CN114692523A - Flow velocity prediction method of self-adaptive high-dimensional hydrodynamic equation based on graph convolution - Google Patents

Flow velocity prediction method of self-adaptive high-dimensional hydrodynamic equation based on graph convolution Download PDF

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CN114692523A
CN114692523A CN202210300040.2A CN202210300040A CN114692523A CN 114692523 A CN114692523 A CN 114692523A CN 202210300040 A CN202210300040 A CN 202210300040A CN 114692523 A CN114692523 A CN 114692523A
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聂婕
时津津
宋宁
赵恩源
王瑞
耿浩冉
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Ocean University of China
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Abstract

The invention belongs to the technical field of fluid dynamic field prediction and discloses a flow velocity prediction method of a self-adaptive high-dimensional fluid dynamic equation based on graph convolution, which comprises the steps of firstly establishing a discretization grid by utilizing a high-dimensional fluid dynamic equation space structure model of a graph convolution neural network and establishing a self-adaptive space structure relation for each grid point in the grid by utilizing the graph convolution neural network; then, based on a finite difference differential equation numerical method, through spatial structure model calculation, predicting and outputting spatial structure coefficients of each grid point; further calculating a spatial derivative; and finally, obtaining a final numerical solution of the high-dimensional hydrodynamic equation based on a physical solving process of the high-dimensional hydrodynamic equation.

Description

Flow velocity prediction method of self-adaptive high-dimensional hydrodynamic equation based on graph convolution
Technical Field
The invention belongs to the technical field of fluid dynamic field prediction, and particularly relates to a flow velocity prediction method of a self-adaptive high-dimensional fluid dynamic equation based on graph convolution.
Background
In the prediction of the fluid dynamic field, the discretization of the fluid dynamic partial differential equation is generally used for solving to obtain an approximate value. The key process of discretization of partial differential equations is to calculate prediction coefficients, and methods such as finite difference or finite volume are mainly used. The conventional method mainly faces the problems that the rough grid division can not obtain an accurate solution and the calculation speed is slow, because the conventional method depends on the size of the grid, and when the division is very fine, the calculation time is prolonged and the calculation speed is slow. Therefore, a data-driven discretization method that systematically derives the discretization of a continuous physical system using machine learning has been proposed. The method has the advantages that the convolutional neural network is used for replacing the computation of the finite difference coefficient, the rest steps are still performed by using the traditional method, the convolutional neural network is integrated into the learning process of the user embedding, and compared with a method without considering comment information, the obtained representation improves the generalization of the user embedding. However, this method still has the following problems:
first, a structured representation of the velocity node association cannot be established (structural relationship missing). Because the method only considers the influence of a single node on the coefficient, the structural relationship between the nodes is not mined. For example, the current method only considers the speed information of the nodes when calculating the coefficients by using the neural network, ignores the relationship between the nodes, and not only lacks rationality of a space mapping mechanism, but also reduces the generalization of the representation.
Secondly, the method only provides a low-dimensional space structure modeling scheme, so that the method is applied to a low-dimensional fluid dynamic equation and cannot solve the problem of a complex space structure of a high-dimensional fluid dynamic equation.
Therefore, in order to solve the above problems, the present invention provides a flow velocity prediction method based on an adaptive high-dimensional fluid dynamic equation of graph convolution.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a flow velocity prediction method of a graph convolution-based self-adaptive high-dimensional hydrodynamic equation, wherein a spatial structure relation is established for each grid point velocity value through a convolution neural network, and then the velocity value at the next moment is obtained through prediction.
In order to solve the technical problems, the invention adopts the technical scheme that:
the flow velocity prediction method of the self-adaptive high-dimensional hydrodynamic equation based on graph convolution comprises the following steps:
s1, node x of fluidnInputting the coarse-grained speed function value into a high-dimensional fluid dynamic equation space structure model of a graph convolution neural network, and constructing an adaptive space structure relation for each grid point in the grid by establishing a discretization grid by using the graph convolution neural network;
s2, predicting and outputting the space structure coefficient of each grid point by a finite difference-based differential equation numerical method through space structure model calculation
Figure BDA0003565034540000021
S3, any grid point x obtained by predicting step S2nCoefficient of spatial structure of
Figure BDA0003565034540000022
And xnX of surrounding pointsn-k,…,xn+kThe coarseness values are combined, i.e. xnPoint and xnK points on the left and k points on the right are adjacent, and x is calculatednSpatial derivatives of (i.e. x)nProcessing the solution of the velocity value after the spatial evolution;
s4, physical solving process based on high-dimensional fluid dynamic equation, namely calculating flux and then calculating x by utilizing the fluxnTime derivative of (i.e. x)nAnd (4) solving the velocity value after the velocity value evolves along with time to finally obtain the final numerical solution of the high-dimensional fluid dynamic equation.
