0094] The loss function corresponding to the boundary condition is:

[0095] (11)

[0096] Where, and are the number of training data for the first and second type boundary conditions of the model when the two ends are fixed support conditions, and are the initial amplitude and the number of initial excitation training respectively. is the neural network output, and As The boundary condition points and initial condition points of the network input, 、 、 and are the actual values ​​at these points.

[0097] Step 3: Separately process the first and second boundary conditions and initial conditions in the pipeline problem, express the Neumann boundary conditions and pipeline initial excitation into the PINN loss function with "soft" constraints, constrain the boundary conditions by minimizing the loss function, and encode the Dirichlet boundary conditions and initial amplitude into the DNN architecture with "hard" constraints. Construct a function that satisfies the initial boundary value conditions and impose hard constraints on the Dirichlet boundary conditions and initial amplitude.

[0098] It should be noted that the specific expression of the function that satisfies the boundary condition special solution constructed by the "hard" constraint in Step 3 is as follows:

[0099] (12)

[0100] in, The Dirichlet boundary conditions and the initial amplitude of the pipeline are: 、 The corresponding conditions are automatically met.

[0101] Under hard constraints, the loss function of the boundary condition is modified to contain only the Neumann boundary condition, that is, the loss function of the boundary condition is:

[0102] (13)

[0103] Similarly, its initial condition loss function is modified to contain only the initial incentive:

[0104] (14)

[0105] Pipeline differential equations The mean square error of the network is:

[0106] (15)

[0107] Therefore, the pipeline loss function under hard constraints is The mean squared error of the network plus The sum of the network's loss functions is:

[0108] (16)

[0109] Where, are the weight coefficients corresponding to each loss function term, and the subscripts ic and bc represent the initial condition and boundary condition, respectively.

[0110] Step 4: In view of the high-order terms and strong nonlinear properties of the flow pipeline equation, internal observation anchor points are added to the network to correct problems such as insufficient peak values ​​generated by the neural network model during prediction.

[0111] It should be noted that the loss function after adding the anchor term in Step 4 is expressed as:

[0112] (17)

[0113] Therefore, the pipeline loss function with anchor point correction added under hard constraints is:

[0114] (18)

[0115] It is worth noting that the weight Typically set to 0 or 1, depending on the governing equation.

[0116] Step 5: Through Fourier feature mapping Function, decomposes the PINN network input data into spatial Fourier feature maps and temporal Fourier feature maps ;

[0117] It should be noted that the Fourier feature map in step 5 The functions are as follows:

[0118] (19)

[0119] Where, is the network input layer vector, including and , M is the number of Fourier feature maps, Represents a Gaussian distribution The mapping matrix obtained by sampling in, b represents the Fourier basis frequency vector, is a hyperparameter that specifies how often the network tends to learn, is the number of neurons in each layer of the PINN network, and d is the input dimension of the network.

[0120] It should be noted that after Fourier mapping, the input data in step 5 is decomposed into spatial Fourier feature maps and temporal Fourier feature maps It is expressed as follows:

[0121] (20)

[0122] (twenty one)

[0123] Step 6: Choose different The value is used to modify the Fourier basis frequency vector, thereby adjusting the neural tangent kernel of the network and controlling the frequency range of neural network learning;

[0124] It should be noted that: Fourier feature map In the function, Represents a Gaussian distribution The mapping matrix obtained by sampling in step 6 is changed The value is used to change the amplitude of the Gaussian distribution, thereby changing the Fourier basis frequency vector b in the mapping matrix to adjust the spectrum of NTK and ultimately control the frequency range of neural network learning. NTK is expressed as follows:

[0125] (twenty two)

[0126] Where, , is the input point data.

[0127] Step 7: Substitute the results of spatial and temporal Fourier feature mapping into the same PINN network for training, and connect the spatial and temporal hidden layers in the network through point multiplication to establish a FFNN network;

[0128] It should be noted that: in step 7, for the spatial Fourier feature map , the first layer and hidden layer of the network are:

[0129] (twenty three)

[0130] (twenty four)

[0131] For the temporal Fourier feature map , the first layer and hidden layer of the network are:

[0132] (25)

[0133] (26)

[0134] The spatial and temporal hidden layers are connected by dot products, that is:

[0135] (27)

[0136] After training, spatial output is obtained and time output , and use point-by-point multiplication to merge the two, and finally output the prediction result through a linear layer connection.

