A Problem-Independent Machine Learning Approach for Large-Scale Topology Optimization Design Based on Fourier Neural Operators

Abstract

This invention, belonging to the field of computational mechanics, presents a problem-independent machine learning method for large-scale topology optimization design based on Fourier neural operators. First, it constructs second-order coarse-scale units to improve the accuracy of structural response. Second, it introduces a neural network based on Fourier neural operators to predict multi-scale shape functions, which improves model training and inference efficiency and enhances generalization ability while preserving input-output spatial information. Simultaneously, it proposes a sample construction method based on the Cahn-Hilliard equation to generate density field data with continuous evolution characteristics, making the training samples more consistent with the real changes in material distribution during topology optimization, thereby improving the model's generalization ability and stability. Finally, through the synergistic effect of the above improvement strategies, this invention can significantly reduce the computational cost in large-scale topology optimization while maintaining computational accuracy.

Description

Technical Field

[0001] This invention belongs to the field of computational mechanics and relates to a machine learning-enhanced method for large-scale topology optimization design. Background Technology

[0002] The demand for high-performance, lightweight structures is widespread in engineering fields such as automotive, aerospace, and energy equipment. Topology optimization, as an advanced numerical design method for seeking the optimal material distribution under given constraints, has become an important means to achieve lightweight and high-performance structural design. Currently, topology optimization has formed various optimization frameworks, represented by the variable density method, level set method, progressive structural optimization method, and moving deformable component / hole method. As topology optimization gradually moves towards practical engineering applications, the scale of design problems continues to increase. In particular, three-dimensional high-resolution topology optimization typically requires the use of extremely fine meshes to discretize the design domain and necessitates a large number of iterative calculations, thus placing higher demands on computational efficiency and hardware performance.

[0003] For large-scale topology optimization problems, the computational bottleneck lies primarily in the need to repeatedly analyze the structural response, which changes with material distribution, during the optimization process. This typically requires dozens or even hundreds of iterations, each involving large-scale computations when using the finite element method (FEM). Especially in three-dimensional problems, the structural degrees of freedom can reach millions or even tens of millions, resulting in enormous memory consumption and severely limiting the widespread application of topology optimization in complex engineering structures. To address these issues, existing research has focused on parallel computing, multi-scale methods, multi-resolution, reduced-order modeling, and adaptive meshes, alleviating the computational burden of large-scale finite element analysis to some extent. However, further improving solution efficiency while maintaining accuracy and applicability remains a crucial problem to be solved in this field.

[0004] In recent years, machine learning methods have demonstrated significant advantages in high-dimensional feature extraction and learning complex mapping relationships, providing new insights for efficient numerical analysis in large-scale topology optimization. Some studies have attempted to utilize machine learning to establish mapping relationships between structures of different resolutions or to directly accelerate the high-resolution topology optimization process, thereby reducing computational costs. However, these methods typically rely on constructing training data specific to the problem. When the design domain, boundary conditions, load types, or material parameters change, the model often needs to be retrained, limiting its reusability and generalization ability. Furthermore, end-to-end machine learning methods usually require a large number of high-fidelity numerical samples as training data, while large-scale three-dimensional topology optimization itself is computationally expensive, further increasing the cost of model construction.

[0005] To overcome the aforementioned shortcomings, researchers have proposed Problem-Independent Machine Learning (PIML) and applied it to large-scale structural analysis and topology optimization. This method combines the ideas of the Extended Multiscale Finite Element Method (EPF), establishing a mapping relationship between the material distribution within coarse-scale elements and their corresponding multiscale shape functions. It replaces the local fine-scale numerical solutions in the EPF with machine learning models, thus significantly improving the efficiency of large-scale problem analysis. While existing PIML methods have achieved two orders of magnitude efficiency improvements, they still face several challenges: First, they typically use linear coarse-scale elements, with boundary displacements varying linearly along the element edges, potentially leading to jagged edges in the optimization results. Second, they often flatten the material distribution and multiscale shape functions into vectors for neural network input and output, failing to fully utilize their spatial distribution information. Third, the training cost of machine learning models remains high. Fourth, existing training samples often use random material distributions, which differ from the density distribution characteristics that evolve gradually during actual topology optimization, potentially affecting the model's generalization performance in real optimization problems.