Further, when the adaptive spatial structure relationship is constructed in step S1, the feature information of each node itself is obtained, and the feature information of the node is obtained from all neighboring nodes of each node, so as to construct a graph structure relationship, including:
2) construction of graph structure relationships for one-dimensional fluid dynamic equations
The one-dimensional spatial equation for Burgers is as follows:
Figure BDA0003565034540000023
let flux
Figure BDA0003565034540000024
Known training data set has T1A time layer, each layer having M1A number of grid points, each having speed information thereon, M1Down sampling at each grid point to obtain m1A plurality of grid points; when training the model, firstly, the m of the first layer is1The speed values of the grid points are used as input, loss is obtained by calculating MAE through the true solution of the speed value of the next time layer obtained through training and the speed value of the next time layer, and when the layer is trained, the next layer is trained;
by constructing a graph structure of a discretization grid, taking spatial grid points as vertexes of the graph, designing relationship weights between every two different grid points as edges of the graph, and establishing spatial structure relationships of all the grid points; the method comprises the following specific steps: when establishing spatial structure relationship, all m of each layer1Establishing a spatial structure relationship at each grid point, wherein the velocity value of each grid point is represented by the velocity values of 2k grid points around the upper layer and the velocity value at the grid point, each grid point has velocity information, and thus the input feature matrix X is obtained from the velocity of each grid point, and the shape of the X is (m)11); since each grid point is represented by 2k points around, i.e. k points to the left and k points to the right of the point; for the first three points, i.e. P1，P2，P3The left side is less than k points, so the three points are represented by the right k points and all the points on the left side of the points, and a space structure relationship is established;
the adjacency matrix A can be obtained by establishing a spatial structure relationship, and the adjacency matrix A can be obtained by the adjacency matrix
Figure BDA0003565034540000031
And
Figure BDA0003565034540000032
degree matrix of
Figure BDA0003565034540000033
Is given by the formula
Figure BDA0003565034540000034
Wherein,
Figure BDA0003565034540000035
i is an identity matrix formed by connecting X,
Figure BDA0003565034540000036
Substituting the speed information into a formula (2), aggregating the speed information of each grid point, predicting to obtain a finite difference coefficient, and calculating a spatial difference and a time derivative;
in addition, the propagation mode between layers of the graph convolution neural network is as follows:
Figure BDA0003565034540000037
wherein, H is a feature matrix of each layer, obtained from the speed of all grid points of each layer, and is X for the input layer; σ is a nonlinear activation function, which both makes full use of the velocity information of the grid points and their relationship, and enhances the correlation between the grid points.
Further, in establishing the spatial structure relationship, m for each layer1Establishing a spatial structure relationship at each grid point, wherein the velocity value of each grid point is represented by the velocity values of 6 grid points around the previous layer and the velocity value at the grid point, each grid point has velocity information, and thus the input feature matrix X is obtained from the velocity of each grid point, and the shape of the X is (m)11); since each grid point is represented by 6 surrounding points, i.e. 3 points to the left and 3 points to the right of the point, for the first three points, i.e. P1，P2，P3The left side is less than 3 points, so the three points are represented by the right 3 points and all the points on the left side of the points, so that the spatial structure relationship is established.
Further, 2) constructing a graph structure relation for a two-dimensional fluid dynamic equation:
the two-dimensional spatial equation for Burgers is as follows:
Figure BDA0003565034540000038
Figure BDA0003565034540000039
known training data set has T2A time layer, each layer having M because of two dimensions of x and y1×M1A plurality of grid points, each grid point having speed information, M1×M1Down sampling at each grid point to obtain m2×m2A plurality of grid points;
when building spatial structure relationship, m for each layer2×m2Establishing a spatial structure relationship at each grid point, wherein the speed value of each grid point is represented by the speed values of 4 grid points around the point at the upper layer and the speed value of the point, and each grid point has speed information, so that an input feature matrix X can be obtained2，X2The shape is (m)2×m2，1)；
The adjacency matrix A can be obtained by establishing a spatial structure relationship2By means of a adjacency matrix
Figure BDA0003565034540000041
And
Figure BDA0003565034540000042
degree matrix of
Figure BDA0003565034540000043
Mixing X2、
Figure BDA0003565034540000044
And substituting the speed information of each grid point into the formula (2), aggregating the speed information of each grid point, predicting to obtain a finite difference coefficient, and calculating a spatial difference and a time derivative.
