A Method for Constructing Manifold Neural Operators for Boundary Value Problems

Abstract

The present invention discloses a method for constructing a manifold neural operator for boundary value problems, which relates to the technical field of machine learning. The steps are as follows: S1. Obtain the Laplace operator eigenfunctions Ψ of the geometry where the solution function is located g and the Laplace operator eigenfunctions Ψ of the geometry where the boundary condition function is located b ; S2. Based on the eigenfunctions Ψ g construct a manifold neural operator to obtain a geometric encoding network, and thereby obtain a set of geometric embedding functions; S3. Based on the eigenfunctions Ψ b construct a manifold neural operator to obtain a boundary condition encoding network, and thereby obtain a boundary condition embedding quantity; S4. Approximate the solution function based on the combination of the set of geometric embedding functions and the boundary condition function embedding vector, so as to construct a manifold neural operator for boundary value problems. The method for constructing a manifold neural operator for boundary value problems adopting the above steps realizes the embedding of complex geometric information in the process of solving boundary value problems and reduces the difficulty of the network in solving boundary value problems.

Description

Technical Field

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

[0002] A boundary value problem refers to finding the solution of a differential equation according to given boundary conditions, which widely exists in various engineering fields. For example, in each iteration of the curing temperature process optimization of composite material components, it is necessary to solve the deformation field (solution) distributed on the component according to the current temperature process (boundary condition) to provide a basis for the next adjustment of the temperature process. For practical engineering problems, differential equations are usually defined on complex geometric domains, such as composite material components with complex structures, aircraft shapes with complex curved surfaces, etc. Therefore, boundary value problems defined on complex geometric domains are of great significance.

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

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

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

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

[0007] S2. Based on the eigenfunction Ψ g Construct a manifold neural operator to obtain a geometric encoding network. Encode the geometric information through the geometric encoding network to obtain a set of geometric embedding functions;

[0008] S3. Based on the eigenfunction Ψ b Construct a manifold neural operator to obtain a boundary condition encoding network. Encode the boundary condition function through the boundary condition encoding network to obtain a boundary condition function embedding vector;

[0009] S4. Combine and approximate the solution function based on the set of geometric embedding functions and the boundary condition function embedding vector, thereby constructing a manifold neural operator for the boundary value problem.

[0010] Preferably, in S1, the method for obtaining the eigenfunction Ψ g and the eigenfunction Ψ b is: Solve the eigenvalue equation defined on the geometry where the solution function is located and the eigenvalue equation defined on 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 characterized by node coordinates.

[0012] In S2 and S3, the manifold neural operator is composed of several Laplacian kernel integral modules. The Laplacian kernel integral module is constructed by combining three sub-modules: frequency domain transformation, linear transformation, or non-linear activation. The combination methods are: only including the frequency domain transformation sub-module, including the frequency domain transformation and linear transformation sub-modules; including the frequency domain transformation and non-linear activation sub-modules; or including one of the frequency domain transformation, linear transformation, and non-linear activation sub-modules.

[0013] The frequency domain transformation sub-module includes three units: encoding, parameterization, and decoding. The encoding unit maps the input function of the module to the frequency domain space using the obtained Laplacian operator eigenfunction to obtain the coordinates of the input under this eigenfunction; the parameterization unit performs a linear transformation or non-linear transformation on the obtained coordinates using a parameterization matrix; the decoding unit restores the coordinates obtained after parameterization to the original function space according to the Laplacian operator eigenfunction.

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

[0015] The non-linear activation sub-module performs non-linear processing on the input function of the module using a non-linear activation function.

[0016] Preferably, in S4, the combination method of the set of geometric embedding functions and the boundary condition function embedding vector includes linear transformation or non-linear transformation.

[0017] Therefore, the manifold neural operator construction method for boundary value problems according to the above steps is adopted in the present invention. 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, improving the stability of the prediction results and further enhancing the solution accuracy for boundary value problems.

[0018] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Description of the Drawings

[0019] Figure 1 It is a schematic diagram of the steps of an embodiment of the manifold neural operator construction method for boundary value problems of the present invention;

[0020] Figure 2 It is a schematic diagram of the boundary value problem of an embodiment of the present invention;

[0021] Figure 3 It is a schematic diagram of the eigenfunction of the given geometric Laplace operator to be solved in heat transfer of an embodiment of the present invention;

[0022] Figure 4 It is a comparison of the true value and the predicted value of the present method under a set of test data of the present invention. Detailed Embodiment

