Flow velocity prediction method for adaptive high-dimensional fluid dynamic equations based on graph convolution

Abstract

The present invention belongs to the technical field of fluid dynamic field prediction, and discloses a flow velocity prediction method for an adaptive high-dimensional fluid dynamic equation based on graph convolution. Firstly, a spatial structure model of a high-dimensional fluid dynamic equation of a graph convolutional neural network is used to establish a discretized grid, and an adaptive spatial structure relationship is constructed for each grid point in the grid using the graph convolutional neural network; then, based on a numerical method of differential equations using finite differences, the spatial structure coefficient of each grid point is predicted and output through calculation of the spatial structure model; and then the spatial derivative is calculated; finally, based on the physical solution process of the high-dimensional fluid dynamic equation, the final numerical solution of the high-dimensional fluid dynamic equation is finally obtained.

Description

Technical Field

[0001] The present invention belongs to the technical field of fluid dynamic field prediction, and in particular relates to a flow velocity prediction method of an adaptive high-dimensional fluid dynamic equation based on graph convolution. Background Art

[0002] When predicting fluid dynamic fields, the discretization of the partial differential equations of fluid dynamics is usually used to solve and obtain approximate values. The key process of discretization of partial differential equations is to calculate the prediction coefficients, mainly using methods such as finite differences or finite volumes. The main problem faced by traditional methods at present is that if the grid division is coarse, an accurate solution cannot be obtained and the calculation speed is slow. This is because the traditional method depends on the grid size, and when the division is very fine, the calculation time will become longer and the calculation speed will be slow. Therefore, a data-driven discretization method that systematically derives the discretization of continuous physical systems using machine learning has been proposed. The advantage of this method is that convolutional neural networks are used to replace the calculation of finite difference coefficients, while the remaining steps still use traditional methods, which are integrated into the learning process of user embedding. The representation obtained improves the generalization of user embedding compared to the method that does not consider comment information. However, this method still has the following problems:

[0003] First, it is impossible to establish a structured representation of the relationship between speed nodes (missing structural relationship). This is because the method only considers the impact of a single node on the coefficient, and does not explore the structural relationship between nodes. For example, when calculating the coefficient using a neural network, the current method only considers the speed information of the node and ignores the relationship between the nodes. Not only does the spatial mapping mechanism lack rationality, but it also reduces the generalization of the representation.

[0004] Second, the method only proposes a low-dimensional spatial structure modeling scheme, so it is applied to low-dimensional fluid dynamics equations and cannot solve the complex spatial structure of high-dimensional fluid dynamics equations.

[0005] Therefore, in response to the above problems, the present invention proposes a flow velocity prediction method for an adaptive high-dimensional fluid dynamics equation based on graph convolution. Summary of the invention

[0006] In view of the shortcomings of the prior art, the present invention provides a flow velocity prediction method for an adaptive high-dimensional fluid dynamics equation based on graph convolution, which establishes a spatial structural relationship for the velocity value of each grid point through a convolutional neural network, and then predicts the velocity value at the next moment.

[0007] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0008] The flow velocity prediction method of the adaptive high-dimensional fluid dynamic equation based on graph convolution includes the following steps:

[0009] S1, the node x of the fluidn The coarse-grained velocity function value is input into the high-dimensional fluid dynamic equation spatial structure model of the graph convolutional neural network. The model establishes a discretized grid and uses the graph convolutional neural network to construct an adaptive spatial structure relationship for each grid point in the grid.

[0010] S2, based on the finite difference differential equation numerical method, the spatial structure coefficients of each grid point are predicted and output through spatial structure model calculation

[0011] S3, any grid point x predicted by step S2 n The spatial structure coefficient With x n x of surrounding points n-k ,…,x n+k The coarse-grained values ​​are combined, that is, x n Point and x n For the adjacent k points on the left and k points on the right, calculate x n The spatial derivative at x n The solution after the velocity value at the position evolves with space;

[0012] S4, based on the physical solution process of the high-dimensional fluid dynamics equation, that is, calculating the flux and then using the flux to calculate x n The time derivative at x n The solution of the velocity value evolving with time at the location is finally obtained as the final numerical solution of the high-dimensional fluid dynamics equation.