Further, 3) constructing a graph structure relation for the high-dimensional fluid dynamic equation, wherein the value of each node is represented by the values of a plurality of nodes around the node at the previous time layer, and establishing a spatial structure relation between each node and the nodes around the node, so that all the nodes are spatially associated with fixed nodes.
Further, when training the graph convolutional neural network model, the loss function is:
the loss function uses the mean square error MSE and is the difference L between the estimated time domain difference and the true value1And the difference L between the estimated future state and the actual state2One of these two loss functions is selected to spread the training,
estimated difference L between time-domain difference and true value1Comprises the following steps:
Figure BDA0003565034540000045
the difference L between the estimated future state and the true state2Comprises the following steps:
Figure BDA0003565034540000046
wherein m represents m nodes per time layer;
Figure BDA0003565034540000047
representing the time-domain difference at a point estimated by the neural network model at a certain time layer, and
Figure BDA0003565034540000048
then the true value of the time domain difference at that point is represented; v. ofiRepresenting neural networks at a certain time levelThe magnitude of the velocity at a certain point estimated by the network model, vi' then represents the true value of the velocity at that point;
if finally obtained is
Figure BDA0003565034540000049
Using a loss function L1To reduce errors; if the velocity value v is to be obtained finally, the loss function L is selected at this time2To optimize model expansion training.
The invention also provides application of the flow velocity prediction method of the self-adaptive high-dimensional fluid dynamic equation based on graph convolution, and the flow velocity prediction method of the self-adaptive high-dimensional fluid dynamic equation based on graph convolution is used for simulating the state of the flow field in the high-pressure pipeline of the nuclear power plant reactor.
The invention also provides application of the flow velocity prediction method of the self-adaptive high-dimensional hydrodynamic equation based on graph convolution.
Compared with the prior art, the invention has the advantages that:
1. when the neural network prediction coefficient is used, the velocity information of the nodes is aggregated by using the information of the edges through graph convolution so as to generate a new node representation, the velocity characteristics (node information) on the grid points and the relation (structure information) between the grid points are fully used, the graph structure relation between the grid points is established, and the graph data takes the node information and the structure information into consideration and fully performs structural representation.
2. And establishing a physical process real relation by fully utilizing the speed characteristics on the grid points and the relation between the grid points. When the graph convolution is used for constructing the graph structure relationship, the incidence relationship of the grid point speed information is fully utilized, the robustness of the speed information is improved, and the rationality of a space mapping mechanism is improved.
3. The method is not only used for a low-dimensional fluid dynamic equation, but also can be used for solving the problem of complex space structure of a high-dimensional fluid dynamic equation, and can be used for predicting, simulating and analyzing a fluid dynamic field, and the method has high calculation speed and high precision.
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In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the description of the embodiments are briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without creative efforts.
FIG. 1 is a graph in which a velocity value is represented by a velocity value of an upper layer in a one-dimensional case in embodiment 1;
FIG. 2 is a spatial structure diagram obtained in the one-dimensional case of example 1;
FIG. 3 is a graph in which the velocity values are represented by the velocity values of the upper layer in the two-dimensional case in embodiment 1;
FIG. 4 is a graph in which the value of a certain point in the three-dimensional case is represented by the value of the node of the upper time layer in example 1;
FIG. 5 is a schematic flow chart in the one-dimensional case of example 1;
FIG. 6 is a schematic flow chart of the two-dimensional case in example 1.
Detailed Description
The invention is further described with reference to the following figures and specific embodiments.
Example 1
The flow velocity prediction method of the adaptive high-dimensional hydrodynamic equation based on graph convolution is combined with the flow velocity prediction method of the adaptive high-dimensional hydrodynamic equation of the graph convolution and comprises the following steps of:
s1, node x of fluidnThe coarse-grained speed function value is input into a high-dimensional fluid dynamic equation space structure model of the graph convolution neural network, and the model constructs an adaptive space structure relation for each grid point in the grid by establishing a discretization grid and utilizing the graph convolution neural network.
S2, predicting and outputting the space structure coefficient of each grid point by a finite difference-based differential equation numerical method through space structure model calculation
Figure BDA0003565034540000061
S3, any grid point x obtained by predicting step S2nCoefficient of spatial structure of
Figure BDA0003565034540000062
And xnX of surrounding pointsn-k,…,xn+k(i.e. x)nPoint and xnK points on the left and k points on the right, k being determined by a space structure model) are combined to calculate xnSpatial derivatives of (i.e. x)nProcessing the solution of the velocity value after the spatial evolution; the calculation formula is as follows:
Figure BDA0003565034540000063
wherein the solution of the formula is a formula in the prior art, and is not described in detail,
Figure BDA0003565034540000064
is the predicted spatial structure coefficient, l is the derivative order, xnRepresented is the coordinate point, v (x)n) Representative is the velocity value at that point, v (x)n-m) Representing the velocity value, x, between two coordinates, point n and point mkRepresenting the accuracy of the difference.