[0006] Therefore, it is necessary to develop a large-scale topology optimization method that combines high accuracy, high efficiency, and good generalization ability, in light of existing large-scale topology optimization methods. Summary of the Invention

[0007] To address the high computational cost of finite element analysis (FEM) in variable density topology optimization methods and the challenges faced by existing PIML methods, this invention innovatively proposes a Problem-Independent Machine Learning (PIML) Method Based on the Fourier Neural Operator for Large-scale Topology Optimization Design. The objectives are: first, to construct second-order coarse-scale elements to alleviate the boundary jaggedness problem caused by the linear displacement assumption and improve the expressive power of the structural response; second, to introduce a neural network based on the Fourier neural operator to predict multi-scale shape functions, which can improve model training and inference efficiency and enhance generalization ability while maintaining input-output spatial information; and finally, to propose a dataset construction method based on the Cahn-Hilliard phase-field equation to generate material distributions with continuous evolution characteristics, making the training samples more consistent with the real changes in material distribution during topology optimization, thereby improving the model's generalization ability and stability. Through the synergistic effect of the above improvement strategies, this invention can significantly reduce the finite element solution cost in large-scale topology optimization while ensuring computational accuracy.

[0008] The technical solution of this invention: A problem-independent machine learning method for large-scale topology optimization design based on Fourier neural operators is proposed, with the following steps: Step 1: Construct multi-scale shape functions: Second-order coarse-scale elements are constructed based on the extended multi-scale finite element framework, and m is represented by the multi-scale shape function matrix N for scaling down. m The physical field of m fine-scale elements, wherein the multi-scale shape function matrix N is obtained through the following finite element analysis: The second-order coarse-scale element consists of 20 nodes, each containing degrees of freedom in the x, y, and z directions. Each node's degree of freedom corresponds to a multi-scale shape function component, and all multi-scale shape function components together form a complete multi-scale shape function matrix N. The calculation process for the multi-scale shape function component corresponding to any degree of freedom of a coarse-scale element node is as follows: the stiffness matrix is ​​calculated using the material distribution of the fine-scale element, and a unit displacement boundary condition is applied to the corresponding degree of freedom. Zero displacement boundary conditions are applied to the degrees of freedom of the remaining coarse-scale element nodes. At the same time, the fine-scale element degrees of freedom corresponding to the boundaries of the coarse-scale element are obtained by quadratic Lagrange interpolation. The corresponding multi-scale shape function component is then obtained through finite element analysis. Step 2: Construct the training dataset: The training dataset includes fine-scale element material distributions generated based on the Cahn-Hilliard phase-field equations, and four multi-scale shape function components of second-order coarse elements. The selected multi-scale shape function components satisfy the following criteria: 1) At any vertex of a coarse-scale unit, select a multi-scale shape function component; 2) Select three multi-scale shape function components at the midpoint of any coarse-scale unit; After selecting four multi-scale shape function components, they are obtained by finite element analysis in step 1; Step 3: Construct a neural network based on Fourier neural operators: The fine-scale unit material distribution generated in step 2 based on the Cahn-Hilliard phase-field equation is used as the input to the neural network, and the four multi-scale shape function components of the second-order coarse unit in step 2 are used as the output of the neural network. The neural network is trained to learn the mapping relationship between the fine-scale unit material distribution and the four multi-scale shape function components in the frequency domain space.