[0013] Furthermore, when constructing the adaptive spatial structure relationship in step S1, the feature information of each node itself is obtained, and the feature information of each node is obtained from all neighboring nodes of each node to construct the graph structure relationship, including:

[0014] 2) Constructing a graphical structure for the one-dimensional fluid dynamics equation

[0015] The one-dimensional space equation for Burgers is as follows:

[0016]

[0017] Let flux

[0018] It is known that there are T in the training data set 1 time layers, each layer has M 1 grid points, each of which has its velocity information, and M 1 The grid points are downsampled to obtain m 1 grid points; when training the model, first 1The speed value of each grid point is used as input, and the loss is calculated by MAE between the speed value of the next time layer obtained through training and the true solution of the speed value of the next time layer. When the layer is trained, the next layer is trained.

[0019] By constructing a graph structure of discretized grids, taking spatial grid points as graph vertices, designing the relationship weights between different grid points as graph edges, and establishing the spatial structural relationship of all grid points; specifically, when establishing the spatial structural relationship, for all m 1 A spatial structure relationship is established among the grid points. The speed value of each grid point is represented by the speed values ​​of the 2k grid points around the previous layer and the speed value at the point. Each grid point has speed information. Therefore, the input feature matrix X is obtained from the speed of each grid point. The shape of X is (m 1 , 1); 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., P 1 , P 2 , P 3 There are less than k points on the left, so these three points are represented by k points on the right and all points on the left of the point to establish a spatial structural relationship;

[0020] By establishing the spatial structure relationship, we can get the adjacency matrix A, and through the adjacency matrix we can get as well as The degree matrix The formula is in, I is the identity matrix, X, Substituting into formula (2), the velocity information of each grid point is aggregated, and then the finite difference coefficients are predicted and the spatial difference and time derivative are calculated;

[0021] In addition, the propagation method between layers of the graph convolutional neural network is:

[0022]

[0023] Among them, H is the feature matrix of each layer, which is obtained by the speed of all grid points in each layer. For the input layer, H is X; σ is a nonlinear activation function, which not only makes full use of the speed information of the grid points and the relationship between them, but also enhances the association between the grid points.

[0024] Furthermore, when establishing the spatial structure relationship, for each layer of m 1 A spatial structure relationship is established among the grid points. The speed value of each grid point is represented by the speed values ​​of the six grid points around the previous layer and the speed value at the point. Each grid point has speed information. Therefore, the input feature matrix X is obtained from the speed of each grid point. The shape of X is (m1 , 1); Since each grid point is represented by 6 surrounding points, namely 3 points to the left and 3 points to the right of the point, for the first three points, namely P 1 , P 2 , P 3 There are less than 3 points on the left, so these three points are represented by the 3 points on the right and all the points to the left of this point to establish the spatial structural relationship.

[0025] Furthermore, 2) construct a graph structure relationship for the two-dimensional fluid dynamics equation:

[0026] The two-dimensional space equation for Burgers is as follows:

[0027]

[0028]

[0029] It is known that there are T in the training data set 2 There are two time layers, x and y, and each layer has M 1 ×M 1 grid points, each of which has velocity information, and M 1 ×M 1 The grid points are downsampled to obtain m 2 ×m 2 grid points;

[0030] When establishing the spatial structure relationship, for each layer of m 2 ×m 2 A spatial structure relationship is established for each grid point. The speed value of each grid point is represented by the speed values ​​of the four grid points around the point in the previous layer and the speed value of the point. Each grid point has speed information, so the input feature matrix X can be obtained. 2 , X 2 The shape is (m 2 ×m 2 , 1);

[0031] By establishing the spatial structure relationship, we can get the adjacency matrix A 2 , we can get the adjacency matrix as well as The degree matrix X 2 , Substituting into formula (2), the velocity information of each grid point is aggregated, and then the finite difference coefficients are predicted and the spatial difference and time derivative are calculated.

[0032] Furthermore, 3) a graph structure relationship is constructed for the high-dimensional fluid dynamics equation. The value at each node is represented by the values ​​of several nodes around the point in the previous time layer. The spatial structure relationship between each node and the nodes around the point is established, so that all nodes are spatially associated with fixed nodes.

