LU600068B1

LU600068B1 - Vibration analysis method for fluid pipeline based on fourier characterization pinn

Info

Publication number: LU600068B1
Application number: LU600068A
Authority: LU
Inventors: Ting Zhang
Original assignee: Univ Fuzhou
Priority date: 2023-11-07
Filing date: 2024-07-22
Publication date: 2025-05-13
Also published as: WO2025007990A1; CN117272570B; CN117272570A; WO2025007990A9

Abstract

The present invention provides a vibration analysis method for fluid pipeline based on Fourier characterization PINN, which includes the following steps. Physical information neural network (PINN) is constructed, initial conditions and boundary conditions are determined according to a pipeline model. The second kction with "soft” constraint, and the first kind of boundary conditions and initial amplitude are encoded in DNN architecture with "hard" constraint. Meanwhile, observation anchor points are added to the network.

Description

DESCRIPTION

VIBRATION ANALYSIS METHOD FOR FLUID PIPELINE BASED ON FOURIER

CHARACTERIZATION PINN

TECHNICAL FIELD

The present invention falls within the technical field of pipeline vibration prediction, and particularly relates to a vibration analysis method for fluid pipeline based on Fourier characterization PINN.

BACKGROUND

Fluid pipelines have widespread applications in numerous fields. When fluid flows within a pipeline, it exerts forces on the pipeline, inducing vibrations that ultimately lead to long-term fatigue failure. Therefore, studying the dynamic response of flow-induced vibrations in pipelines during fluid conveyance is of practical significance. However, due to the presence of fluid-solid interaction (FSI) in the fluid pipelines, pipeline vibrations exhibit multi-scale and high-low frequency superposition characteristics, making the analysis of pipeline vibrations challenging. Hence, there is a need to develop an algorithm that can stably learn high-frequency information while accurately predicting governing equations with high-order nonlinear terms across spatio-temporal multi-scales, without increasing computational costs.

PINN leverages the universal approximation capabilities of neural networks by simultaneously minimizing the sum of data fitting errors and partial differential equation residuals to train the neural network parameters, effectively solving PDEs (partial differential equations) with high-order nonlinear terms. However, the fully connected architecture commonly used in the PINN suffers from a spectral bias issue, causing the network to tend to learn low-frequency components during training and ignore high- frequency components, making it difficult to ensure correct learning of high-frequency functions. When the solutions to partial differential equations exhibit high-frequency and multi-scale characteristics, the PINN often struggles to train stably and fail to produce 099068 accurate predictions.

SUMMERY

In view of the problem of spectrum deviation and insufficient learning ability of high- frequency information in the traditional deep fully connected network, a vibration analysis method for fluid pipeline based on Fourier characterization PINN provided by the present invention is an effective solution, which slows down the rapid decline of NTK in the network caused by the increase of frequency, reduces the attenuation degree of characteristic values with the increase of frequency, and improves the characteristic values of high-frequency information, improving the convergence speed of high- frequency information training and enhancing the ability of the fully connected network to learn high-frequency functions.

The present invention provides the vibration analysis method for fluid pipeline based on Fourier characterization PINN. An object of the present invention is to establish a numerical algorithm that can accurately and effectively solve a fluid pipeline model, so that the algorithm has strong high-frequency learning ability and is applied to predict the change of pipeline vibration.

The solution includes constructing physical information neural network (PINN), determining the initial conditions and boundary conditions according to the pipeline model, and obtaining their corresponding loss functions; The second kind of boundary conditions and initial excitation of pipeline are expressed in PINN loss function with "soft" constraint, and the first kind of boundary conditions and initial amplitude are encoded in DNN architecture with "hard" constraint. At the same time, observation anchor points are added to the network to correct when there is insufficient peak value in the network model training process. The y function of Fourier characteristic map is constructed, and the data input by PINN feedforward neural network is decomposed into spatial Fourier characteristic map ya(&) and time Fourier characteristic map yy(7). By choosing different o values to modify the Fourier basis frequency vector and adjust the neural tangent kernel of the network, the frequency range of neural network learning is controlled. After the

. . . LU600068 above processing, the two mapping results are trained through the set network respectively, and finally the hidden layers of space and time are connected through point multiplication in the network to establish FFNN network; Space output z{, and time output z“} are obtained through training and learning, and they are merged by point-by- point multiplication. Finally, the prediction results are output through linear layer connection.

