Deep coordinate transformation network enhanced partial differential equation neural network solving method

Abstract

The application provides a partial differential equation neural network solving method enhanced by a deep coordinate transformation network, relates to the technical field of applied mathematics and computational physics, and comprises the following steps: generating a calculation grid of a target calculation region or shape by using a grid generation tool; training a deep coordinate transformation network based on the generated calculation grid to obtain a trained deep coordinate transformation network; embedding the trained deep coordinate transformation network as a preprocessing module into a neural network solver to obtain an embedded neural network solver; and solving a partial differential equation by using the embedded neural network solver to obtain a solving enhancement result and complete the solving enhancement of the neural network partial differential equation. The application solves the problem of insufficient local modeling expression capability when complex geometric structures and multi-scale flow characteristics are processed.

Description

TECHNICAL FIELD

[0001] The present specification relates to the field of applied mathematics and computational physics, and in particular, to a deep coordinate transformation network enhanced partial differential equation neural network solving method. BACKGROUND

[0002] In recent years, with the development of neural network solvers such as Physics-Informed Neural Networks (PINNs), Deep Ritz Method and Deep Galerkin Method, the partial differential equation (PDE) solving method based on deep learning has shown broad application prospects in computational physics and engineering problems. This method introduces PDE residual terms and boundary / initial condition constraints into the loss function of the neural network, thereby approximating the PDE solution in a meshless framework, and has good parameterization ability and inverse problem adaptability.

[0003] However, since the fully connected neural network is essentially a global function approximator, its expression ability is relatively uniform in the entire calculation domain, and it lacks the ability to locally analyze local high gradient regions (such as boundary layers, shock waves, and geometric corners). This global modeling characteristic results in slow convergence and poor accuracy when dealing with complex flow problems with obvious multi-scale features and local drastic changes, which severely limits the practicality of neural network methods in engineering high Reynolds number turbulent flow simulation.

[0004] Traditional numerical methods (such as finite volume and finite element methods) achieve analysis of local complex structures by constructing non-uniform grids, increasing computational degrees of freedom in high gradient regions, and other means, thereby effectively improving simulation accuracy. This "grid prior" strategy is one of the key factors for the success of traditional CFD. However, current neural network solvers lack a modeling mechanism comparable to "adaptive grids", and only rely on adjusting the sampling point density (such as adaptive sampling) or domain decomposition, which still cannot fundamentally improve their ability to analyze local complex physical features. In addition, existing methods that attempt to incorporate grid information, such as grid transformation methods, have achieved some results, but are limited to single block structured grids and cannot adapt to arbitrary complex geometries, and often introduce numerical errors. SUMMARY

[0005] To solve the above problems in the prior art, the deep coordinate transformation network enhanced partial differential equation neural network solving method provided by the present application solves the problem of insufficient local modeling expression ability when dealing with complex geometric structures and multi-scale flow characteristics.

[0006] In order to achieve the above-mentioned purposes, the technical scheme adopted by the present application is as follows: a partial differential equation neural network solving method enhanced by a deep coordinate transformation network, comprising:

[0007] S1: using a mesh generation tool to construct a deep coordinate transformation network of multiple types;

[0008] S2: training the constructed deep coordinate transformation network to obtain a trained deep coordinate transformation network;

[0009] S3: embedding the trained deep coordinate transformation network as a preprocessing module into a neural network solver to obtain an embedded neural network solver;

[0010] S4: using the embedded neural network solver to solve the partial differential equation to obtain a solving enhancement result, and completing the solving enhancement of the neural network partial differential equation.

[0011] Further, the deep coordinate transformation network comprises a mapping function and a forward propagation, wherein the expression of the mapping function is:

[0012] ;

[0013] wherein, represents the mapped high-dimensional calculation space coordinate, represents neural network mapping, represents a physical space coordinate;

[0014] The expression of the forward propagation is:

[0015] ;

[0016] wherein, represents the output of the 0th layer, represents the output of the lth hidden layer, and represents a network parameter, represents the output of the 7th layer, represents the number of hidden layers.