[0033] Furthermore, when training the graph convolutional neural network model, the loss function is:

[0034] The loss function uses the mean squared error MSE and is the difference L between the estimated time domain difference and the true value 1 and the difference between the estimated future state and the actual state L 2 Choose one of these two loss functions to start training.

[0035] The difference between the estimated time domain difference and the true value L 1 for:

[0036]

[0037] The difference between the estimated future state and the actual state L 2 for:

[0038]

[0039] Where m means that there are m nodes in each time layer; represents the time domain difference at a certain point estimated by the neural network model at a certain time layer, and It represents the true value of the time domain difference at this point; v i represents the velocity value at a certain point estimated by the neural network model at a certain time layer, and v i ′ represents the true value of the velocity at that point;

[0040] If the final result is Use the loss function L 1 To reduce the error; if the final value is the speed value v, then choose the loss function L 2 To optimize the model training.

[0041] The present invention also provides an application of the flow rate prediction method of the adaptive high-dimensional fluid dynamic equation based on graph convolution, and uses the flow rate prediction method of the adaptive high-dimensional fluid dynamic equation based on graph convolution to simulate the flow field state in the high-pressure pipeline of a nuclear power plant reactor.

[0042] The present invention also provides an application of the flow velocity prediction method of the adaptive high-dimensional fluid dynamic equation based on graph convolution. The flow velocity prediction method of the adaptive high-dimensional fluid dynamic equation based on graph convolution is used to construct a spatial structural relationship and predict, obtain the seismic wave velocity, and simulate the seismic wave field.

[0043] Compared with the prior art, the present invention has the following advantages:

[0044] 1. When using the neural network prediction coefficient, the present invention aggregates the speed information of the node using the edge information through graph convolution to generate a new node representation, fully utilizes the speed characteristics (node ​​information) on the grid points and the relationship between the grid points (structural information), and establishes a graph structural relationship between the grid points. The graph data takes into account both the node information and the structural information, and fully performs structured representation.

[0045] 2. Make full use of the velocity characteristics on the grid points and the relationship between the grid points to establish the real connection of the physical process. When using graph convolution to construct the graph structure relationship, the correlation between the grid point velocity information is fully utilized to improve the robustness of the velocity information and increase the rationality of the spatial mapping mechanism.

[0046] 3. The method of the present invention is not only used for low-dimensional fluid dynamics equations, but can also be used to solve the complex spatial structure of high-dimensional fluid dynamics equations. The method of the present invention can be used to predict, simulate, and analyze fluid dynamic fields with fast calculation speed and high accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings required for use in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other accompanying drawings can be obtained based on these accompanying drawings without paying creative work.

[0048] Figure 1 It is a diagram showing that the velocity value in the one-dimensional case in Example 1 is represented by the velocity value of the previous layer;

[0049] Figure 2 is a spatial structure relationship diagram obtained in the one-dimensional case in Example 1;

[0050] Figure 3 It is a diagram showing that the velocity value in the two-dimensional case in Example 1 is represented by the velocity value of the previous layer;

[0051] Figure 4 It is a diagram showing that the value of a certain point in the three-dimensional case in Example 1 is represented by the value of the node of the previous time layer;

[0052] Figure 5This is a schematic diagram of the process in one dimension in Example 1;

[0053] Figure 6 This is a schematic diagram of the process in the two-dimensional case in Example 1. DETAILED DESCRIPTION

[0054] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0055] Example 1

[0056] A flow velocity prediction method based on graph convolution for adaptive high-dimensional fluid dynamic equations, combined with Figure 5 , Figure 6 , including the following steps:

[0057] S1, the node x of the fluid n A high-dimensional fluid dynamics equation spatial structure model is constructed by inputting the coarse-grained velocity function value into the graph convolutional neural network. The model establishes a discretized grid and uses the graph convolutional neural network to construct an adaptive spatial structure relationship for each grid point in the grid.