[0023] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0024] Embodiment

[0025] As Figure 2 shown, this embodiment focuses on solving the boundary value problem of heat transfer. The input is the boundary temperature function a(x′) on the boundary, as shown in (a) of Figure 2 . The output is the temperature function u(x) over the entire geometric domain. The geometric domain where this problem lies is relatively complex, as shown in (b) of Figure 2 . When solving this problem, the geometric domain is meshed, and the boundary condition function and the solution function are discretized into temperature functions at corresponding nodes. The number of nodes on the boundary and the global geometric domain are 186 and 7199 respectively. 210 groups of function pairs are obtained as training data through numerical simulation methods, 90 groups of function pairs are used as test data for the method, and 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] As Figure 1 shown, the steps are as follows:

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

[0028] Two sets of eigenfunctions of the Laplace operator are obtained by solving the eigenvalue equations of the Laplace operator on the global grid and the boundary grid respectively. At the same time, when using the two sets of eigenfunctions to construct the kernel integral module, the first 128 are selected, that is, the mathematical structures of the two sets of eigenfunctions are: Figure 3 In (a), it represents the eigenfunction of the Laplace operator of the geometry where the solution function is located, and in (b), it represents the eigenfunction of the Laplace operator of the geometry where the boundary condition function is located.

[0029] S2. Based on the eigenfunction Ψ g Construct a manifold neural operator to obtain a geometric encoding network, and encode the geometric information through the geometric encoding network to obtain a set of geometric embedding functions;

[0030] S3. Based on the eigenfunction Ψ b Construct a manifold neural operator to obtain a boundary condition encoding network, and encode the boundary condition function through the boundary condition encoding network to obtain a boundary condition function embedding vector;

[0031] Based on the idea of iterative kernel integration, use the corresponding eigenfunctions to construct a geometric embedding network and a boundary condition analysis network, which can project the input function onto the corresponding eigenfunctions, and then perform parameterization, thereby realizing operator mapping.

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

[0033] S4. Based on the combination of the set of geometric embedding functions and the boundary condition function embedding vector, approximate the solution function, thereby constructing a manifold neural operator for the boundary value problem.

[0034] In this embodiment, the combination method of the set of geometric embedding functions and the boundary condition function embedding vector is selected as a linear transformation, that is, the approximation of the solution function

[0035] 300 sets are obtained through numerical calculation methods The function pairs are used to construct the dataset for machine learning. 210 groups are randomly selected for the training of the model, and 90 groups are used for testing the model effect. The test set data does not participate in the training of the model. During the training process, the classical neural network training method is adopted. 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. Among them, (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 a method for constructing a manifold neural operator for boundary value problems with the above structure. By constructing two sub-networks to encode geometric information and boundary condition functions respectively, the iterative kernel integral classification neural operator model can be directly applied without introducing additional modifications, improving the stability of the prediction results, and further improving the solution accuracy for boundary value problems.

[0037] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify or equivalently replace the technical solutions of the present invention, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for constructing a manifold neural operator for boundary value problems, characterized in that: The steps are as follows: S1. Obtain the Laplace operator eigenfunction Ψ of the geometry where the solution function is located g and the Laplace operator eigenfunction Ψ of the geometry where the boundary condition function is located b ; Two sets of Laplace eigenfunctions are obtained by solving the eigenvalue equations of the Laplace operator on the global grid and the boundary grid respectively, and the mathematical structures of the two sets of eigenfunctions are set as follows: S2. Based on the feature function Ψ g Construct a manifold neural operator to obtain a geometric encoding network, and encode the geometric information through the geometric encoding network to obtain a set of geometric embedding functions; S3. Based on the feature function Ψ b Construct a manifold neural operator to solve the heat transfer boundary value problem, obtain a boundary condition encoding network, encode the boundary condition function through the boundary condition encoding network, and obtain a boundary condition function embedding vector; Based on the idea of iterative kernel integration, use the corresponding eigenfunctions to construct a geometric embedding network and a boundary condition analysis network, project the input function onto the corresponding eigenfunctions, parameterize it, and then realize the operator mapping; The input boundary condition function is f bc = a(x'), that is, the boundary temperature function a(x'); Discretize the boundary temperature function a(x') into values corresponding to the nodes, i.e., The global geometric information to be embedded is characterized by the global grid node coordinates g, that is Obtain a geometric embedding function group through encoding and a boundary condition function embedding vector S4. Based on the combined approximation solution function of the geometric embedding function group and the boundary condition function embedding vector, construct a manifold neural operator for the boundary value problem; The combination method of the geometric embedding function group and the boundary condition function embedding vector is selected as a linear transformation, that is, the approximate function of the solution function Obtained by numerical calculation method {(a i , u i )} function pair, where u(x) represents the temperature function over the entire geometric domain, construct a machine learning data set, randomly divide the data set, and use one group for model training and one group for testing the model effect; The test set data does not participate in the training of the model. During the training process, a classical neural network training method is adopted for iterative training.

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

3. A 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 characterized by node coordinates.

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

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