S4, a physical solving process based on the high-dimensional fluid dynamic equation, namely calculating the flux and then calculating x by using the fluxnTime derivative of (i.e. x)nAnd (4) solving the velocity value after the velocity value evolves along with time to finally obtain the final numerical solution of the high-dimensional fluid dynamic equation. Wherein x isnThe flux calculation formula of (b) is:
Figure BDA0003565034540000065
the time derivative calculation formula is:
Figure BDA0003565034540000066
the above formula integrates the time derivative to obtain the predicted velocity value, Δ x represents the displacement between the two grids n and m,
Figure BDA0003565034540000067
and
Figure BDA0003565034540000068
are flux values that all refer to half grid points.
In step S1, when constructing the adaptive spatial structure relationship, obtaining the characteristic information of each node, obtaining the characteristic information of the node from all the neighboring nodes of each node, constructing the graph structure relationship, and respectively introducing the specific construction process to the one-dimensional, two-dimensional and high-dimensional fluid dynamic equations as follows;
1) construction of graph structure relationships for one-dimensional fluid dynamic equations
The embodiment adopts a one-dimensional space equation of Burgers, as follows:
Figure BDA0003565034540000071
make flux
Figure BDA0003565034540000072
10000 time layers are known in the training data set, each layer has 512 grid points, each grid point has its speed information, and 512 grid points are down sampled to obtain 32 grid points. When the model is trained, the velocity values of the 32 grid points of the first layer are used as input, loss is obtained by calculating MAE through the true solution of the velocity values of the next time layer obtained through training and the velocity values of the next time layer, and when the layer is trained, the next layer is trained.
Firstly, a graph structure of a discretization grid is constructed, spatial grid points are used as vertexes of the graph, and relation weights between different grid points are designed to be used as edges of the graphAnd establishing the spatial structure relationship of all grid points. The method comprises the following specific steps: when establishing spatial structure relationship, all m of each layer1The grid points establish a spatial structure relation, the speed value of each grid point is represented by the speed values of 2k grid points around the upper layer and the speed value at the grid point, each grid point has speed information, and therefore an input feature matrix X is obtained according to the speed of each grid point, and the shape of the X is (m)11); since each grid point is represented by 2k points around it, i.e. k points to the left and k points to the right of the point. For the first three points, i.e. P1，P2，P3There are less than k points to the left, so these three points are represented by k points to the right and all points to the left of this point, establishing a spatial structure relationship. Here, k is the optimal number of surrounding points obtained through experiments, that is, the error between the central point speed value obtained through prediction of k surrounding points and the true solution is the minimum.
In this embodiment, when the spatial structure relationship is established, one spatial structure relationship is established for 32 grid points of each layer, and the velocity value of each grid point is represented by the velocity values of 6 grid points around the previous layer and the velocity value at that point, as shown in fig. 1. Each grid point has speed information, so that an input feature matrix X is obtained according to the speed of each grid point, and the shape of the X is (32, 1); the adjacent matrix A can be obtained by establishing a spatial structure relationship, and the adjacent matrix A can be obtained by the adjacent matrix
Figure BDA0003565034540000073
And
Figure BDA0003565034540000074
degree matrix of
Figure BDA0003565034540000075
Is given by the formula
Figure BDA0003565034540000076
Wherein,
Figure BDA0003565034540000077
Figure BDA0003565034540000078
i is a unit matrix formed by connecting X,
Figure BDA0003565034540000079
And substituting the speed information of each grid point into the formula (2), aggregating the speed information of each grid point, predicting to obtain a finite difference coefficient, and calculating a spatial difference and a time derivative. The following illustrates the specific calculation process:
suppose that the n-th time layer grid point under coarse granularity is 32, and the 32 grid points are respectively P1，P2，P3，...，P31，P32Indicates that the velocity at each grid point is v1，v2，v3，...，v31，v32From which a feature matrix H can be derived(1)(32 × 1) is:
Figure BDA0003565034540000081
since each grid point is represented by 6 surrounding points, i.e. 3 points to the left and 3 points to the right of the point, for the first three points, i.e. P1，P2，P3There are less than 3 points to the left, so these three points are represented by the 3 points to the right and all points to the left of that point. The finally established spatial structure relationship is shown in fig. 2.