[0058] S2, based on the finite difference differential equation numerical method, the spatial structure coefficients of each grid point are predicted and output through spatial structure model calculation

[0059] S3, any grid point x predicted by step S2 n The spatial structure coefficient With x n x of surrounding points n-k ,…,x n+k (i.e. x n Point and x n The adjacent left k points and right k points (k is determined by the spatial structure model) are combined with the coarse-grained values ​​to calculate x n The spatial derivative at x n The solution after the velocity value at the position evolves with space; the calculation formula is:

[0060]

[0061] The solution of this formula is a prior art formula and will not be described in detail. is the predicted spatial structure coefficient, l is the derivative order, x n represents the coordinate point, v(x n ) represents the velocity value at that point, v(x n-m ) represents the velocity value between the two coordinate points n and m, x k Represents the accuracy of the difference.

[0062] S4, based on the physical solution process of the high-dimensional fluid dynamics equation, that is, calculating the flux and then using the flux to calculate x n The time derivative at x n The solution of the velocity value at the position evolving with time is finally obtained as the final numerical solution of the high-dimensional fluid dynamics equation. n The flux calculation formula at is:

[0063]

[0064] The time derivative calculation formula is:

[0065]

[0066] The above formula integrates the time derivative to obtain the predicted velocity value. Δx represents the displacement between the two grids n and m. and It refers to the flux value at half grid point.

[0067] In step S1, when constructing the adaptive spatial structural relationship, the characteristic information of each node itself is obtained, and the characteristic information of each node is obtained from all neighboring nodes of each node to construct the graph structural relationship. The specific construction process for the one-dimensional, two-dimensional and high-dimensional fluid dynamic equations is introduced as follows;

[0068] 1) Constructing a graphical structure for the one-dimensional fluid dynamics equation

[0069] This embodiment uses Burgers' one-dimensional space equation, as shown below:

[0070]

[0071] Let flux

[0072] It is known that there are 10,000 time layers in the training data set, each layer has 512 grid points, each grid point has its speed information, and the 512 grid points are downsampled to 32 grid points. When training the model, the speed values ​​of the 32 grid points in the first layer are first used as input, and the loss is calculated by MAE between the speed values ​​of the next time layer obtained through training and the true solution of the speed values ​​of the next time layer. When the layer is trained, the next layer is trained.

[0073] First, by constructing a graph structure of discretized grids, taking spatial grid points as graph vertices, designing the relationship weights between different grid points as graph edges, and establishing the spatial structural relationship of all grid points. The details are as follows: When establishing the spatial structural relationship, for all m in each layer 1A spatial structure relationship is established among the grid points. The speed value of each grid point is represented by the speed values ​​of the 2k grid points around the previous layer and the speed value at the point. Each grid point has speed information. Therefore, the input feature matrix X is obtained from the speed of each grid point. The shape of X is (m 1 , 1); 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., P 1 , P 2 , P 3 There are less than k points on the left, so these three points are represented by k points on the right and all points on the left of the point to establish a spatial structural relationship. Here k is the optimal number of surrounding points obtained by experiment, that is, the error between the center point velocity value predicted by k surrounding points and the true solution is the smallest.

[0074] In this embodiment, when establishing a spatial structural relationship, a spatial structural relationship is established for each layer of 32 grid points, and the speed value of each grid point is represented by the speed values ​​of the 6 grid points around the previous layer and the speed value at the point, such as Figure 1 As shown. Each grid point has speed information, so the input feature matrix X is obtained from the speed of each grid point, and the shape of X is (32, 1); by establishing the spatial structure relationship, the adjacency matrix A can be obtained, and the adjacency matrix can be obtained as well as The degree matrix The formula is in, I is the identity matrix, X, Substitute into formula (2), aggregate the velocity information of each grid point, and then predict the finite difference coefficients and calculate the spatial difference and time derivative. The following example illustrates the specific calculation process:

[0075] Assume that the number of grid points in the nth time layer under coarse granularity is 32, and these 32 grid points are represented by P 1 , P 2 , P 3 , ..., P 31 , P 32 To represent, the velocity at each grid point is v 1 , v 2 , v 3 , ..., v 31 , v 32 , thus we can get the characteristic matrix H (1) (32×1) is:

[0076]

[0077] Since each grid point is represented by 6 surrounding points, namely 3 points to the left and 3 points to the right of the point, for the first three points, namely P 1 , P 2 , P 3 There are less than 3 points on the left, so these three points are represented by the 3 points on the right and all the points to the left of this point. The final spatial structure relationship is as follows Figure 2 shown.

