Manifold neural operator construction method for boundary value problem

Abstract

The invention discloses a manifold neural operator construction method for a boundary value problem, and relates to the technical field of machine learning, and the method comprises the following steps: S1, obtaining a geometric Laplacian operator feature function psi g where a solution function is located and a geometric Laplacian operator feature function psi b where a boundary condition function is located; s2, constructing a manifold neural operator based on the characteristic function psi g, and obtaining a geometric coding network so as to obtain a geometric embedding function group; s3, constructing a manifold neural operator based on the characteristic function psi b, and obtaining a boundary condition coding network so as to obtain a boundary condition embedding amount; and S4, embedding a combined approximation solution function of the vectors based on the geometric embedding function group and the boundary condition function so as to construct a boundary value problem-oriented manifold neural operator. According to the manifold neural operator construction method for the boundary value problem, the embedding of complex geometric information in the boundary value problem solving process is realized, and the difficulty of solving the boundary value problem by a network is reduced.

Description

Technical Field

[0001] The present invention relates to the technical field of machine learning, and in particular to a method for constructing a manifold neural operator for boundary value problems. Background Art

[0002] Boundary value problems refer to finding solutions to differential equations based on given boundary conditions, and are widely found in various engineering fields. For example, in each iteration of the optimization of the curing temperature process of composite components, it is necessary to solve the deformation field (solution) distributed on the components based on the current temperature process (boundary conditions) to provide a basis for the next round of adjustments to the temperature process. For practical engineering problems, differential equations are usually defined on complex geometric domains, such as composite components with complex structures, aircraft shapes with complex surfaces, etc. Therefore, boundary value problems defined on complex geometric domains are of great significance.

[0003] The solution to the boundary value problem can be regarded as establishing a mapping between the boundary condition function and the solution function. The traditional numerical calculation solution method has low computational efficiency and is difficult to support the large-scale iterative optimization required in the actual engineering design process. Once the data-driven model is trained, the efficiency of solving the boundary value problem can be significantly improved. Among them, the neural operator can learn the operator mapping between functions, which is an important trend in the data-driven solution of boundary value problems. Since boundary value problems in the engineering field often involve complex geometric domains, traditional neural operators rely on Fourier transform or wavelet transform and can only handle operator learning problems on simple regular geometric domains. Patent CN116187386A proposes a neural operator construction method for complex geometric shapes, which extends the applicable scenarios of operators from Euclidean space to Riemann manifolds, and realizes operator learning on complex geometric domains. However, for boundary value problems where the input (boundary conditions) and output (solution function) are defined on different geometric domains, the above-mentioned manifold neural operator cannot be directly applied, and basis transformations need to be introduced in the model, which can easily lead to unstable prediction results. Summary of the invention

[0004] The purpose of the present invention is to provide a method for constructing a manifold neural operator for boundary value problems. According to the characteristics that the solution function and boundary condition function of boundary value problems in engineering are located in different complex geometric domains, two sub-networks are constructed based on the manifold neural operator to respectively encode the geometric domain information of the solution function and the boundary condition function, and then the solution function is approximated based on the combination of the sub-network outputs.

[0005] To achieve the above object, the present invention provides a method for constructing a manifold neural operator for a boundary value problem, the steps of which are as follows:

[0006] S1. Get the Laplace operator characteristic function Ψ of the geometry where the solution function is located g and the Laplace operator characteristic function Ψ of the geometry where the boundary condition function is locatedb ;

[0007] S2, based on characteristic function Ψ g Construct a manifold neural operator to obtain a geometric coding network, encode geometric information through the geometric coding network, and obtain a geometric embedding function group;

[0008] S3, based on characteristic function Ψ b Construct a manifold neural operator, obtain a boundary condition encoding network, encode the boundary condition function through the boundary condition encoding network, and obtain an embedding vector of the boundary condition function;

[0009] S4. Based on the combination of geometric embedding function group and boundary condition function embedding vector, the solution function is approximated to construct a manifold neural operator for boundary value problems.