[0017] Further, the expression of the loss function of the deep coordinate transformation network training is:

[0018] ;

[0019] ;

[0020] ;

[0021] ;

[0022] wherein, represents a loss function result, , and represents an adjustable hyperparameter balancing the influence of each constraint, represents an edge length loss, represents an isotropic loss, represents a regularization loss, represents the number of edges, represents the length of the ith edge in the computational space, represents the target edge length of the mesh, represents the total number of line elements from the cell center of each face element to its vertices, represents the length of the jth line element in the computational space, represents the target length of the jth line element, represents a quadrilateral mesh, represents a triangular mesh, represents the number of face elements, represents the coordinates of the face element center of the mth face element in the computational space, represents the number of vertices of the mth face element, represents the coordinates of the 0th vertex of the mth face element in the computational space, represents the sum of squares of vectors.

[0023] Further, the S3 comprises:

[0024] The trained deep coordinate transformation network is embedded into the neural network solver as a preprocessing module, and the non-uniform mesh coordinates of the physical space are mapped to the computational space through the preprocessing module, realizing adaptive magnification of the local encryption region, improving the expression ability of the neural network to the local features of the complex flow field, and obtaining the embedded neural network solver.

[0025] Further, the S4 comprises:

[0026] The partial differential equation is solved by using the embedded neural network solver, and in the solving process, all automatic differentiation and residual calculation are performed in the computational space coordinates of the trained deep coordinate transformation network, to obtain a solving enhancement result, and complete the solving enhancement of the neural network partial differential equation.

[0027] The beneficial effects of the present application are: the present application provides a partial differential equation neural network solving method enhanced by a deep coordinate transformation network, a trained deep coordinate transformation network is used as a preprocessing module and embedded in a neural network solver to obtain an embedded neural network solver; the embedded neural network solver is used to solve the partial differential equation to obtain a solving enhancement result. (1) The deep coordinate transformation network is trained to learn the grid distribution information, a continuous and differentiable coordinate transformation mechanism is constructed, the key area is enlarged in the calculation space, and the local fitting capability of the neural network is enhanced; (2) The method is not limited to structured grids, but can also be applied to unstructured grids, has good geometric generality and expandability, and can process complex three-dimensional shapes widely existing in typical engineering; (3) The proposed coordinate transformation mechanism is a continuous and differentiable deep model module, which can be seamlessly integrated with the neural network solver, and the training process still retains the advantages of automatic differentiation and end-to-end back propagation; (4) The method shows superior error convergence and prediction accuracy in high Reynolds number turbulent flow and three-dimensional airfoil flow problems, and significantly expands the application range of the neural network solver. BRIEF DESCRIPTION OF DRAWINGS

[0028] The present specification will be further illustrated in the form of exemplary embodiments, which will be described in detail with reference to the accompanying drawings. These embodiments are not limiting, and in these embodiments, the same numbers represent the same structures, wherein:

[0029] Figure 1 is an exemplary flowchart of a partial differential equation neural network solving method enhanced by a deep coordinate transformation network according to some embodiments of the present specification;

[0030] Figure 2 is an exemplary schematic diagram of a neural network solver framework based on a deep coordinate transformation network (DCTN) according to some embodiments of the present specification;

[0031] Figure 3 is an exemplary schematic diagram of different types of grid samples for verifying the effect of the present application according to some embodiments of the present specification;

[0032] Figure 4 is an exemplary schematic diagram of the loss function convergence history curve of the DCTN solving PDE based on different grid training according to some embodiments of the present specification;

[0033] Figure 5 is an exemplary schematic diagram of the error convergence history curve of the DCTN solving PDE based on different grid training according to some embodiments of the present specification. DETAILED DESCRIPTION

[0034] The specific embodiments of the present application are described below to enable those skilled in the art to understand the present application, but it should be clear that the present application is not limited to the scope of the specific embodiments, and for those skilled in the art, any changes that are obvious within the spirit and scope of the present application defined and determined by the appended claims are included in the protection of the present application.