The adjacency matrix a (32 × 32) obtained from the spatial structural relationship is:
Figure BDA0003565034540000082
can be calculated by the adjacency matrix A
Figure BDA0003565034540000083
And degree matrix
Figure BDA0003565034540000084
Wherein,
Figure BDA0003565034540000085
Figure BDA0003565034540000086
i is an identity matrix;
Figure BDA0003565034540000087
is that
Figure BDA0003565034540000088
Degree matrix (degree matrix) of (1), the formula is
Figure BDA0003565034540000089
Obtained
Figure BDA00035650345400000810
And degree matrix
Figure BDA00035650345400000811
Respectively as follows:
Figure BDA0003565034540000091
Figure BDA0003565034540000092
and the speed information of the nodes is aggregated by substituting the speed information into the following formula (2), the GCN is also a neural network layer, and the propagation mode between the layers is as follows:
Figure BDA0003565034540000093
wherein, H is a feature matrix of each layer, obtained from the speed of all grid points of each layer, and is X for the input layer; σ is a nonlinear activation function, which both makes full use of the velocity information of the grid points and their relationship, and enhances the correlation between the grid points.
Second, the machine-learned prediction coefficients can be forced to satisfy the m-th order polynomial constraint, thereby attenuating the approximation error to O (Δ x)m) If the learned discretization is adapted to a smooth solution on the grid scale, the polynomial precision constraint will ensure that the classical finite difference method can be recovered.
2) And (3) constructing a graph structure relation for a two-dimensional fluid dynamic equation:
the two-dimensional spatial equation for Burgers is as follows:
Figure BDA0003565034540000094
Figure BDA0003565034540000101
10000 temporal layers are known in the training dataset because there are two dimensions x and y, each layer has 512 × 512 grid points, each grid point has velocity information, and the 512 × 512 grid points are down sampled to obtain 32 × 32 grid points.
A spatial structure relationship is established for 32 × 32 grid points of each layer, and the velocity value of each grid point is represented by the velocity values of 4 grid points around the point and the velocity value of the point in the previous layer, as shown in fig. 3. Each grid point has speed information, so that an input feature matrix X can be obtained2，X2The shape is (32 × 32, 1).
The adjacency matrix A can be obtained by establishing a spatial structure relationship2By means of a adjacency matrix
Figure BDA0003565034540000102
And
Figure BDA0003565034540000103
degree matrix of
Figure BDA0003565034540000104
Mixing X2、
Figure BDA0003565034540000105
And substituting the speed information of each grid point into the formula (2), aggregating the speed information of each grid point, predicting to obtain a finite difference coefficient, and calculating a spatial difference and a time derivative.
The specific calculation process is as follows:
assuming that the number of grid points in the nth time layer is 32 × 32 in the coarse granularity in the two-dimensional case, the 32 × 32 grid points are respectively represented by P1,1，P1,2，...，P1,31，P1,32，P2,1，P2,2，...，P2,31，P2,32，...，P32,1，P32,2，...，P32,31，P32,32Indicates that the velocity at each grid point is v1,1，v1,2，...，v1,31，v1,32，v2,1，v2,2，...，v2,31，v2,32，...，v32,1，v32,2，...，v32,31，v32,32From which a feature matrix H can be derived(1)(1024 × 1) is:
Figure BDA0003565034540000106
the adjacent matrix A can be obtained from the space structure relationship2(1024 × 1024) is:
Figure BDA0003565034540000111
by means of an adjacency matrix A2Can be calculated to obtain
Figure BDA0003565034540000112
(1024 × 1024) and
Figure BDA0003565034540000113
degree matrix of
Figure BDA0003565034540000114
Is given by the formula
Figure BDA0003565034540000115
Obtained
Figure BDA0003565034540000116
And degree matrix
Figure BDA0003565034540000117
Respectively as follows:
Figure BDA0003565034540000118
Figure BDA0003565034540000119
and substituting the speed information into the formula (2) to aggregate the speed information of the nodes, predicting to obtain a finite difference coefficient, and calculating a spatial difference and a time derivative.
3) High-dimensional fluid dynamic equation construction diagram structural relation
The one-dimensional and two-dimensional Burgers equations can better show the structural relationship between each time layer and all nodes of one time layer on the time layer by constructing graph convolution space structural relationship, and the method is also suitable for other partial differential equations except the Burgers equation and is also suitable for higher-dimensional partial differential equations except the one-dimensional and two-dimensional equations.