[0078] From the spatial structure relationship, the adjacency matrix A (32×32) can be obtained as:

[0079]

[0080] The adjacency matrix A can be calculated and the degree matrix in, I is the identity matrix; yes The degree matrix of What you get and the degree matrix They are as follows:

[0081]

[0082]

[0083] Substitute them into the following formula (2) to aggregate the speed information of the nodes. GCN is also a neural network layer, and the propagation method between its layers is:

[0084]

[0085] Among them, H is the feature matrix of each layer, which is obtained by the speed of all grid points in each layer. For the input layer, H is X; σ is a nonlinear activation function, which not only makes full use of the speed information of the grid points and the relationship between them, but also enhances the association between the grid points.

[0086] Secondly, we can force the machine learning prediction coefficients to satisfy the m-order polynomial constraint, so that the approximation error decays to O(Δx m ), if the learned discretization is suitable for a solution that is smooth on the grid scale, the polynomial accuracy constraints ensure that the classical finite difference method can be recovered.

[0087] 2) Construct a graphical structure for the two-dimensional fluid dynamics equation:

[0088] The two-dimensional space equation for Burgers is as follows:

[0089]

[0090]

[0091] It is known that there are 10,000 time layers in the training data set. Because there are two dimensions, x and y, each layer has 512×512 grid points. Each grid point has speed information. The 512×512 grid points are downsampled to 32×32 grid points.

[0092] A spatial structure relationship is established for the 32×32 grid points in each layer. The speed value of each grid point is represented by the speed values ​​of the four grid points around the point in the previous layer and the speed value of the point, such as Figure 3 As shown. Each grid point has velocity information, so the input feature matrix X can be obtained 2 , X 2 The shape is (32×32, 1).

[0093] By establishing the spatial structure relationship, we can get the adjacency matrix A 2 , we can get the adjacency matrix as well as The degree matrix X 2 , Substituting into formula (2), the velocity information of each grid point is aggregated, and then the finite difference coefficients are predicted and the spatial difference and time derivative are calculated.

[0094] The specific calculation process is as follows:

[0095] Assume that the number of grid points in the nth time layer is 32×32 in the coarse-grained two-dimensional case. These 32×32 grid points are represented by P 1,1 , P 1,2 , ..., P 1,31 , P 1,32 , P 2,1 , P 2,2 , ..., P 2,31 , P 2,32 , ..., P 32,1 , P 32 ,2,...,P 32,31 , P 32,32 To represent, the velocity at each grid point is v 1,1 , v 1 ,2,...,v 1,31 , v 1,32 , v 2,1 , v 2,2 , ..., v 2,31 , v 2,32 , ..., v 32 ,1,v 32,2,...,v 32 ,31,v 32,32 , thus we can get the characteristic matrix H (1) (1024×1) is:

[0096]

[0097] The adjacency matrix A can be obtained from the spatial structure relationship 2 (1024×1024) is:

[0098]

[0099] Through the adjacency matrix A 2 It can be calculated (1024×1024) and The degree matrix The formula is What you get And the degree matrix They are as follows:

[0100]

[0101]

[0102] Substitute them into formula (2) to aggregate the node velocity information, then predict the finite difference coefficients and calculate the spatial difference and time derivative.

[0103] 3) Constructing graph structure relationships using high-dimensional fluid dynamics equations

[0104] One-dimensional and two-dimensional Burgers equations can better demonstrate the structural relationship between each time layer and all nodes of the previous time layer by constructing the spatial structural relationship of graph convolution. This method is also applicable to other partial differential equations except Burgers equation, and also to higher-dimensional partial differential equations except one-dimensional and two-dimensional equations.