[0137] The final prediction result output by the linear layer is:

[0138] (28)

[0139] The following describes the implementation of the present invention in detail with reference to specific implementation cases:

[0140] In this case, the spatial domain is considered to be , the time domain is , by inputting the vector Different applications Initialized multiple Fourier feature embedding, based on the numerical solution characteristics of the pipeline vibration problem, takes ,Right now Mapping, taking ,Right now Mapping, decomposing it into spatial Fourier feature map and temporal Fourier feature maps , and then embed the spatial and temporal input coordinates respectively. After Fourier characterization, the embedded input is trained through a 3-layer fully connected neural network with 100 neurons per hidden layer. The network batch size is set to , , By choosing appropriate hyperparameters , controlling the neural network to learn vibrations in a specific frequency range, thereby accurately learning the relatively low frequency of the dynamic response of the flow pipeline vibration on a macroscopic time scale and the relatively high frequency on a microscopic scale, to solve the problem that PINN cannot accurately learn high-frequency signals.

[0141] In order to evaluate the accuracy of the results, a reference dataset was created using the Generalized Finite Difference Method (GFDM).

[0142] Firstly, the PINN algorithm is used to analyze the flow pipeline model. The short-term vibration prediction results obtained by learning and predicting in the time domain are compared with the GFDM numerical solution as shown in Figure 3. It can be seen that due to the nonlinear spatiotemporal and multi-scale nature of the pipeline vibration model, even with the application of hard constraints, the PINN model is still completely unable to learn the pipeline's vibration dynamic response. In sharp contrast, the FF-PINN model successfully solves the spectral deviation problem by applying separate Fourier feature embeddings to the input coordinates of space and time, and independently propagating them through the neural network one by one. ) stably learns the low-frequency components of the objective function and makes the high-frequency components converge faster. Comparing the simulation results with the reference data, it can be seen that the absolute error between the amplitude learned by FF-PINN in the entire space and time domain and the GFDM reference solution at high frequencies is only 10 -3 ,The simulation results are very close to the reference data, and the training and learning effect of ,the vibration frequency is good.

[0143] To further evaluate the accuracy of the results, we selected = 0.10, 0.25, 0.5 (dashed lines in Figure 3), plot the time domain Pipeline amplitude The time history diagram is shown in Figure 4. It can be seen that the FF-PINN simulation results are very close to the experimental data and can accurately learn the relatively low-frequency dynamic response of the flow pipeline vibration on the macro time scale and the relatively high-frequency dynamic response on the micro time scale, thus solving the problem that PINN cannot accurately learn high-frequency signals.

[0144] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware. Furthermore, the present application may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to magnetic disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0145] This application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the application. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate a device for implementing the functions specified in one or more processes in the flowchart and / or one or more blocks in the block diagram.

[0146] These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce a product including an instruction device that implements the functions specified in one or more processes in the flowchart and / or one or more boxes in the block diagram.

[0147] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, so that the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in one or more processes in the flowchart and / or one or more boxes in the block diagram.

[0148] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other manner. Any person skilled in the art may utilize the above-disclosed technical content to modify or modify the present invention into equivalent embodiments. However, any simple modifications, equivalent variations, and modifications to the above embodiments that do not depart from the technical content of the present invention and are based on the technical essence of the present invention remain within the scope of protection of the present invention.

[0149] This patent is not limited to the above-mentioned best embodiment. Anyone can derive various other forms of a flow pipeline vibration analysis method based on Fourier-characterized PINN under the inspiration of this patent. All equal changes and modifications made within the scope of the patent application of this invention should fall within the scope of this patent.