To construct graph structure relationships for high dimensional hydrodynamic equations, for higher dimensional equations, the nodes at each time level are increased many times compared to one and two dimensions due to the increase in dimensions, and the values at each node become more complex when represented by the values of several nodes around the point at the previous time level. In the embodiment, a spatial structure relationship between each node and nodes around the node is established, so that all nodes are spatially associated with fixed nodes, and the value of each node is represented by the values of several nodes around the node at the previous time layer. Fig. 4 is an example of a value represented by a value of several nodes around a point at an upper time layer on a three-dimensional partial differential equation.
In addition, when training the graph convolutional neural network model, the loss function is:
the loss function uses the mean square error MSE and is the difference L between the estimated time domain difference and the true value1(Time derivative loss) and the difference L between the estimated future state and the true state2(Integrated solution loss) selects one of the two loss functions to spread the training,
estimated difference L between time-domain difference and true value1Comprises the following steps:
Figure BDA0003565034540000121
the difference L between the estimated future state and the true state2Comprises the following steps:
Figure BDA0003565034540000122
wherein m represents m nodes per time layer;
Figure BDA0003565034540000123
representing the time-domain difference at a point estimated by the neural network model at a certain time layer, and
Figure BDA0003565034540000124
then the true value of the time domain difference at that point is represented; v. ofiRepresenting the velocity values at a certain point estimated by the neural network model at a certain time layer, and vi' then represents the true value of the velocity at that point.
If finally obtained is
Figure BDA0003565034540000125
Using a loss function L1To reduce errors; if the velocity value v is to be obtained finally, the loss function L is selected at this time2To optimize model expansion training.
Example 2
This example describes a specific method/application of the present invention for simulating the flow field condition in a high pressure pipeline of a nuclear power plant reactor and analyzing the pressure loss in the pipeline. In the operation process of a nuclear power plant, reactor coolant with high temperature, high pressure and high flow rate in a reactor coolant pipeline (hereinafter referred to as a main pipeline) connecting a reactor pressure vessel and a steam generator transfers heat generated by nuclear fuel in the reactor pressure vessel to the steam generator. At the moment, the flow field state of the reactor needs to be simulated by adopting a plurality of methods, and the pressure loss of the main pipeline is analyzed, so that the proper instruments such as the temperature, the pressure, the flow and the like of the main pipeline are selected, and the reference can be provided for the performance parameters of other related equipment such as a reactor coolant pump and the like.
In the prior art, a finite element method of computational fluid mechanics is generally adopted, a flow variable to be solved is approximated by a simple equation, then the approximate relation is substituted into a continuous control equation to form a discrete equation set, and finally an algebraic equation set is solved. However, the method has the problems that the rough grid division can not be accurately solved, when the grid division is fine, the calculation speed becomes slow, at the moment, the flow velocity prediction method of the self-adaptive high-dimensional fluid dynamic equation based on graph convolution of the invention is utilized, the graph convolution neural network is utilized to establish the incidence relation between the nodes, and the convolution neural network is utilized to replace the calculation of the finite difference coefficient, so that the calculation speed is accelerated, the rationality of space mapping is increased, the steady-state analysis is better carried out on the reactor coolant pipeline between the pressure vessel and the steam generator of the nuclear power plant, so that the fluid state of the reactor coolant pipeline under the normal working condition of the nuclear power plant is simulated (it needs to be noted that the steps of obtaining the flow field state by the method except the prediction speed, and other steps such as a model modeling method and the like can be realized by the prior art), the obtained simulation model can well restore the flow condition of the coolant of the main pipeline, can obtain flow field data richer in theoretical calculation, provides reference for subsequent manufacturing and material selection of main pipeline equipment and model selection design of instruments and control systems, and provides data support for the framework of the control system.
Example 3
This example describes another specific use/application of the present invention for seismic wave numerical simulation, where the velocity value v (x) obtained by applying the method of the present invention is usedn) Is the wave velocity. The seismic wave numerical simulation is an important base stone for seismic exploration, a mathematical theoretical model is formed by effectively simplifying actual problems, the propagation rule of seismic waves in corresponding media is simulated and researched, and seismic records at earth surface or well medium wave detection points are calculated. Therefore, the seismic wave numerical simulation is not only an important tool for understanding the propagation law of the seismic wave and explaining the seismic data, but also a basis of seismic imaging. The seismic wave numerical simulation can simulate acoustic waves, elastic waves, SH waves and the like, and for example, the formula (9) is a two-dimensional isotropic uniform medium acoustic wave equation.
Figure BDA0003565034540000131
Wherein u isPThe displacement is represented by a displacement of the displacement,
Figure BDA0003565034540000141
the longitudinal wave velocity of the medium is shown, rho is the density of the elastomer, and lambda and mu are Lame coefficients.