[0105] Constructing a graph structure relationship for high-dimensional fluid dynamics equations. For higher-dimensional equations, due to the increase in dimensions, the number of nodes on each time layer increases many times compared to one-dimensional and two-dimensional ones. When the value at each node is represented by the values ​​of several nodes around the point in the previous time layer, it becomes more complicated. This embodiment establishes a spatial structure relationship between each node and the nodes around the point, so that all nodes are spatially associated with fixed nodes, and the value at each node is represented by the values ​​of several nodes around the point in the previous time layer. Figure 4 This is an example where a point on a three-dimensional partial differential equation is represented by the values ​​of several nodes around the point in the previous time layer.

[0106] In addition, it should be noted that when training the graph convolutional neural network model, the loss function is:

[0107] The loss function uses the mean squared error MSE and is the difference L between the estimated time domain difference and the true value 1 (Time derivative loss) and the difference between the estimated future state and the actual state L 2 (Integrated solution loss) Choose one of these two loss functions to start training.

[0108] The difference between the estimated time domain difference and the true value L 1 for:

[0109]

[0110] The difference between the estimated future state and the actual state L 2 for:

[0111]

[0112] Where m means that there are m nodes in each time layer; represents the time domain difference at a certain point estimated by the neural network model at a certain time layer, and It represents the true value of the time domain difference at this point; v i represents the velocity value at a certain point estimated by the neural network model at a certain time layer, and v i ′ represents the true value of the velocity at that point.

[0113] If the final result is Use the loss function L 1 To reduce the error; if the final value is the speed value v, then choose the loss function L 2 To optimize the model training.

[0114] Example 2

[0115] This embodiment introduces a specific method of use / application of the present invention, which is used to simulate the flow field state in the high-pressure pipeline of a nuclear power plant reactor and analyze the pipeline pressure loss. During the operation of a nuclear power plant, the high-temperature, high-pressure, and high-flow rate reactor coolant in the reactor coolant pipeline (hereinafter referred to as the main pipeline) connecting the reactor pressure vessel and the steam generator transfers the heat generated by the nuclear fuel in the reactor pressure vessel to the steam generator. At this time, some methods need to be used to simulate its flow field state and analyze the main pipeline pressure loss, so as to select appropriate main pipeline temperature, pressure, flow and other instruments, and also provide a reference for the performance parameters of other related equipment such as the reactor coolant pump.

[0116] The existing technology generally adopts the computational fluid dynamics finite element method, which first uses simple equations to approximate the flow variables to be calculated, then substitutes the approximate relationship into the continuity control equation to form a discrete equation group, and finally solves the algebraic equation group. However, the problem with this method is that if the grid division is rough, an accurate solution cannot be obtained. When the grid division is fine, the calculation speed will become slow. At this time, the flow velocity prediction method of the adaptive high-dimensional fluid dynamic equation based on graph convolution of the present invention is used, and the association relationship between nodes is established by using a graph convolution neural network. The convolution neural network is used to replace the calculation of the finite difference coefficient, thereby speeding up the calculation speed and increasing the rationality of the spatial mapping, so as to better perform a steady-state analysis of the reactor coolant pipeline between the pressure vessel and the steam generator of the nuclear power plant, thereby simulating the fluid state of the reactor coolant pipeline under normal operating conditions of the nuclear power plant (it should be noted that, except for the steps of predicting the velocity and obtaining the flow field state using the method of the present invention, other steps such as model building methods can be achieved using existing technologies). The obtained simulation model can well restore the coolant flow of the main pipeline, and can obtain more abundant flow field data for theoretical calculation, which can provide a reference for the subsequent main pipeline equipment manufacturing, material selection, and instrument and control system selection design, and provide data support for the architecture of the control system.

[0117] Example 3

[0118] This embodiment introduces another specific use method / application of the present invention for numerical simulation of seismic waves. In this case, the velocity value v(x n ) is the wave velocity. Numerical simulation of seismic waves is an important cornerstone of seismic exploration. By effectively simplifying practical problems, a mathematical theoretical model is formed to simulate and study the propagation law of seismic waves in the corresponding medium, and calculate the seismic records at the surface or well detection points. Therefore, numerical simulation of seismic waves is not only an important tool for understanding the propagation law of seismic waves and interpreting seismic data, but also the basis of seismic imaging. Numerical simulation of seismic waves can simulate acoustic waves, elastic waves, SH waves, etc. For example, formula (9) is the acoustic wave equation of a two-dimensional isotropic homogeneous medium.

