A manifold neural operator training method, device and application suitable for non-specific geometric shapes

Abstract

The present invention discloses a manifold neural operator training method, device and application applicable to non-specific geometric shapes, which relates to the field of machine learning technology. The steps are as follows: S1, obtaining a manifold neural operator defined in N S1, input functions and output functions on different geometric shapes as training data; S2, obtain the Laplace operator feature function set corresponding to different geometric shapes; S3, construct a manifold neural operator model including one or more Laplace kernel integration modules; S4, train the constructed manifold neural operator model, and dynamically use the Laplace operator feature function set matching the geometric shape in the Laplace kernel integration module according to the geometric shape corresponding to the training data input each time. The present invention adopts a manifold neural operator training method, device and application suitable for non-specific geometric shapes with the above structure. The trained model is suitable for non-specific geometric shapes, and there is no need to train the model separately for each specific geometric shape.

Description

Technical Field

[0001] The present invention relates to the field of machine learning technology, and in particular to a manifold neural operator training method, device and application suitable for non-specific geometric shapes. Background Art

[0002] With the development of machine learning, various data-driven models are moving towards the direction of "more general large models". For neural operator models, the demand for them in many fields has also expanded from operator learning tasks for specific geometric shapes to operator learning tasks for non-specific geometric shapes. For example, in the field of structural optimization design, neural operator models are needed to predict the structural response performance under different geometries to reduce the cost of physical experiments. The existing invention patent with application number 2023101222893 proposes a method for constructing neural operators for complex geometric shapes. However, this method only uses the characteristic function set of the Laplace operator corresponding to a specific geometric shape, which results in the construction of neural operator models suitable for specific geometric shapes, and it is difficult to apply to other non-specific geometric shapes. For new geometric shapes, new models need to be retrained, which is difficult to meet the requirements of tasks such as structural optimization design. Summary of the invention

[0003] The purpose of the present invention is to provide a manifold neural operator training method, device and application suitable for non-specific geometric shapes. The trained model is suitable for non-specific geometric shapes and there is no need to train the model separately for each specific geometric shape.

[0004] To achieve the above object, the present invention provides a training method for manifold neural operators applicable to non-specific geometric shapes, the steps of which are as follows:

[0005] S1. For Geometric shapes , respectively, to obtain the Group input functions and output functions together as training data;

[0006] S2. Acquisition The geometric shapes corresponding to Group Laplacian operator characteristic function set: ,in Indicates that each set of Laplacian operator characteristic functions contains the corresponding geometric shape Laplacian operator characteristic function;

[0007] S3, constructing a manifold neural operator model including one or more Laplace kernel integration modules;

[0008] S4. Use the training data obtained by S1 to train the manifold neural operator model constructed by S3. During the training process, according to the geometric shape corresponding to the training data input each time, dynamically use the Laplace operator feature function set matching the geometric shape in the Laplace kernel integral module.

[0009] Preferably, in S2, obtaining the Laplace operator characteristic function set corresponding to different geometric shapes specifically includes: firstly meshing the geometric shape, then defining the Laplace operator on the mesh, and obtaining the previous Laplace operator by solving the Laplace operator characteristic equation. Laplace operator characteristic functions constitute the Laplace operator characteristic function set.

[0010] Preferably, in S3, the process of constructing the Laplace kernel integration module specifically includes: each Laplace kernel integration module includes: a frequency domain transformation submodule, a linear transformation submodule and a nonlinear activation submodule, wherein the frequency domain transformation submodule includes a coding submodule. , parameterization ,decoding Three steps:

[0011] coding The input function of the frequency domain transformation submodule is decomposed by reducing the dimension using the Laplace operator characteristic function set to obtain a coefficient vector;

[0012] Parameterization Perform linear or nonlinear transformations on coefficient vectors;

[0013] decoding Parameterization using the Laplace operator characteristic function set The vector after dimensionality increase is reconstructed to obtain the output function of the frequency domain transformation submodule.

[0014] Preferably, in S4, dynamically using a Laplace operator feature function set that matches the geometric shape in a Laplace kernel integral module, specifically comprising: when constructing a Laplace kernel integral module during training, encoding using a Laplace operator feature function set that matches the geometric shape in a frequency domain transform submodule and decoding , parameterized Keep it common to all training data; the linear transformation submodule and the nonlinear activation submodule are kept common to all training data.

[0015] A training device for manifold neural operators of non-specific geometric shapes, comprising a data acquisition module, a Laplace operator feature function set acquisition module, a model building module, and a model training module.