[0035] Embodiment one

[0036] Figure 1 is an exemplary flowchart of a partial differential equation neural network solving method enhanced by a deep coordinate transformation network according to some embodiments of the present specification. As shown in Figure 1 , the flow includes the following steps. In some embodiments, the flow can be executed by a processor.

[0037] S1: Use a mesh generation tool to construct multiple types of deep coordinate transformation networks.

[0038] The deep coordinate transformation network (DCTN) is a fully connected neural network for realizing continuous coordinate mapping from physical space to computational space. For example, the deep coordinate transformation network can include a mapping function and forward propagation, realizing continuous coordinate mapping from physical space (two-dimensional) or (three-dimensional) to computational space ; wherein, represents a physical space coordinate vector, represents the i-th component, represents a coordinate vector of the computational space, represents the i-th component.

[0039] In some embodiments, the deep coordinate transformation network structure is a fully connected neural network containing 7 hidden layers, each containing 128 neurons. The activation function uses a continuous and differentiable hyperbolic tangent (tanh) function to ensure the smoothness and differentiability of the transformation function; wherein the mesh in the deep coordinate transformation network contains the regional scale information contained in the traditional high-quality mesh.

[0040] Make full use of the regional scale information contained in the traditional high-quality mesh as an auxiliary input of the deep neural network to guide the neural network to improve the analytical ability in high gradient regions such as boundary layers, shock waves, etc.

[0041] In some embodiments, the processor can use a professional mesh generation tool to construct three different types of meshes, such as the attached Figure 3The used grids are one structured grid and two unstructured grids, respectively. All the grids are carefully designed and locally refined, especially in the leading edge, trailing edge and boundary layer regions of the airfoil, to ensure sufficient resolution and accurate flow field capture in these high gradient flow regions.

[0042] In some embodiments, the expression of the mapping function can be:

[0043] ;

[0044] wherein, represents the mapped high-dimensional computational space coordinate, represents the neural network mapping, represents the physical space coordinate.

[0045] In some embodiments, the expression of the forward propagation can be:

[0046] ;

[0047] wherein, represents the output of the 0th layer, represents the output of the lth hidden layer, and represents the network parameters, represents the output of the 7th layer, represents the number of hidden layers.

[0048] S2: training the constructed depth coordinate transformation network to obtain a trained depth coordinate transformation network.

[0049] In some embodiments, the depth coordinate transformation network takes existing high-quality grids as training samples, and by minimizing the loss function constructed by the edge length and shape difference of the cells in the grid, it ensures that the grid distribution in the computational space after coordinate transformation tends to be uniform, and the optimization algorithm uses the L-BFGS optimizer to ensure efficient convergence of network training.

[0050] In some embodiments, the processor can constrain all edge lengths to be equal, setting them to prevent excessively large coordinate values, to obtain an edge length loss.

[0051] In some embodiments, the processor can define the center of the surface cell as the average of its vertex coordinates to obtain an isotropic loss. It can prevent the tilt of the surface cell and ensure that the distance from the center of the surface cell to its vertex remains almost equal, promoting the shape to approach an equilateral triangle or a square.

[0052] In some embodiments, the processor can impose a constraint that the cell center of each face element in the calculation space is aligned with the average of its vertices, resulting in a regularization loss that prevents the mesh from being distorted in the high-dimensional calculation space.