Claims

1. A method for analyzing fluid pipeline vibration based on Fourier-characterized PINN, characterized in that: The following steps are involved: Step 1: Introduce the residual network PDE of the flow pipeline equation and construct the corresponding PINN network; Step 2: According to the flow pipeline model, determine the initial conditions and boundary conditions, and obtain the corresponding loss function in the PINN network, where the initial conditions include the initial amplitude and initial excitation of the pipeline, and the boundary conditions include the first-class Dirichlet conditions and the second-class Neumann conditions; Step 3: Separately process the first and second boundary conditions and initial conditions in the pipeline problem, express the Neumann boundary conditions and pipeline initial excitation into the PINN loss function with "soft" constraints, constrain the boundary conditions by minimizing the loss function, encode the Dirichlet boundary conditions and initial amplitude into the DNN architecture with "hard" constraints, construct a function that satisfies the initial boundary value conditions, and impose hard constraints on the Dirichlet boundary conditions and initial amplitude; Step 4: In view of the high-order terms and strong nonlinear properties of the flow pipeline equation, internal observation anchor points are added to the network to correct the problem of insufficient peak values ​​generated by the neural network model during prediction; Step 5: Through Fourier feature mapping Function, decomposes the input data of the PINN network into spatial Fourier feature maps and the temporal Fourier feature map ； Step 6: By selecting different The value modifies the Fourier basis frequency vector and adjusts the neural tangent kernel of the network, thereby controlling the frequency range of neural network learning; Step 7: Substitute the results of spatial and temporal Fourier feature mapping into the same PINN network for training, and connect the spatial and temporal hidden layers in the network through point multiplication to establish Fourier features neural networks (FFNN); Step 8: Get spatial output after training and learning and time output , and use point-by-point multiplication to merge the two, and finally output the prediction result through a linear layer connection; In step 1, the residual network of the equation is introduced for the flow pipeline model The specific expression is as follows: ； ； In the formula, a is the input quantity, Indicates that the network can learn hyperparameters, including weights W and bias b, is the output vector, is the dimensionless component of the ratio of the mass of the fluid per unit length in the pipe to the total mass of the pipe, is the dimensionless component of the flow velocity in the pipe, is the output of the hidden layer of the neural network, and the superscript indicates the number of network layers; ； is an L-layer neural network with an input vector Including spatial coordinates in piping vibration models and time coordinates , the output vector is the pipeline amplitude , the network is a feed-forward neural network; In step 2, the loss function of the second type of Neumann boundary condition under soft constraints is: ； Its initial condition, i.e. the initial excitation loss function of the pipeline, is: ； Pipeline differential equations The mean square error of the network is: ； Therefore, the pipeline loss function under soft constraints is: ； In the formula, The number of training data for the Neumann boundary condition of the model when the support condition is fixed at both ends, is the number of initial incentive training, is the number of internal configuration points, is the neural network output, and are the boundary condition points and initial condition points used as network input, 、 is the actual value of the corresponding point; is passed to Configuration points of the network, parameters Corresponding to the weight coefficients in the loss function, effectively assigning different learning rates to each individual loss term; these weights can be specified by the user or automatically adjusted during network training; In step 3, the loss function of the first type of Dirichlet boundary condition under hard constraints is: ； Its initial condition, i.e. the initial amplitude loss function of the pipeline, is: ； Pipeline differential equations The mean square error of the network is: ； Therefore, the pipeline loss function under hard constraints is: FNN ； In the formula, the function of the special solution that satisfies the boundary conditions constructed by the "hard" constraint is: ， The number of training data for the Dirichlet boundary conditions of the model when the two ends are fixed support conditions, is the number of initial amplitude training, is the weight coefficient corresponding to each loss function term, and the subscripts ic and bc represent the initial condition and boundary condition respectively; In step 4, for the high-order terms and strong nonlinear properties of the flow pipeline equation, internal observation anchor points are added to the network, corresponding to the subscript domain, and the loss function of the anchor point term is: ； The pipeline loss function with anchor point correction term added under hard constraints is: ； In the formula, the weight Depending on the governing equation itself, it is set to 0 or 1; Step 5: Spatial Fourier feature mapping and the temporal Fourier feature map It is expressed as follows: ； ； In the formula, Represents a Gaussian distribution The mapping matrix obtained by sampling in represents the Fourier basis frequency vector, is a hyperparameter that specifies how often the network prefers to learn, is the number of neurons in each layer of the PINN network, and d is the input dimension of the network; The neural tangent kernel in step 6 is expressed as follows: ； In the formula, ， is the input point data; In step 7, for the spatial Fourier feature map , the first layer and hidden layer of the network are: ； ； For the temporal Fourier feature map , the first layer and hidden layer of the network are: ； ； The spatial and temporal hidden layers are combined as follows: ； In the formula, is the activation function, and the superscript in the formula represents the number of network layers.

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