[0010] Preferably, in S1, the characteristic function Ψ is obtained g and the characteristic function Ψ b The method is to solve the characteristic equation defined in the geometry where the solution function is located and the characteristic equation defined in the geometry where the boundary condition function is located.

[0011] Preferably, in S2, the geometric information represents the geometric information of the geometric domain where the solution function is located, and is represented by node coordinates.

[0012] In S2 and S3, the popular neural operator is composed of several Laplace kernel integration modules, which are constructed by combining three sub-modules: frequency domain transformation, linear transformation or nonlinear activation. The combination method adopts: only including frequency domain transformation sub-module, including frequency domain transformation and linear transformation sub-module; including frequency domain transformation and nonlinear activation sub-module; or including one of frequency domain transformation, linear transformation and nonlinear activation sub-module.

[0013] The frequency domain transformation submodule includes three units: encoding, parameterization, and decoding. The encoding unit uses the obtained Laplace operator characteristic function to map the input function of the module to the frequency domain space to obtain the coordinates of the input under the characteristic function; the parameterization unit uses the parameterization matrix to perform linear or nonlinear transformation on the obtained coordinates; the decoding unit restores the coordinates obtained after parameterization to the original function space according to the Laplace operator characteristic function.

[0014] The linear transformation submodule performs a linear transformation on the input function of the module.

[0015] The nonlinear activation submodule uses a nonlinear activation function to perform nonlinear processing on the input function of the module.

[0016] Preferably, in S4, the combination of the geometric embedding function group and the boundary condition function embedding vector includes linear transformation or nonlinear transformation.

[0017] Therefore, the present invention adopts the above-mentioned steps to construct a manifold neural operator method for boundary value problems. By constructing two sub-networks to encode geometric information and boundary condition functions respectively, the iterative kernel integral neural operator model can be directly applied without introducing additional modifications, thereby improving the stability of the prediction results and further improving the accuracy of solving boundary value problems.

[0018] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 A schematic diagram of the steps of an embodiment of a method for constructing a manifold neural operator for boundary value problems according to the present invention;

[0020] Figure 2 A schematic diagram of a boundary value problem according to an embodiment of the present invention;

[0021] Figure 3 A schematic diagram of a given geometric Laplace operator characteristic function solved in heat transfer according to an embodiment of the present invention;

[0022] Figure 4 The figure is a comparison between the actual value of an embodiment of the present invention under a set of test data and the predicted value of the method. DETAILED DESCRIPTION

[0023] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.

[0024] Example

[0025] like Figure 2 As shown, this embodiment focuses on solving the heat transfer boundary value problem, and the input is the boundary temperature function a(x′) on the boundary, such as Figure 2 The output is the temperature function u(x) over the entire geometric domain. The geometric domain of this problem is relatively complex, such as Figure 2 As shown in (b), when solving this problem, the geometric domain is meshed, and the boundary condition function and the solution function are discretized into the temperature function on the corresponding nodes, where the number of nodes on the boundary and the global geometric domain are 186 and 7199, respectively. 210 groups of The function pairs are used as training data, and 90 groups of function pairs are used as test data for the method. At the same time, the global geometric node coordinates are extracted to represent the global geometric information, and the geometric information is embedded in the learning process of the network.

[0026] like Figure 1 As shown, the steps are as follows:

[0027] S1. Get the Laplace operator characteristic function Ψ of the geometry where the solution function is locatedg and the Laplace operator characteristic function Ψ of the geometry where the boundary condition function is located b .

[0028] The two sets of Laplace operator characteristic functions are obtained by solving the characteristic equations of the Laplace operator on the global grid and the boundary grid respectively. At the same time, the first 128 sets of characteristic functions are selected when constructing the kernel integration module using the two sets of characteristic functions. That is, the mathematical structure of the two sets of characteristic functions is: Figure 3 (a) represents the Laplace operator characteristic function of the geometry where the solution function is located, and (b) represents the Laplace operator characteristic function of the geometry where the boundary condition function is located.