[0053] In some embodiments, the expression of the loss function of the deep coordinate transformation network training can be:

[0054] ;

[0055] ;

[0056] ;

[0057] ;

[0058] wherein, represents the loss function result, , and represents an adjustable hyperparameter balancing the influence of each constraint, represents an edge length loss, represents an isotropic loss, represents a regularization loss, represents the number of edges, represents the length of the ith edge in the calculation space, represents the target edge length of the mesh, represents the total number of line elements from the cell center of each face element to its vertices, represents the length of the jth line element in the calculation space, represents the target length of the jth line element, represents a quadrilateral mesh, represents a triangular mesh, represents the number of face elements, represents the coordinates of the face element center of the mth face element in the calculation space, represents the number of vertices of the mth face element, represents the coordinates of the 0th vertex of the mth face element in the calculation space, represents the sum of squares of vectors.

[0059] S3: Embed the trained deep coordinate transformation network as a preprocessing module into the neural network solver to obtain an embedded neural network solver.

[0060] In some embodiments, the trained deep coordinate transformation network can be embedded into a neural network solver as a preprocessing module, and the non-uniform grid coordinates of the physical space are mapped to the computational space through the preprocessing module to realize adaptive magnification of the local encryption area, improve the expression ability of the neural network to the local characteristics of the complex flow field, and obtain an embedded neural network solver.

[0061] In some embodiments, the trained deep coordinate transformation network supports integration with multiple types of neural network partial differential equation solvers, including but not limited to: physical information neural network (PINN) and time step oriented neural network (TSONN).

[0062] S4: using the embedded neural network solver to solve the partial differential equation to obtain a solution enhancement result, and completing the solution enhancement of the neural network partial differential equation.

[0063] The solution enhancement result is capable of improving the analytical ability of the neural network solver in the high gradient area, such as Figure 4 and Figure 5 As shown in the figures, for the high Reynolds number airfoil flow problem, the use of DCTN reduces the loss by one order of magnitude, and the error is reduced from 15% to within 3%; wherein Mesh represents the grid.

[0064] In some embodiments, the processor can use the embedded neural network solver to solve the partial differential equation, and all automatic differentiation and residual calculation are performed in the computational space coordinates of the trained deep coordinate transformation network during the solving process, to obtain a solution enhancement result and complete the solution enhancement of the neural network partial differential equation.

[0065] In some embodiments, as shown in Figure 4 , the loss function convergence curves with and without DCTN are shown, and it is obvious that the loss decreases faster after introducing DCTN, and the training process is more stable. The Figure 5 error convergence curve shows that DCTN significantly reduces the solution error and improves the prediction accuracy. The experimental results fully verify the effectiveness and superiority of the method in improving the processing ability of the neural network solver for complex engineering flow problems.

[0066] In some embodiments, the embedded neural network solver is suitable for incompressible flow, laminar and turbulent flow, heat conduction, electromagnetic wave propagation and other complex engineering problems based on PDE modeling, and is especially suitable for high Reynolds number flow, multi-scale structure, complex boundary and geometry numerical simulation scenarios.

[0067] In this way, not only has the local analysis ability of high gradient areas such as boundary layer and shock wave, but also is suitable for structured grid and unstructured grid, and can process complex shape geometry, so as to be widely adapted to various engineering application scenarios. By integrating the high-quality grid information designed by human experience into the deep neural network, the expression ability, convergence performance and solving accuracy of the deep neural network for multi-scale complex flow problems can be significantly improved, thereby opening up a new development direction for numerical simulation based on neural network. The expression ability and calculation accuracy of the deep neural network for engineering fluid mechanics, multi-scale modeling and high Reynolds number turbulent flow simulation are improved.

[0068] In this way, the area scale information contained in the traditional high-quality grid can be fully utilized as an auxiliary input of the deep neural network, so as to improve the analysis ability of the neural network solver in high gradient areas such as boundary layer, and specifically to promote the loss reduction and accuracy improvement. As shown in the convergence curves. Figure 4 and Figure 5 The performance improvement before and after introducing the DCTN is shown in the convergence curves. The effectiveness of the method in improving the solving accuracy and accelerating the convergence speed in high Reynolds number boundary layer flow is verified. The method also shows good accuracy and convergence performance in a plurality of representative engineering examples (including multi-section airfoils, three-dimensional wings, turbulent boundary layers, etc.), and is significantly superior to the neural network solver without using the DCTN in the case where the traditional method is prone to failure, such as high Reynolds number flow, thereby having definite engineering application value.