[0009] The neural network based on Fourier neural operators consists of three parts: an input mapping layer, a Fourier feature update layer, and an output mapping layer. First, the input mapping layer projects the fine-scale unit material distribution onto a high-dimensional feature space. Then, the Fourier layer performs a fast Fourier transform on the high-dimensional feature space and performs weighted learning of low-spectral information in the frequency domain. At the same time, it combines local linear transformation and nonlinear activation functions to achieve the fusion and update of global and local information. Finally, the output mapping layer projects the transformed high-dimensional feature space back into the target space to obtain multi-scale shape function components. Step 4: Preprocessing for topology optimization design: The objective condition is to minimize the structural flexibility; the parameters for topology optimization design are defined (design domain, discretization method, Young's modulus, Poisson's ratio, penalty coefficient, displacement / external force boundary conditions, filter radius, and volume fraction); the design domain is discretized into coarse-scale elements and fine-scale elements; the stiffness matrix, filter coefficient matrix, and element density are initialized. Step 5: Iterative solution of topology optimization design: The specific implementation process will be demonstrated through the following steps: 5.1) Based on the material distribution of the discrete fine-scale units, select the second-order coarse-scale units that need to solve the multi-scale shape function, and perform several spatial rotations on the corresponding material distributions to form the input of the neural network. 5.2) Call the neural network based on Fourier neural operators that has been trained in step 2, and use the fine-scale unit material distribution before and after each spatial rotation in step 5.1) as input to predict the corresponding four multi-scale shape function components. 5.3) Rotate the multiscale shape function components predicted in step 5.2) after each spatial rotation back to their original positions, and together with the multiscale shape function components before the spatial rotation, form a complete multiscale shape function matrix. 5.4) Correct the complete multi-scale shape function matrix in step 5.3) according to the rigid body displacement constraints; 5.5) Calculate the stiffness matrix and load vector of all second-order coarse-scale elements based on the corrected complete multi-scale shape function matrix in step 5.4), and assemble them. 5.6) Apply displacement / external force boundary conditions; 5.7) Solve for the displacement field of coarse-scale nodes; 5.8) Based on the corrected complete multi-scale shape function matrix in step 5.4), the displacement field of the coarse-scale node is mapped back to the displacement field of the fine-scale node. 5.8) Calculate the sensitivity of fine-scale elements; 5.9) Update the material distribution of fine-scale elements using optimization criteria; 5.10), return to step 5.1), until the maximum number of iterations is reached.

[0010] Furthermore, in step 2, by randomly generating initial perturbation conditions and model parameters, the Cahn-Hilliard phase-field equation is solved in a three-dimensional uniform computational domain to obtain a phase-field distribution with continuity and physical consistency. This distribution is then divided into multiple uniform subdomains, and the phase-field distribution of the subdomains is used as the distribution of fine-scale unit materials. Simultaneously, spatial rotation is performed to increase the diversity of neural network inputs.

[0011] Furthermore, regarding the spatial rotation in steps 5.1), 5.2), and 5.3): to reduce the complexity of the neural network, only the four multi-scale shape function components mentioned in step 2 are selected as the output of the neural network. When the selected four multi-scale shape function components are... and hour: The distribution of fine-scale materials is determined by the coordinate system. Rotate space to a new coordinate system The new coordinate system is obtained by predicting using the neural network trained in step 3. The four multi-scale shape function components below and Rotate them back to their original coordinate system The four multi-scale shape function components in the original coordinate system are obtained. and By repeating the above operations with different spatial rotation methods, all multi-scale shape function components can be obtained, which together form a complete multi-scale shape function matrix.

[0012] The beneficial effects of this invention are: (1) This invention constructs a second-order coarse-scale element and introduces a multi-scale shape function modeling method, which enables the coarse-scale element to describe more complex deformation modes, thereby improving the calculation accuracy of structural response and reducing the common boundary jaggedness and non-smoothness phenomena in topology optimization results. At the same time, this method has good scalability and can be extended to higher-order extended multi-scale finite element modeling and complex structural analysis problems.

[0013] (2) This invention uses Fourier neural operators to replace the repetitive numerical solution process in the extended multi-scale finite element method. By globally modeling the material distribution in the frequency domain, it achieves efficient mapping learning from the material distribution to the multi-scale shape function matrix. This method fully preserves the geometric and topological features of the structure field. At the same time, based on the low-dimensional representation in the frequency domain, it has stronger feature extraction capabilities and higher training efficiency, and has better computational efficiency and generalization ability in large-scale three-dimensional topology optimization problems.

[0014] (3) This invention proposes a multi-scale shape function dimensionality reduction and reconstruction strategy based on spatial rotation. It directly predicts only the multi-scale shape function components corresponding to the key degrees of freedom, and recovers the remaining responses through spatial rotation, thereby significantly reducing the model output dimension and training complexity. While ensuring computational accuracy, this strategy greatly reduces the scale of network parameters, improves the trainability and stability of the model, and enhances the model's adaptability under different loading directions and structural configurations.