[0119]

[0120] where u P represents displacement, represents the longitudinal wave velocity of the medium, ρ is the density of the elastic body, and λ and μ are the Lame coefficients.

[0121] Finite difference method, pseudo-spectral method and finite element method are used for numerical simulation of seismic wave fields. Among them, the finite difference method uses uniform or non-uniform format to solve the wave equation, and the accuracy of the finite difference method is easy to improve and easy to implement, but when the model scale is large and the model is finely divided, the simulation time of seismic waves will increase exponentially. At this time, the solution proposed in this patent is used, that is, the velocity prediction method of adaptive high-dimensional fluid dynamic equations based on graph convolution, and the graph convolutional neural network is used to construct the spatial structure relationship and predict the coefficients, while the remaining steps still use the finite difference method and existing technology. At this time, not only does it not need to divide the grid finely, but it can also accurately simulate the seismic waves, and the simulation time is also greatly reduced, which improves the efficiency of seismic wave simulation.

[0122] Of course, the above description is not a limitation of the present invention, and the present invention is not limited to the above examples. Any changes, modifications, additions or substitutions made by ordinary technicians in this technical field within the essential scope of the present invention should fall within the protection scope of the present invention.

Claims

1. A flow velocity prediction method based on graph convolution for adaptive high-dimensional fluid dynamic equations, characterized in that: The following steps are involved: S1, the node x of the fluid n The coarse-grained velocity function value is input into the high-dimensional fluid dynamic equation spatial structure model of the graph convolutional neural network. The model establishes a discretized grid and uses the graph convolutional neural network to construct an adaptive spatial structure relationship for each grid point in the grid. When constructing the adaptive spatial structure relationship in step S1, the feature information of each node itself is obtained, and the feature information of each node is obtained from all neighboring nodes of each node to construct the graph structure relationship, including: 1) Constructing a graphical structure for the one-dimensional fluid dynamics equation Take the Burgers equation as an example. The Burgers equation is a nonlinear partial differential equation used in the field of fluid mechanics to simulate shock wave propagation and reflection. The one-dimensional space equation of Burgers is as follows: Traffic volume It is known that there are T1 time layers in the training data set, each layer has M1 grid points, each grid point has its speed information, and M1 grid points are downsampled to obtain m1 grid points; when training the model, the speed value of the m1 grid points of the first layer is first used as input, and the loss is calculated by MAE between the speed value of the next time layer obtained through training and the true solution of the speed value of the next time layer. When the layer is trained, the next layer is trained; By constructing a graph structure of discretized grids, taking spatial grid points as graph vertices, designing the relationship weights between different grid points as graph edges, and establishing the spatial structural relationship of all grid points; specifically, when establishing the spatial structural relationship, a spatial structural relationship is established for all m1 grid points of each layer, and the speed value of each grid point is represented by the speed values ​​of the surrounding 2k grid points of the previous layer and the speed value at the point. Each grid point has speed information, so the input feature matrix X is obtained from the speed of each grid point, and the shape of X is (m1, 1); since each grid point is represented by the surrounding 2k points, 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, and P3, there are less than k points on the left, so these three points are represented by the k points on the right and all the points on the left of the point, and the spatial structural relationship is established; By establishing the spatial structure relationship, we can get the adjacency matrix A, and through the adjacency matrix we can get as well as The degree matrix The formula is in, I is the identity matrix, X, Substituting into formula (2), the velocity information of each grid point is aggregated, and then the finite difference coefficients are predicted and the spatial difference and time derivative are calculated; In addition, the propagation method between layers of the graph convolutional neural network is: Among them, H is the feature matrix of each layer, which is obtained by the speed of all grid points in each layer. For the input layer, H is X; σ is a nonlinear activation function, which not only makes full use of the speed information of the grid points and the relationship between them, but also enhances the association between the grid points; S2, based on the finite difference differential equation numerical method, the spatial structure coefficients of each grid point are predicted and output through spatial structure model calculation S3, any grid point x predicted by step S2 n The spatial structure coefficient With x n x of surrounding points n-k ,…,x n+k The coarse-grained values ​​are combined, that is, x n Point and x n For the adjacent k points on the left and k points on the right, calculate x n The spatial derivative at x n The solution after the velocity value at the position evolves with space; S4, based on the physical solution process of the high-dimensional fluid dynamics equation, that is, calculating the flux and then using the flux to calculate x n The time derivative at x n The solution of the velocity value evolving with time at the location is finally obtained as the final numerical solution of the high-dimensional fluid dynamics equation.