To solve the above technical problems, the present invention adopts the following technical solutions.

The vibration analysis method for fluid pipeline based on Fourier characterization

PINN includes the following steps:

Step 1, introducing a residual network PDE of a fluid pipeline equation to construct corresponding PINN;

Step 2, determining initial conditions and boundary conditions according to a fluid pipeline model, and obtaining a corresponding loss function in the PINN, where the initial conditions include initial amplitude and initial excitation of a pipeline, and the boundary conditions include the first Dirichlet condition and the second Neumann condition:

Step 3, treating the first and second kinds of the boundary conditions and the initial conditions in a pipeline problem separately, expressing Neumann boundary conditions and pipeline initial excitation in a PINN loss function with "soft" constraint to constrain the boundary conditions by minimizing the loss function; and encoding Dirichlet boundary conditions and the initial amplitude into DNN architecture with "hard" constraint, and constructing a function satisfying a special solution of initial boundary conditions to hard constrain the Dirichlet boundary conditions and the initial amplitude;

Step 4: adding internal observation anchor points in terms of high-order terms and strong nonlinearity of the fluid pipeline equation to correct a problem of insufficient peak value in the prediction of a neural network model;

Step 5: decomposing input data of the PINN into spatial Fourier characteristic mapping yo(€) and time Fourier characteristic mapping yo) by a Fourier characteristic

; ; LU600068 mapping y function;

Step 6: modifying a Fourier basis frequency vector by selecting different 0 values, adjusting a neural tangent kernel of the network, and controlling a frequency range of neural network learning;

Step 7: substituting the results of Fourier characteristic mapping of space and time into the same PINN for training, and connecting hidden layers of space and time through point multiplication in the network to establish a FFNN network; and

Step 8: obtaining space output z, and time output through z‘? training and learning and combining the space output and the time output, and outputting the prediction results through linear layer connection.

Further, in Step1, an equation residual network PDE (€ ,7) is introduced into the fluid pipeline model, and an specific expression is as follows: 4 1 2 2 2

PDE: (£7) > 0 mo (S-7) +2ß2u on, ($,T) + u? 0 7.7) + 0 (5.7)

DE 0&0 o& or no (Et) = zz =W at) 4 bt

Where a is an input quantity, 6 represents a learnable superparameter of the network; the network contains weight W and deviation b, ne (€, 7) is an output vector, B is a dimensionless component of a mass ratio of the fluid in the pipe per unit length, U is a dimensionless component of the flow velocity in the pipe; and superscript represents the number of layers of the network; and ne (n, 7) is an L-layer neural network, in which input vector a (€ 7)" includes a space coordinate € and a time coordinate 7 in a pipeline vibration model, an output vector is a pipeline amplitude n, and this network is a feedforward neural network.

Further, in Step2, the loss function of the second kind of Neumann boundary conditions under soft constraint is 1 & 0) Wy

Le =— In, > Te )- nés >The )

Na i=1

Co i. . Ce oo ; ; LU600068

The initial condition, that is, a pipeline initial excitation loss function, is:

I< @ OD @ „61

L, = mE >The )-n(. Fr ) ie i=l

A mean square error of a f network of a pipeline differential equation is: 1 Nppg ; ;

Lppp = No >, |PDE : (Ete)

PDE i=l

The pipeline loss function under soft constraint is:

L=24,L, + Âge Pre + ZppgLppg where Nec is the number of training data for the Neumann boundary condition of the model when the two ends are fixed, Nic is the number of initial excitation training, NPoe is the number of internal configuration points, ne is an output of the neural network, (€°,1.) and (#9 7”) is a boundary condition point and initial condition point input by the network, and 7”. 7) nE,7”) is an actual value of the corresponding point; (&X.,70.) is a configuration point transferred to PDE(€ ,7) network, and parameters {i ,4, Appp} correspond to the weight coefficient in the loss function, to effectively assign different learning rates to each individual loss term; and these weights are allowed to be specified by users or adjusted automatically during network training.