[0069] In some embodiments of the present specification, a deep coordinate transformation network enhanced partial differential equation neural network solving method is provided. A trained deep coordinate transformation network is used as a preprocessing module and embedded in a neural network solver to obtain an embedded neural network solver. The embedded neural network solver is used to solve a partial differential equation to obtain a solving enhanced result. (1) The deep coordinate transformation network is trained to learn grid distribution information, and a continuous and differentiable coordinate transformation mechanism is constructed to magnify key areas in the calculation space to enhance the local fitting ability of the neural network; (2) The method is not limited to structured grid, but can also be applied to unstructured grid, and has good geometric generality and expandability, and can process complex three-dimensional shapes widely existing in typical engineering; (3) The proposed coordinate transformation mechanism is a continuous and differentiable deep model module, which can be seamlessly integrated with the neural network solver, and the training process still retains the advantages of automatic differentiation and end-to-end back propagation; (4) The method shows superior error convergence and prediction accuracy in high Reynolds number turbulent flow and three-dimensional airfoil flow problems, and significantly expands the application range of the neural network solver.

[0070] Embodiment two

[0071] Due to the high Reynolds number and thin boundary layer problems in engineering, the neural network solver without DCTN is often difficult to capture the boundary layer structure and correctly analyze the flow field characteristics.

[0072] In some embodiments, a deep coordinate transformation network enhanced partial differential equation neural network solving method comprises:

[0073] S1: using a mesh generation tool, constructing a multi-type deep coordinate transformation network;

[0074] S2: training the constructed deep coordinate transformation network to obtain a trained deep coordinate transformation network;

[0075] S3: embedding the trained deep coordinate transformation network as a preprocessing module into a neural network solver to obtain an embedded neural network solver;

[0076] S4: using the embedded neural network solver to solve the typical incompressible Reynolds-averaged Navier-Stokes equation in the field of aeronautical engineering to obtain pressure coefficient solving enhancement results, and complete the enhancement of the neural network partial differential equation.

[0077] In some embodiments, the deep coordinate transformation network comprises a mapping function and a forward propagation, wherein the expression of the mapping function is:

[0078] ;

[0079] wherein, represents the mapped high-dimensional calculation space coordinates, represents neural network mapping, represents the physical space coordinates;

[0080] The expression of the forward propagation is:

[0081] ;

[0082] wherein, represents the output of the 0th layer, represents the output of the lth hidden layer, and represent network parameters, represents the output of the 7th layer, represents the number of hidden layers.

[0083] In some embodiments, the expression of the loss function of the deep coordinate transformation network training is:

[0084] ;

[0085] ;

[0086] ;

[0087] ;

[0088] wherein, denotes the loss function result, 、 and denotes the adjustable hyperparameters balancing the influence of each constraint, denotes the edge length loss, denotes the isotropic loss, denotes the regularization loss, denotes the number of edges, denotes the length of the i-th edge in the computational space, denotes the target edge length of the mesh, denotes the total number of line elements from the cell center of each face element to its vertices, denotes the length of the j-th line element in the computational space, denotes the target length of the j-th line element, denotes a quadrilateral mesh, denotes a triangular mesh, denotes the number of face elements, denotes the coordinates of the face element center of the m-th face element in the computational space, denotes the number of vertices of the m-th face element, denotes the coordinates of the 0-th vertex of the m-th face element in the computational space, denotes the sum of squares of vectors.

[0089] In some embodiments, the S3 comprises:

[0090] The trained deep coordinate transformation network is embedded into the neural network solver as a preprocessing module, and the non-uniform mesh coordinates of the physical space are mapped to the computational space through the preprocessing module, realizing adaptive magnification of the local encryption region, improving the expression ability of the neural network to the local features of the complex flow field, and obtaining the embedded neural network solver.