Numerical simulation finite difference method, pseudo-spectrum method, finite element method and the like of seismic wave fields. The finite difference method solves the wave equation by using a uniform or non-uniform format, is easy to improve in precision and realize, but can multiply increase the seismic wave simulation time when the model is large in scale and fine in model subdivision. At the moment, by utilizing the scheme provided by the patent, namely the flow velocity prediction method of the graph convolution-based self-adaptive high-dimensional fluid dynamic equation, the graph convolution neural network is used for constructing the space structure relationship and predicting the coefficient, and the other steps still adopt the finite difference method and the prior art, so that not only is the grid division not needed to be fine, but also the seismic waves can be accurately simulated, the simulation time is greatly reduced, and the seismic wave simulation efficiency is improved.
It is understood that the above description is not intended to limit the present invention, and the present invention is not limited to the above examples, and those skilled in the art should understand that they can make various changes, modifications, additions and substitutions within the spirit and scope of the present invention.

Claims (8)

1. The flow velocity prediction method of the self-adaptive high-dimensional hydrodynamic equation based on graph convolution is characterized by comprising the following steps of:
s1, node x of fluidnInputting the coarse-grained speed function value into a high-dimensional fluid dynamic equation space structure model of a graph convolution neural network, and constructing an adaptive space structure relation for each grid point in the grid by establishing a discretization grid by using the graph convolution neural network;
s2, predicting and outputting the space structure coefficient of each grid point by a finite difference-based differential equation numerical method through space structure model calculation
Figure FDA0003565034530000011
S3, any grid point x obtained by predicting step S2nCoefficient of spatial structure of
Figure FDA0003565034530000012
And xnX of surrounding pointsn-k,…,xn+kThe value of the coarse particle size is combined, i.e. xnPoint and xnK points on the left and k points on the right are adjacent, and x is calculatednSpatial derivatives of (i.e. x)nProcessing the solution of the velocity value after the spatial evolution;
s4, a physical solving process based on the high-dimensional fluid dynamic equation, namely calculating the flux and then calculating x by using the fluxnTime derivative of (i.e. x)nAnd (4) solving the velocity value after the velocity value evolves along with time to finally obtain the final numerical solution of the high-dimensional fluid dynamic equation.
2. The method for predicting the flow velocity of the adaptive high-dimensional fluid dynamic equation based on graph convolution according to claim 1, wherein when the adaptive spatial structure relationship is constructed in step S1, the feature information of each node is obtained, and the feature information of each node is obtained from all neighboring nodes of the node, so as to construct the graph structure relationship, including:
1) construction of graph structure relationships for one-dimensional fluid dynamic equations
Take Burgers' equation as an example. The Burgers equation is a nonlinear partial differential equation that is applied to the field of fluid mechanics to simulate shock wave propagation and reflection. The one-dimensional spatial equation for Burgers is as follows:
Figure FDA0003565034530000013
let flux
Figure FDA0003565034530000014
Known training data set has T1A time layer, each layer having M1A number of grid points, each having speed information thereon, M1Down sampling at each grid point to obtain m1A plurality of grid points; when training the model, firstly, the m of the first layer is1The speed values of the grid points are used as input, loss is obtained by calculating MAE through the true solution of the speed value of the next time layer obtained through training and the speed value of the next time layer, and when the layer is trained, the next layer is trained;
by constructing a graph structure of a discretization grid, taking spatial grid points as vertexes of the graph, designing relationship weights between every two different grid points as edges of the graph, and establishing spatial structure relationships of all the grid points; the method comprises the following specific steps: when establishing spatial structure relationship, all m of each layer1Establishing a spatial structure relationship at each grid point, wherein the velocity value of each grid point is represented by the velocity values of 2k grid points around the upper layer and the velocity value at the grid point, each grid point has velocity information, and thus the input feature matrix X is obtained from the velocity of each grid point, and the shape of the X is (m)11); since each grid point is represented by 2k points around, i.e. k points to the left and k points to the right of the point; for the first three points, i.e. P1，P2，P3The left side is less than k points, so the three points are represented by the right k points and all the points on the left side of the points, and a space structure relationship is established;
the adjacent matrix A can be obtained by establishing a spatial structure relationship, and the adjacent matrix A can be obtained by the adjacent matrix
Figure FDA0003565034530000021
And
Figure FDA0003565034530000022
degree matrix of
Figure FDA0003565034530000023
Is given by the formula
Figure FDA0003565034530000024
Wherein,
Figure FDA0003565034530000025
i is an identity matrix formed by connecting X,
Figure FDA0003565034530000026
Substituting the speed information into a formula (2), aggregating the speed information of each grid point, predicting to obtain a finite difference coefficient, and calculating a spatial difference and a time derivative;
in addition, the propagation mode between layers of the graph convolution neural network is as follows:
Figure FDA0003565034530000027
wherein, H is a feature matrix of each layer, obtained from the speed of all grid points of each layer, and is X for the input layer; σ is a nonlinear activation function, which both makes full use of the velocity information of the grid points and their relationship, and enhances the correlation between the grid points.