Further, in Step3, the loss function of the first kind of Dirichlet boundary conditions under hard constraint is: ar 1 = 0 pod i i 0 i i :

Li “ = Ea) ce Mec)

The initial condition, that is, the pipeline initial amplitude loss function, is: ar 1 = 0 pod i i 0 i i :

Li “ = Ea) ce Mec)

A mean square error of PDE (€, 7) network of the pipeline differential equation is:

, LU600068 pret __À pur (Erbe Erbe) LE Om ; a

Nom 5 dé OéÔr dé or

The pipeline loss function under soft constraint is:

J rard — AL + Ag Lie + AL

In this equation the function constructed with "hard" constraint and satisfying the special solution of boundary conditions is no (ET) =n, ET) 7-6-1) , Nec is the number of training data of Dirichlet boundary conditions of the model under the condition of two- end fixed support, Nic is the number of initial amplitude training, A is the weight coefficient corresponding to each loss function term, and subscripts ic and bc respectively represent the initial conditions.

Further, in Step 4, according to the high-order term and strong nonlinearity of the fluid pipeline equation, an internal observation anchor point is added to the network, corresponding to the subscript domain, and the loss function of the anchor point term is: hard 1 "er hard ( #(i) ©) ©) ©) 2

Lorman = > I} (S domain > domain) 7 NS domain 7 domain )

N domain i=1

The pipeline loss function with anchor correction under hard constraint is:

Is = A Le + Ao Li? + Âppe Lips + À roman Lm where the weight À domain depends on the control equation itself and is set to 0 or 1.

Further, in Step 5, the spatial Fourier characteristic map ya(€) and the temporal

Fourier characteristic map ya(7) are expressed as follows: cos(2zB’.&)

Yo) =| © 4=12,.,M, sin(27B/,5) cos(2zB’. 7) yo@=| VL J=L200M, sin(2zB 7) where B=[b,b... ,b,] el 2 represents the mapping matrix sampled from the

Gaussian distribution N(0,0*), bm represents the Fourier basis frequency vector, 0 >0 is a i . ; LU600068 hyperparameter that specifies the learning frequency of the network, m is the number of neurons in each layer of the PINN, and d is the input dimension of the network.

Further, in Step 6, the neural tangent kernel of the network is expressed as follows:

K,(yv2)=7 (7 y y(v,)= > cos(27b] (v =v, )) 7 where "| and ”> are input point data.

Further, in Step 7, for spatial Fourier characteristic mapping y(i)(€), the first layer and hidden layer of the network are respectively: zZ = GW py (6) +60) 1 =1,2,...M. zZ =p 22) +50) 1 =1,2,..,M,,1=2,..,1

For the time Fourier characteristic map yo(7), the first layer and hidden layer of the network are respectively: z) = 6(W“ V+ pV), j=12,...M =p 279 +60), j=1,2,.,M,,1=2,..,L

The hidden layers of space and time are merged by the following equation: zZ = 22/0 zz ‚7 TJ where ¢ is the activation function, and the superscript in the equation indicates the number of network layers.

Compared with the prior art, the present invention and its preferred solutions overcome the deficiency of fully connected neural network in learning and predicting high- frequency functions, and can accurately capture high-frequency oscillation aiming at a fluid pipeline model by combining the utilization of physical information by PINN, to better learn and predict the vibration of the pipeline.

BRIEF DESCRIPTION OF THE DRAWINGS

. . . . . oo LU600068

The present invention will be further described in detail with reference to the attached drawings and specific implementation.

FIG. 1 is a method flowchart of examples according to the present invention;

FIG. 2 is a schematic diagram of a structural framework of examples according to the present invention;

FIG. 3 is a cloud diagram comparing a numerical solution of GFDM with predicted solutions of PINN and FF-PINN of examples according to the present invention;

FIG. 4 is a comparison diagram of a numerical solution of GFDM and the predicted amplitude n of PINN and FF-PINN of examples according to the present invention.

DETAILED DESCRIPTION

In order to make the characteristics and advantages of this patent more obvious and understandable, the following examples are given in detail as follows:

It is to be pointed out that the following detailed description is exemplary and is intended to provide further explanation for this application. Unless otherwise specified, all technical and scientific terms used in this specification have the same meaning as commonly understood by ordinary technicians in the technical field to which this application belongs.