[0091] In some embodiments, the S4 comprises:

[0092] The embedded neural network solver is used to solve the partial differential equation, and all automatic differentiation and residual calculation are performed in the computational space coordinates of the trained deep coordinate transformation network during the solving process, obtaining a solving enhancement result, and completing the solving enhancement of the neural network partial differential equation.

[0093] In the simulation of high Reynolds number airfoil flow, the use of DCTN makes the loss decrease by one order of magnitude, and the final pressure coefficient error is reduced from 15% to within 3%. As Figure 4 and Figure 5 The comparative results show that the use of DCTN makes the loss decrease by one order of magnitude, and the final pressure coefficient error is reduced from 15% to within 3%. This fully verifies the effectiveness and advantage of DCTN in solving high Reynolds number complex flow with large gradients.

Claims

1. A deep coordinate transformation network enhanced partial differential equation neural network solving method, characterized in that, The application relates to a method for solving a partial differential equation by using a neural network, and belongs to the technical field of computational fluid dynamics. The method comprises the following steps: S1: generating a calculation grid of a target calculation region or shape by using a grid generation tool; the method comprises the following steps: constructing three different types of grids by using a professional grid generation tool, wherein the three different types of grids are one set of structured grids and two sets of unstructured grids, and the grids are encrypted in the regions of airfoil leading edges, trailing edges and boundary layers; S2: training a depth coordinate transformation network based on the generated calculation grid, so as to obtain a trained depth coordinate transformation network; ； ； ； ； wherein, represents a loss function result, , and represents a tunable hyperparameter balancing the influence of each constraint, represents an edge length loss, represents an isotropic loss, represents a regularization loss, represents the number of edges, represents the length of the ith edge in the computational space, represents the target edge length of the mesh, represents the total number of wire elements from the cell center of each face element to its vertices, represents the length of the jth wire element in the computational space, represents the target length of the jth wire element, represents a quadrilateral mesh, represents a triangular mesh, represents the number of face elements, represents the coordinates of the face element center of the mth face element in the computational space, represents the number of vertices of the mth face element, represents the coordinates of the 0th vertex of the mth face element in the computational space, represents the sum of squares of vectors; The expression of a loss function of the depth coordinate transformation network training is as follows: S3: embedding the trained depth coordinate transformation network into a neural network solver as a preprocessing module, so as to obtain an embedded neural network solver; 2. The deep coordinate transformation network enhanced partial differential equation neural network solving method according to claim 1, characterized in that, S4: solving a partial differential equation by using the embedded neural network solver, so as to obtain a solving enhancement result and complete solving enhancement of the neural network partial differential equation. ； wherein, represents the mapped high-dimensional computational space coordinate, represents the neural network mapping, represents the physical space coordinate; The depth coordinate transformation network comprises a mapping function and forward propagation, wherein the expression of the mapping function is as follows: ； wherein, represents the output of the 0th layer, represents the output of the lth hidden layer, and represents the network parameters, represents the output of the 7th layer, represents the number of hidden layers.

3. The deep coordinate transformation network enhanced partial differential equation neural network solving method according to claim 1, wherein, The expression of the forward propagation is as follows: The S3 comprises:

4. The deep coordinate transformation network enhanced partial differential equation neural network solving method of claim 1, wherein, The trained depth coordinate transformation network is embedded into the neural network solver as the preprocessing module, the non-uniform grid coordinates of a physical space are mapped to a calculation space through the preprocessing module, adaptive amplification of a locally encrypted region is realized, the expression capability of the neural network for local characteristics of a complex flow field is improved, and the embedded neural network solver is obtained. The S4 comprises: The partial differential equation is solved by using the embedded neural network solver, all automatic differentiation and residual error calculation are performed in the calculation space coordinates of the trained depth coordinate transformation network in the solving process, a solving enhancement result is obtained, and solving enhancement of the neural network partial differential equation is completed.

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