[0029] S2, based on characteristic function Ψ g Construct a manifold neural operator to obtain a geometric coding network, encode geometric information through the geometric coding network, and obtain a geometric embedding function group;

[0030] S3, based on characteristic function Ψ b Construct a manifold neural operator, obtain a boundary condition encoding network, encode the boundary condition function through the boundary condition encoding network, and obtain an embedding vector of the boundary condition function;

[0031] Based on the idea of ​​iterative kernel integration, the corresponding characteristic functions are used to construct geometric embedding networks and boundary condition analysis networks. The input function can be projected onto the corresponding characteristic function and then parameterized to achieve operator mapping.

[0032] The boundary condition function f input in this embodiment is bc =a(x), that is, the boundary temperature function a(x′), is discretized into the values ​​at the corresponding nodes, that is, The global geometric information to be embedded is represented by the global grid node coordinates g: The encoded geometric embedding function set and the boundary condition function embedding vector

[0033] S4. Based on the combination of geometric embedding function group and boundary condition function embedding vector, the solution function is approximated to construct a manifold neural operator for boundary value problems.

[0034] In this embodiment, the combination of the geometric embedding function group and the boundary condition function embedding vector adopts linear transformation, that is, the approximation of the solution function

[0035] 300 sets of The function constructs a data set for machine learning. 210 groups are randomly selected for model training and 90 groups are used to test the model effect. The test set data does not participate in the training of the model. The classic neural network training method is used in the training process. After 2000 iterations, the test error on the test data is 0.0079% (relative L2 error). Figure 4 The prediction effect and prediction error of a set of test data in the test set are given, where (a) represents the boundary condition function, (b) represents the true value, (c) represents the predicted value, and (d) represents the prediction error. It can be seen that the method of the present invention can achieve good prediction results.

[0036] Therefore, the present invention adopts the above-mentioned structure to construct a manifold neural operator method for boundary value problems. By constructing two sub-networks to encode geometric information and boundary condition functions respectively, the iterative kernel integral neural operator model can be directly applied without introducing additional modifications, thereby improving the stability of the prediction results and further improving the accuracy of solving boundary value problems.

[0037] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for constructing a manifold neural operator for boundary value problems, characterized by: Here are the steps: S1. Get the Laplace operator characteristic function Ψ of the geometry where the solution function is located g and the Laplace operator characteristic function Ψ of the geometry where the boundary condition function is located b ; S2, based on characteristic function Ψ g Construct a manifold neural operator to obtain a geometric coding network, encode geometric information through the geometric coding network, and obtain a geometric embedding function group; S3, based on characteristic function Ψ b Construct a manifold neural operator, obtain a boundary condition encoding network, encode the boundary condition function through the boundary condition encoding network, and obtain an embedding vector of the boundary condition function; S4. Based on the combination of geometric embedding function group and boundary condition function embedding vector, the solution function is approximated to construct a manifold neural operator for boundary value problems.

2. A method for constructing a manifold neural operator for boundary value problems according to claim 1, characterized in that: In S1, obtain the characteristic function Ψ g and the characteristic function Ψ b The method is to solve the characteristic equation of the Laplace operator defined in the geometry where the solution function is located and the characteristic equation of the Laplace operator defined in the geometry where the boundary condition function is located.

3. The method for constructing a manifold neural operator for boundary value problems according to claim 1, characterized in that: In S2, the geometric information represents the geometric information of the geometric domain where the solution function is located, and is represented by node coordinates.

4. The method for constructing a manifold neural operator for boundary value problems according to claim 1, characterized in that: In S4, the combination of the geometric embedding function group and the boundary condition function embedding vector includes linear transformation or nonlinear transformation.

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