It is to be noted that the terminology used here is only for describing specific implementation, and is not intended to limit exemplary implementation according to the present invention. As used herein, the singular form is also intended to include the plural form unless the context clearly indicates otherwise. Furthermore, it is to be understood that when the terms "comprising" and/or "including" are used in this specification, they specify the presence of characteristics, steps, operations, devices, components and/or combinations thereof.

As shown in FIGS 1-4, the present invention provides a vibration analysis method for fluid pipeline based on Fourier characterization PINN, including the following steps.

In Step 1, a residual network PDE of a fluid pipeline equation was introduced to construct corresponding PINN.

. 4 ; ; ; ; ; LU600068

The model of the fluid pipeline selected in the present invention is an improved equation of the classical Euler-Bernoulli beam model.

Consider a straight pipe with fixed supports at both ends. The length of the pipe is L, with x axis along the central axis of the pipe and y axis along the transverse direction of the pipe. Ignoring the influence of gravity, internal damping and fluid pressure, and not considering the influence of temperature load and its axial tension, the differential equation of lateral motion may be expressed as: 0"y oy 2 0% oy _

Elo 2m Uo PAU az OM EMI =0 (1) where E is the elastic modulus of the pipeline, / is the moment of inertia of the cross section of the pipeline, mr and mp are the mass of the fluid in the pipeline per unit length and the mass of the pipeline, respectively, and U is the fluid flow rate.

An expression of pipeline boundary condition is: oy(0,1) Qy(1,1) 0,1) =0, (11) = 0,———7 = 0,+——— = 0 y(0,1)=0, y(1,1)=0, p= = (2)

A equation is dimensionless, and the differential equation of lateral motion of straight pipe with fixed ends is obtained: 4 1 2 2 2

ON Bu 1 7 p20 dé déôr 08? or (3)

Similarly, corresponding dimensionless boundary conditions and initial conditions with fixed ends may be written as follows: n(0,7)=0.n(1,7) = 0,910.7) = 0.0.7) =0 oc oc (4)

On(0,7) ; n(#,0)= 0, —— = 0.01sin(7é) or (5)

It is to be noted that the network ne (€, T) in PINN established in Step1 is an L-layer neural network, and its input vector a‘) (€ ,T)T includes the spatial coordinate & and the time coordinate T in the pipeline vibration model, and its output vector is the pipeline amplitude n. Because the network is a feedforward neural network, that is, each layer

. ; LU600068 creates data for the next layer through the following nested transformation: 70 — p(w a + 6%) (6) = WO. a 4+b0),1=2,..,L (7)

An expression of the output vector is:

No (&.7) = 7) WO at 4 pr (8) where a is the input quantity, © represents the learnable hyperparameter of the network, which contains the weight W and the deviation b, ¢ represents the activation function, superscript represents the number of network layers, and ne (€, T) is the output vector.

Further, an equation residual network PDE (€, 7) is introduced, that is 4 1 2 2 2

PDE: (&,7) > 0 ny(&.7) YE ny (&.7) +u? 0 n,(&.7) + 0 n,(é,T)

BE 0&0 0&? or? (9)

In Step 2, initial conditions and boundary conditions are determined according to a fluid pipeline model, and a corresponding loss function is obtained in the PINN, where the initial conditions include initial amplitude and initial excitation of a pipeline, and the boundary conditions include the first Dirichlet condition and the second Neumann condition.

It is to be noted that the loss function corresponding to the initial condition in Step 2 is:

LEE HE 0.0)

N. = 1 1 1 1 N. = or 2 2 or 2 2 ic icy (10)

A loss function corresponding to the boundary condition is:

Ya 4 4 1 “la La On, al

I = El ap mat + Sad) Mer)

N,, = Ne 1106 og (11)

Where N,, and N, are the number of training data for the first and second boundary conditions of the model when the two ends are fixed and supported,

respectively. N,, and N, are the initial amplitude and the initial excitation training number, respectively. Where n, is an output of neural network, and ue à) and

MES.) are the boundary condition points and initial condition points as an input of n network. m(é2.r;)) hen ‚ED 72) and FEE are the actual values at these points.