[0016] A data acquisition module, used for acquiring training data, each set of training data includes an input function, an output function and a geometric shape;

[0017] A Laplace operator characteristic function set acquisition module is used to acquire Laplace operator characteristic function sets corresponding to different geometric shapes;

[0018] A model building module, used for building a manifold neural operator model including one or more Laplace kernel integration modules;

[0019] The model training module is used to train the constructed manifold neural operator model using training data.

[0020] An application of a training method for a manifold neural operator for non-specific geometric shapes, specifically comprising: first obtaining a Laplacian operator feature function set of a desired geometric shape, then inputting an input function of the desired geometric shape into a manifold neural operator model trained by S4, and encoding the Laplacian operator feature function set matching the desired geometric shape in a frequency domain transform submodule of a Laplacian kernel integral module. and decoding , and finally the output function is predicted by the manifold neural operator.

[0021] Preferably, the geometric shape to be determined is a geometric shape that is identical to or similar to a geometric shape included in the training data.

[0022] Therefore, the present invention adopts a manifold neural operator training method, device and application suitable for non-specific geometric shapes using the above structure and steps. The trained manifold neural operator model is not limited to a specific shape. After the training is completed, the output function can be predicted for a variety of geometric shapes similar to the training geometric shape, and there is no need to train a model for each shape separately. When there are enough types of geometric shapes used for training, it can even be achieved that for any geometric shape input, the output within the allowable error range can be obtained.

[0023] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 A flowchart of an embodiment of a training method for manifold neural operators of different geometric shapes according to the present invention;

[0025] Figure 2 It is a schematic diagram of a manifold neural operator and a Laplace kernel integration module in an embodiment of the present invention;

[0026] Figure 3 is a schematic diagram of the structure of a training device according to an embodiment of the present invention;

[0027] Figure 4 Schematic diagrams of six different geometric shapes of embodiments of the present invention;

[0028] Figure 5 A schematic diagram of the definition of the Laplacian operator on a triangular mesh in an embodiment of the present invention. DETAILED DESCRIPTION

[0029] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.

[0030] Example

[0031] This embodiment takes the two-dimensional Darcy seepage problem as an example, and applies Figure 3 The training device shown in FIG. 1 and the training steps are as follows: Figure 1 shown.

[0032] S1. Obtaining the permeability field defined on different geometric shapes As the input function, and its corresponding pressure field As the output function, as the training data.

[0033] The governing equation for the Darcy seepage problem is:

[0034] ;

[0035] in is the permeability field, which is the input function of this embodiment; is the pressure field, is the output function of this embodiment; is the source term, which is set to 1 in this embodiment. This embodiment uses a Gaussian random field and piecewise functions To generate the conductivity field As shown in FIG4 , six geometric shapes are designed in this embodiment, and 300 sets of permeability field input functions are generated by sampling from the Gaussian random field for each geometric shape, and then the corresponding pressure field output function is calculated using the PDE solver of MATLAB. The training data contains a total of 1800 sets of input-output function pairs.

[0036] .

[0037] S2. Obtain the Laplace operator characteristic function set corresponding to different shapes.

[0038] The geometric shapes are meshed using triangular meshes, such as Figure 4 and Figure 5 As shown, Figure 5 Can be seen as Figure 4The magnified view of the local area of ​​the node, where the "point" represents the mesh vertex and the "line" represents the edge of the triangular mesh unit. The definition and formula of the Laplacian operator of the triangular mesh are as follows, with the vertex Take this as an example (can be extended to other nodes):

[0039] ;

[0040] Represents a vertex All adjacent vertices of Figure 5 Midpoint Contains vertices The 6 adjacent vertices. and vertices Connected to form an edge, there are two units with this edge as a common edge. and represents the angles in the two cells opposite to the common side, and is the corresponding cotangent function value. is a scalar function defined on a geometric grid, and is a scalar function At the node and nodes Therefore, the vertex Superscalar Function The Laplace operator All adjacent nodes The cotangent function of , and function value To obtain.

[0041] After defining the Laplace operator on the triangular mesh, the Galerkin method is used in this implementation to solve the Laplace characteristic equation The solution is performed to obtain the Laplace operator characteristic function sets corresponding to different geometric shapes, and each characteristic function set contains 32 characteristic functions.

[0042] S3. Construct a manifold neural operator model including one or more Laplace kernel integration modules.

[0043] In this embodiment, the manifold neural operator includes four Laplace kernel integration modules, among which the encoding of the frequency domain transform submodule is and decoding The number of cutoffs is consistent with the number of characteristic functions contained in the Laplace operator characteristic function set, that is, 32; parameterization Linear transformation is adopted. The nonlinear activation submodule adopts gelu activation function. In addition, in order to enhance the learning ability of the manifold neural operator, this embodiment sets a fully connected neural network with 1 layer of 32 neurons before the Laplace kernel integration module, and sets 2 layers of fully connected neural networks after the Laplace kernel integration module, with the numbers of neurons being 128 and 1 respectively, and the gelu activation function is adopted for the fully connected layer.