2. The flow velocity prediction method based on the adaptive high-dimensional fluid dynamic equation based on graph convolution according to claim 1 is characterized in that: When establishing the spatial structural relationship, a spatial structural relationship is established for the m1 grid points of each layer. The speed value of each grid point is represented by the speed values ​​of the 6 surrounding grid points in the previous layer and the speed value at the point. Since each grid point is represented by the 6 surrounding points, namely the 3 points on the left and the 3 points on the right of the point, there are less than 3 points on the left of the first three points, namely P1, P2, and P3. Therefore, these three points are represented by the 3 points on the right and all the points on the left of the point, so as to establish the spatial structural relationship.

3. The flow velocity prediction method based on the adaptive high-dimensional fluid dynamic equation based on graph convolution according to claim 1, characterized in that: 2) Constructing a graphical structure for the two-dimensional fluid dynamics equations: The two-dimensional space equation for Burgers is as follows: It is known that there are T2 time layers in the training data set. Because there are two dimensions, x and y, each layer has M1×M1 grid points. Each grid point has velocity information. The M1×M1 grid points are downsampled to obtain m2×m2 grid points. When establishing a spatial structural relationship, a spatial structural relationship is established for the m2×m2 grid points of each layer. The speed value of each grid point is represented by the speed values ​​of the four grid points around the point in the previous layer and the speed value of the point. Each grid point has speed information, so the input feature matrix X2 can be obtained, and the shape of X2 is (m2×m2, 1); By establishing the spatial structure relationship, we can get the adjacency matrix A2. as well as The degree matrix X2, Substituting into formula (2), the velocity information of each grid point is aggregated, and then the finite difference coefficients are predicted and the spatial difference and time derivative are calculated.

4. The flow velocity prediction method based on the adaptive high-dimensional fluid dynamic equation based on graph convolution according to claim 3 is characterized in that: 3) Construct a graph structure relationship for the high-dimensional fluid dynamics equation. The value at each node is represented by the values ​​of several nodes around the point in the previous time layer. Establish a spatial structural relationship between each node and the nodes around the point, so that all nodes are spatially associated with fixed nodes.

5. The flow velocity prediction method based on the adaptive high-dimensional fluid dynamic equation based on graph convolution according to any one of claims 1 to 4, characterized in that: When training the graph convolutional neural network model, the loss function is: The loss function uses the mean square error MSE, and one of the two loss functions, L1, the difference between the estimated time domain difference and the true value, and L2, the difference between the estimated future state and the true state, is selected for training. The difference L1 between the estimated time domain difference and the true value is: The difference L2 between the estimated future state and the actual state is: Where m means that there are m nodes in each time layer; represents the time domain difference at a certain point estimated by the neural network model at a certain time layer, and It represents the true value of the time domain difference at this point; v i represents the velocity value at a certain point estimated by the neural network model at a certain time layer, and v i ' represents the true value of the velocity at that point; If the final result is Use loss function L1 to reduce the error; if the final value is the speed value v, then select loss function L2 to optimize the model for training.

6. Application of the flow velocity prediction method based on the adaptive high-dimensional fluid dynamic equation based on graph convolution as claimed in claim 5, characterized in that: The flow field state in the high-pressure pipeline of a nuclear power plant reactor is simulated using the flow velocity prediction method of the adaptive high-dimensional fluid dynamic equation based on graph convolution.

7. The application of the flow velocity prediction method based on the graph convolution adaptive high-dimensional fluid dynamic equation as claimed in claim 5, characterized in that: The velocity prediction method of the adaptive high-dimensional fluid dynamic equation based on graph convolution is used to construct and predict the spatial structural relationship, obtain the seismic wave velocity, and simulate the seismic wave field.

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