In Step 3, the first and second kinds of the boundary conditions and the initial conditions in a pipeline problem are treated separately, Neumann boundary conditions and pipeline initial excitation are expressed in a PINN loss function with "soft" constraint to constrain the boundary conditions by minimizing the loss function; and Dirichlet boundary conditions and the initial amplitude are encoded into DNN architecture with "hard" constraint, and a function satisfying a special solution of initial boundary conditions is constructed to hard constrain the Dirichlet boundary conditions and the initial amplitude.

It is to be noted that the specific expression of the function constructed by the "hard" constraint in Step 3 that satisfies the special solution of the boundary condition is as follows: ny" (ET) =n, 7) EE) (12) where n"(5, r) is the corresponding condition that is automatically satisfied when

Dirichlet boundary condition and initial amplitude of pipeline are: € = 0,1 and 1 = 0.

Under the hard constraint, the loss function of the boundary condition is modified to contain only Neumann boundary condition, that is, the loss function of the boundary condition is:

N,, hard 2

MS

N,. i=1 og 05 (13)

Similarly, its initial conditional loss function is modified to contain only initial excitation: 2 hard ls on," (EV 19) AM (zo 79) ic N, = or ie ? ie or ic ? Tic (14)

A mean square error of PDE (€ ,7) network of the pipeline differential equation is:

N 4 hard (i) (i) 1 2 hard 2 hard 2 hard 2

Lee _ 1 >“ No (Eros ppp) +2ß?u 0 ur +u’ 0 To. + 0 To.

Nee À 05 OSOT og or (15)

Therefore, the pipeline loss function under hard constraint is the sum of the mean square error of n network plus the loss function of PDE (€ ,7) network, namely:

LD = AL + 2 Ly + Appe Lp (16) where A is the weight coefficient corresponding to each loss function term, and the subscripts ic and bc respectively represent the initial condition and the boundary condition.

In Step 4, in terms of high-order terms and strong nonlinearity of the fluid pipeline equation, internal observation anchor points are added to the network to correct the problem of insufficient peak value in the prediction of the neural network model.

It is to be noted that the loss function after adding the anchor item in Step 4 is expressed as: hard 1 "er hard ( #(i) ©) ©) ©) 2

Lorman = > I} (S domain > domain) 7 NS domain 7 domain )

N domain i=1 (17)

The pipeline loss function with anchor correction under hard constraint is: ree = Ae Le + Ap Ly? + Appr Lis + toma Loan (1 8)

Where the weight Adomain depends on the control equation itself and is set to 0 or 1.

Step 5: input data of the PINN are decomposing into spatial Fourier characteristic mapping yo(€) and time Fourier characteristic mapping yo) by a Fourier characteristic mapping y function.

It is to be noted that the gamma function of Fourier characteristic mapping y in Step5 is as follows: cos(2zB a")

Yo =| — J=12,...M sin2zB a) (19)

Where a is the input layer vector of the network, including € and 7, M is the number of Fourier characteristic mapping, B=[b,.5,,... ,b,]" eL 2 represents the mapping matrix sampled from the Gaussian distribution N(0 ,0*), b is the Fourier base frequency vector, co >0 is the hyperparameter that specifies the learning frequency of the network, m is the number of neurons in each layer of the PINN, and d is the input dimension of the network.

It is to be noted that after Fourier mapping, the spatial Fourier characteristic map vo(&) and the temporal Fourier characteristic map ya(7) into which the input data is decomposed in Step5 are expressed as follows: cos(2zB’.&) re] pe. eva sin(27B/,$) (20) cos(2zB’. 7) wie I. rman sin(2zB 7) (21)

Step 6: a Fourier basis frequency vector is modified by selecting different 0 values, a neural tangent kernel of the network is adjusted, and a frequency range of neural network learning is controlled;

It is to be noted that in the gamma function of Fourier characteristic map y,

B=[b,,b… ,b,] el 7° represents Gaussian distribution. For the mapping matrix sampled in N(0,02). In Step 6, the amplitude of Gaussian distribution is changed by changing the o value, changing the Fourier base frequency vector b in the mapping matrix to adjust the frequency spectrum of NTK, and finally controlling the frequency range of neural network learning.