[0044] S4. The constructed manifold neural operator model is trained using the training data. During the training process, according to the geometric shape corresponding to the training data input each time, a Laplace operator feature function set matching the geometric shape is dynamically used in the Laplace kernel integration module.

[0045] In this implementation, the manifold neural operator is trained using the classic neural network training method based on 1800 sets of training data. The batch size is 5, the learning rate is 0.001, the optimizer is Adam, and the loss function is defined as follows:

[0046] ;

[0047] in, is the number of training data, represents the true value of the pressure field, Represents the predicted value of the pressure field. After 1000 iterations of training, the test errors of 1200 test data sets (each shape contains 200 test data sets) are shown in Table 1. It can be seen that the test errors on the six geometric shapes are all less than 1.7%, proving that the model trained by the method of the present invention is applicable to different geometric shapes.

[0048] Table 1 Test error table of different shapes

[0049] .

[0050] 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 training method for manifold neural operators suitable for non-specific geometric shapes, characterized by: Here are the steps: S1. For Geometric shapes , respectively, to obtain the Group input functions and output functions together as training data; S2. Acquisition The geometric shapes corresponding to Group Laplacian operator characteristic function set: ,in Indicates that each set of Laplacian operator characteristic functions contains the corresponding geometric shape Laplacian operator characteristic function; S3, constructing a manifold neural operator model including one or more Laplace kernel integration modules; S4, using the training data obtained by S1 to train the manifold neural operator model constructed by S3, during the training process, according to the geometric shape corresponding to the training data input each time, dynamically using the Laplace operator characteristic function set matching the geometric shape in the Laplace kernel integral module; In S3, the process of constructing the Laplace kernel integration module specifically includes: each Laplace kernel integration module includes: a frequency domain transformation submodule, a linear transformation submodule and a nonlinear activation submodule, wherein the frequency domain transformation submodule includes a coding , parameterization ,decoding Three steps: coding The input function of the frequency domain transformation submodule is decomposed by reducing the dimension using the Laplace operator characteristic function set to obtain a coefficient vector; Parameterization Perform linear or nonlinear transformations on coefficient vectors; decoding Parameterization using the Laplace operator characteristic function set The vector after dimensionality increase and reconstruction is performed to obtain the output function of the frequency domain transformation submodule; In S4, a Laplace operator feature function set matching the geometric shape is dynamically used in a Laplace kernel integral module, specifically including: when constructing a Laplace kernel integral module during training, a Laplace operator feature function set matching the geometric shape is used for encoding in a frequency domain transform submodule and decoding , parameterized Keep it common to all training data; the linear transformation submodule and the nonlinear activation submodule are kept common to all training data.

2. The method for training a manifold neural operator applicable to non-specific geometric shapes according to claim 1, characterized in that: In S2, the Laplace operator characteristic function set corresponding to different geometric shapes is obtained, which specifically includes: firstly, the geometric shape is meshed, then the Laplace operator is defined on the mesh, and the previous function is obtained by solving the Laplace operator characteristic equation. Laplace operator characteristic functions constitute the Laplace operator characteristic function set.

3. A training device for manifold neural operators of non-specific geometric shapes as described in any one of claims 1 to 2, characterized in that: It includes data acquisition module, Laplace operator feature function set acquisition module, model construction module and model training module. A data acquisition module, used for acquiring training data, each set of training data includes an input function, an output function and a geometric shape; A Laplace operator characteristic function set acquisition module is used to acquire Laplace operator characteristic function sets corresponding to different geometric shapes; A model building module, used for building a manifold neural operator model including one or more Laplace kernel integration modules; The model training module is used to train the constructed manifold neural operator model using training data.

4. The application of a training method for a manifold neural operator of non-specific geometric shapes as claimed in any one of claims 1 to 2, characterized in that: Specifically include: First, the Laplace operator feature function set of the desired geometric shape is obtained, and then the input function of the desired geometric shape is input to the manifold neural operator model trained by S4, and the Laplace operator feature function set matching the desired geometric shape is used for encoding in the frequency domain transform submodule of the Laplace kernel integral module. and decoding , and finally the output function is predicted by the manifold neural operator.

5. The application of the training method for manifold neural operators of non-specific geometric shapes according to claim 4, characterized in that: The required geometric shape is a geometric shape that is the same as or similar to a geometric shape included in the training data.

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