3. According toThe method of flow velocity prediction for adaptive high-dimensional fluid dynamic equations based on graph convolution of claim 2, wherein the spatial structure relationship is established for m of each layer1The grid points establish a spatial structure relationship, the velocity value of each grid point is represented by the velocity values of the 6 grid points around the previous layer and the velocity value at the point, and since each grid point is represented by 6 points around the grid point, namely 3 points at the left and 3 points at the right of the grid point, the spatial structure relationship is established for the first three points, namely P1，P2，P3The left side is less than 3 points, so the three points are represented by the right 3 points and all the points on the left side of the points, so that the spatial structure relationship is established.
4. The method for flow rate prediction for an adaptive high-dimensional fluid dynamic equation based on graph convolution of claim 2), wherein 2) a graph structure relationship is constructed for a two-dimensional fluid dynamic equation:
the two-dimensional spatial equation for Burgers is as follows:
Figure FDA0003565034530000028
Figure FDA0003565034530000029
known training data set has T2A time layer, each layer having M because of two dimensions of x and y1×M1A plurality of grid points, each having speed information, and M1×M1Down sampling at each grid point to obtain m2×m2A plurality of grid points;
when building spatial structure relationship, m for each layer2×m2Establishing a spatial structure relationship at each grid point, wherein the speed value of each grid point is represented by the speed values of 4 grid points around the point at the upper layer and the speed value of the point, and each grid point has speed information, so that an input feature matrix X can be obtained2，X2The shape is (m)2×m2，1)；
The adjacency matrix A can be obtained by establishing a spatial structure relationship2By means of a adjacency matrix
Figure FDA0003565034530000031
And
Figure FDA0003565034530000032
degree matrix of
Figure FDA0003565034530000033
Mixing X2、
Figure FDA0003565034530000034
And substituting the speed information of each grid point into the formula (2), aggregating the speed information of each grid point, predicting to obtain a finite difference coefficient, and calculating a spatial difference and a time derivative.
5. The method for predicting flow rate of an adaptive high-dimensional fluid dynamic equation based on graph convolution of claim 4, wherein 3) a graph structure relationship is constructed for the high-dimensional fluid dynamic equation, the value of each node is represented by the values of several nodes around the node at the previous time layer, and a spatial structure relationship between each node and the nodes around the node is established, so that all the nodes are spatially associated with a fixed node.
6. The method for flow velocity prediction of adaptive high-dimensional fluid dynamic equations based on graph convolution according to any one of claims 1 to 5, wherein when training the graph convolution neural network model, the loss function is:
the loss function uses the mean square error MSE and is the difference L between the estimated time domain difference and the true value1And the difference L between the estimated future state and the actual state2One of these two loss functions is selected to spread the training,
estimated difference L between time-domain difference and true value1Comprises the following steps:
Figure FDA0003565034530000035
the difference L between the estimated future state and the true state2Comprises the following steps:
Figure FDA0003565034530000036
wherein m represents m nodes per time layer;
Figure FDA0003565034530000037
representing the time-domain difference at a point estimated by the neural network model at a certain time layer, and
Figure FDA0003565034530000038
then the true value of the time domain difference at that point is represented; v. ofiRepresenting the velocity values at a certain point estimated by the neural network model at a certain time layer, and vi' then represents the true value of the velocity at that point;
if finally obtained is
Figure FDA0003565034530000041
Using a loss function L1To reduce errors; if the final value to be obtained is the velocity value v, the penalty function L is then selected2To optimize model expansion training.
7. The application of the flow velocity prediction method based on the graph convolution adaptive high-dimensional fluid dynamic equation in claim 6, characterized in that the flow velocity prediction method based on the graph convolution adaptive high-dimensional fluid dynamic equation is used for simulating the flow field state in the high-pressure pipeline of the nuclear power plant reactor.
8. The use of the method for flow velocity prediction of the graph convolution based adaptive high-dimensional hydrodynamic equation according to claim 6, wherein the method for flow velocity prediction of the graph convolution based adaptive high-dimensional hydrodynamic equation is used to construct a spatial structure relationship and predict, obtain seismic wave velocities, and model seismic wavefields.
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