Where the NTK is expressed as follows:

K, (vw) = y(n y 7(v,) = 3 cos 2b! (5 —v,)) / (22)

Where Yı and "> are input point data.

In Step 7, the results of Fourier characteristic mapping of space and time are substituted into the same PINN for training, and hidden layers of space and time are connected through point multiplication in the network to establish a FFNN; and

4: ; ; ; yr ; LU600068 it is to be noted that in Step 7, for spatial Fourier characteristic mapping yw(é), the first layer and hidden layer of the network are: zZ = GW y, (6) +60) 1 =1,2,...M. (23) zZ =p 22) +50) 1 =1,2,..,M,,1=2,..,1 (24)

For the time Fourier characteristic map yo(7), the first layer and hidden layer of the network are respectively: zy, = GW Ya) +0), j =1,2,..,M, (25) 0) _ 0, 0-0 dy 7 —

ZO = GW 279 +60), j=1,2,.,M,,1=2,..,L (26)

Connect the hidden layers of space and time by point multiplication, namely: (L) _ 4 (L) (L) zZ =z, Uz (27)

In Step 8, space output z“, and time output z;/ are obtained through training and learning, and they are combined by point-by-point multiplication, and finally the prediction results are output through linear layer connection.

Finally, the prediction result output through the linear layer is:

Ne ($,7) =W [22.20 u, | +60 (28)

The implementation of the present invention will be described in detail with specific cases as follows: in this case, the space domain is £e[0,1] and the time domain is re[0,15]. By embedding the input vector v=(£,7) with multiple Fourier characteristics initialized with different 0, according to the numerical solution characteristics of the vibration problem of the pipeline. When M, =1 is taken in the space direction and that is o° =[1] mapping, and M_=2 is taken in the time direction and that is co” =[1,30] mapping, it is decomposed into spatial Fourier characteristic map yw(&) and temporal Fourier characteristic map yga(7), and then the spatial and temporal input coordinates gis 090068 embedded respectively. After Fourier characterization, the embedded input is trained and learned through a three-layer fully connected neural network with 100 neurons in each hidden layer. The network batch size is set to N,=4000, N, =320 and N, =80. By selecting the appropriate hyperparameter o, the neural network is controlled to learn the vibration in a specific frequency range, to accurately learn the relatively low frequency of the dynamic response of the vibration of the pipeline on the macro time scale and the relatively high frequency on the micro scale, to solve the problem that the PINN is unable to accurately learn the high frequency signal.

In order to evaluate the accuracy of the results, the generalized finite difference method (GFDM) is used to create a reference data set.

Firstly, PINN algorithm is configured to predict the fluid pipeline model in ze[0,1] time domain, and the comparison between the short-time domain vibration prediction result and GFDM numerical solution is shown in FIG. 3. It may be seen that, due to the nonlinear time-space multi-scale nature of the pipeline vibration model, the PINN is still unable to learn the dynamic response of the pipeline vibration even if the hard constraint method is applied. In contrast, the FF-PINN model successfully solves the problem of spectrum deviation by embedding Fourier characteristics into the input coordinates of space and time and propagating them independently through neural networks one by one.

It not only learns the low-frequency components of the objective function stably for a long time ( 7e[0,15] ), but also makes the high-frequency components converge faster.

Comparing the simulation results with the reference data, it may be seen that the absolute error of the amplitude obtained by FF-PINN in the whole space and time domain is only 107 at the highest frequency of GFDM reference solution, and the simulation results are very close to the reference data, which has a good training and learning effect on vibration frequency.

In order to further evaluate the accuracy of the results, three positions (¢ = 0.10, 0.25 and 0.5) are selected (dashed line in Figure 3), and the time-history diagram of pipeline amplitude 1€[12,15] is drawn, as shown in Figure 4. It may be seen that the simulation 99958 results of FF-PINN are very close to the experimental data, which may accurately learn the relative low frequency of the dynamic response of the pipeline vibration on the macro time scale and the relative high frequency on the micro scale, to solve the problem that the PINN may not accurately learn the high frequency signal.

It is to be understood by those skilled in the art that examples of the present invention may be provided as a method, a system, or a computer program product. Therefore, the present invention may take the form of an entirely hardware example, an entirely software embodiment, or a practical example combining software and hardware aspects. Moreover, the present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage,

CD-ROM, optical storage, etc.) containing computer-usable program codes.

The present invention is described with reference to flowcharts and/or block diagrams of methods, devices (systems), and computer program products according to examples of the present invention. It is to be understood that each process and/or block in the flowcharts and/or block diagrams, and combinations of processes and/or blocks in the flowcharts and/or block diagrams may be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor or other programmable data processing apparatus to produce a machine, such that the instructions, which are executed by the processor of the computer or other programmable data processing apparatus, produce means for implementing the functions specified in one or more flow charts and/or block diagrams.

These computer program instructions may also be stored in a computer-readable memory that may direct a computer or other programmable data processing apparatus to operate in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means that implement the functions specified in one or more flow charts and/or block diagrams.

These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus, such that a series of operational steps gis 090068 performed on the computer or other programmable apparatus to produce a computer- implemented process, such that the instructions executed on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block or blocks in the block diagram.

What has been described above is only a preferred example of the present invention, and it is not a restriction to the present invention in other forms. Any person familiar with this field may make use of the technical contents provided above to change or modify it into an equivalent example. However, any simple modification, equivalent change and modification made to the above examples according to the technical essence of the present invention without departing from the technical scheme of the present invention still falls with the protection scope of the technical scheme of the present invention.

This patent is not limited to the above-mentioned best mode. Under the inspiration of this patent, anyone may come up with other forms of the vibration analysis method for fluid pipeline based on Fourier characterization PINN, and all the equal changes and modifications made according to the patent application scope of the present invention is be to fall with the scope of this patent.

Claims

1. À vibration analysis method for fluid pipeline based on Fourier characterization PINN comprising the following steps: Step 1, introducing a residual network PDE of a fluid pipeline equation to construct corresponding PINN; Step 2, determining initial conditions and boundary conditions according to a fluid pipeline model, and obtaining a corresponding loss function in the PINN, wherein the initial conditions include initial amplitude and initial excitation of a pipeline, and the boundary conditions include the first Dirichlet condition and the second Neumann condition; Step 3, treating the first and second kinds of the boundary conditions and the initial conditions in a pipeline problem separately, expressing Neumann boundary conditions and pipeline initial excitation in a PINN loss function with "soft" constraint to constrain the boundary conditions by minimizing the loss function; and encoding Dirichlet boundary conditions and the initial amplitude into DNN architecture with "hard" constraint, and constructing a function satisfying a special solution of initial boundary conditions to hard constrain the Dirichlet boundary conditions and the initial amplitude; Step 4: adding internal observation anchor points in terms of high-order terms and strong nonlinearity of the fluid pipeline equation to correct a problem of insufficient peak value in the prediction of a neural network model; Step 5: decomposing input data of the PINN into spatial Fourier characteristic mapping yo(€) and time Fourier characteristic mapping yo) by a Fourier characteristic mapping y function; Step 6: modifying a Fourier basis frequency vector by selecting different 0 values, adjusting a neural tangent kernel of the network, and controlling a frequency range of neural network learning; Step 7: substituting the results of Fourier characteristic mapping of space and time into the same PINN for training, and connecting hidden layers of space and time through point multiplication in the network to establish Fourier features neural networks (FFNN); and Step 8: obtaining space output z, and time output Zz} through training and learning and combining the space output and the time output, and outputting tHe/600068 prediction results through linear layer connection; wherein in Step1, an equation residual network PDE(E ,7) is introduced into the fluid pipeline model, and an specific expression is as follows: 4 1 2 2 2 PDE: (&,7) > mer) 76.7) + 2p (5-7) 76.7) +2 PME) (6.7) + anos.) (57) DE 0&0 o& or n,(E, 0) =z" =W at) 4 bt wherein a is an input quantity, 6 represents a learnable superparameter of the network; the network contains weight w and deviation b, ne (€, 7) is an output vector, ß is a dimensionless component of a mass ratio of the fluid in the pipe per unit length, u is a dimensionless component of the flow velocity in the pipe; and z is the output of the hidden layer of the neural network, and the superscript indicates the number of layers of the network; . . 4: © ro Ne (nN, T) is an L-layer neural network, in which input vector “ (57) includes a space coordinate n and a time coordinate T in a pipeline vibration model, an output vector is a pipeline amplitude n, and this network is a feedforward neural network; in Step2, the loss function of the second kind of Neumann boundary conditions under soft constraint is 1 & 0) Wy Le =— In, > Te )- nés >The ) Na i=1 the initial condition, that is, a pipeline initial excitation loss function, is: I & 0 0) @ „61 L, =— In, (£8 Tr )- nés re ) N, i=1 a mean square error of a / network of a pipeline differential equation is: 1 = 0 0 5 Lepy =— > |PDE : GA Nope i=1 the pipeline loss function under soft constraint is: L=2,L,+A4.Ly + Appr Lops wherein Nec is the number of training data for the Neumann boundary condition of the model when the two ends are fixed, Nic is the number of initial excitation training, Neos is the number of internal configuration points, ne is an output of the neural network/600068 OC) OC) (G>T) and (Si >": ) is a boundary condition point and initial condition point input by the @ 40) @ 40) network, and nT) MG 7) is an actual value of the corresponding point; @ @ (eerie) js a configuration point transferred to PDE(§ ,7) network, and parameters ios Ayo Appi § correspond to the weight coefficient in the loss function, to effectively assign different learning rates to each individual loss term; and these weights are allowed to be specified by users or adjusted automatically during network training; in Step3, the loss function of the first kind of Dirichlet boundary conditions under hard constraint is: 1 Nb onl on 2 BEE the initial condition, that is, the pipeline initial amplitude loss function, is: 1 Ne ont on 2 [hard _ 6 570) Ze 70 ic N. > or (CC Tr ) or (CC > Tie ) a mean square error of PDE(& , 1) network of the pipeline differential equation is: pri _ 1 = onl (ED TO) . DB One or on . onl 2 EON, 5 DE“ OéÔr dE? or? a pipeline loss function under hard constraint is: FNN J'erd — AL + Ag Li + A De wherein the function constructed with "hard" constraint and satisfying the special solution of boundary conditions is ne (ET) EME TEEN Nec is the number of training data of Dirichlet boundary conditions of the model under the condition of two-end fixed support, Nic is the number of initial amplitude training, A is the weight coefficient corresponding to each loss function term, and subscripts ic and bc respectively represent the initial conditions; in Step 4, according to the high-order term and strong nonlinearity of the fluid pipeline equation, an internal observation anchor point is added to the network, corresponding to the subscript domain, and the loss function of the anchor point term is:

1 Neer LU600068 L'an Tr > ri (Eoin Toman) 7 ME main» domain ) N domain i=1 the pipeline loss function with anchor correction under hard constraint is: re = Ae Le + Âge LE + Appi Libs + À romain Lion wherein the weight Adomain depends on the control equation itself and is set to 0 or 1; in Step5, the spatial Fourier characteristic map yo(é) and the temporal Fourier characteristic map yo(7) are expressed as follows: cos(27 B5 €) Yo) =| © 1=12,.,M. sin(27B/,5) cos(2x B/.T) = VD j=12,.M, sin(27 B/,,T) wherein B=[b.b,...b |" el 7 represents the mapping matrix sampled from the Gaussian distribution N(0,02), bm represents the Fourier basis frequency vector, 0 >0 is a hyperparameter that specifies the learning frequency of the network, m is the number of neurons in each layer of the PINN, and d is the input dimension of the network; in Step 6, the neural tangent kernel of the network is expressed as follows: K,(y.v.)=7(7, )" y(v,)= > cos(27b] (y —v, )) 7 wherein V1 and ”z are input point data; and in Step 7, for spatial Fourier characteristic mapping ya(€), the first layer and hidden layer of the network are respectively: zZ = GW py (6) +60) 1 =1,2,...M. =p? +") 7=1,2,.,M,,1=2,.,L wherein for the time Fourier characteristic map yy(7), the first layer and hidden layer of the network are respectively: z) = 6(W“ V+ pV), j=12,...M 20 = 6WO U +60), j=1,2,.,M,,1=2,..,L the hidden layers of space and time are merged by the following equation: LU600068 ZB _ 7) 7) 7) ét TJ wherein © is the activation function, and the superscript in the equation indicates the number of network layers.