[0015] (4) This invention employs a physical consistency dataset generation method based on the Cahn-Hilliard equation to construct training samples that better conform to the density evolution characteristics of topology optimization. Compared to randomly generated "mosaic" material distributions, the material field generated by this method exhibits continuity and phase separation characteristics, more closely resembling the structural evolution laws in actual topology optimization processes. This effectively reduces training data distribution bias and improves the generalization ability and robustness of machine learning models in real optimization problems. Furthermore, this dataset generation strategy can be extended to various multi-scale structure learning tasks. Attached Figure Description

[0016] Figure 1 This is a schematic diagram of the second-order coarse-scale unit of the present invention, wherein (a) is a schematic diagram before flipping and (b) is a schematic diagram after flipping; Figure 2 This is a schematic diagram of the Fourier neural operator network for predicting multi-scale shape functions according to the present invention, wherein (a) is a schematic diagram of the network input and output, (b) is a schematic diagram of the Fourier neural operator network, and (c) is a schematic diagram of the Fourier layer. Figure 3 This is a flowchart illustrating the operation of the problem-independent machine learning-based large-scale topology optimization design method based on Fourier neural operators of the present invention. Figure 4 This is a schematic diagram of the design domain and boundary conditions of an embodiment of the present invention; Figure 5 The diagram shows the topology optimization results of linear / second-order coarse-scale units under different discrete conditions according to the embodiments of the present invention. (a) is a linear coarse-scale unit: coarse-scale unit: 40x10x20, fine-scale unit: 8,000,000, total degrees of freedom: 24,422,103; (b) is a second-order coarse-scale unit: coarse-scale unit: 40x10x20, fine-scale unit: 8,000,000, total degrees of freedom: 24,422,103; (c) is a linear coarse-scale unit: coarse-scale unit: 80x20x40, fine-scale unit: 64,000,000, total degrees of freedom: 193,684,203; (d) is a second-order coarse-scale unit: coarse-scale unit: 80x20x40, fine-scale unit: 64,000,000, total degrees of freedom: 193,684,203. Detailed Implementation

[0017] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0018] A problem-independent machine learning method for large-scale topology optimization based on Fourier neural operators.

[0019] First, second-order coarse-scale elements are constructed based on the extended multi-scale finite element framework, such as... Figure 1 The diagram shown illustrates a second-order coarse-scale element. Each coarse-scale element consists of 20 nodes, each containing degrees of freedom in the x, y, and z directions. Each node's degree of freedom corresponds to a multi-scale shape function component, and all multi-scale shape function components together form a complete multi-scale shape function matrix. The calculation process for the multi-scale shape function component corresponding to any degree of freedom of a coarse-scale element node is as follows: the stiffness matrix is ​​calculated using the material distribution of the fine-scale element, and a unit displacement boundary condition is applied to the corresponding degree of freedom. Zero displacement boundary conditions are applied to the degrees of freedom of the remaining coarse-scale element nodes. Simultaneously, the fine-scale element degrees of freedom corresponding to the boundaries of the coarse-scale element are obtained using quadratic Lagrange interpolation. The corresponding multi-scale shape function component is then calculated through finite element analysis. This method can more accurately describe the nonlinear deformation modes within the element, thus effectively alleviating the boundary jaggedness problem caused by linear coarse-scale elements.

[0020] Secondly, a neural network based on Fourier neural operators is used to replace the repetitive finite element calculation process described above. Specifically, a uniformly discrete material distribution is used as input, and the corresponding multi-scale shape function components are used as output. To improve the learning efficiency and generalization ability of this mapping, this invention introduces Fourier neural operators, achieving efficient learning of continuous field information by globally modeling features in the frequency domain. The model consists of three parts: input mapping, Fourier feature update layer, and output mapping, so that both input and output are represented in the form of spatial fields, thereby fully preserving their geometric and topological information. In specific implementation, such as... Figure 2 The diagram shown illustrates a neural network based on Fourier neural operators. First, the Fourier neural operator projects the material distribution into a high-dimensional feature space through an input mapping layer. Then, in a multi-layer Fourier feature update module, a fast Fourier transform is performed on the high-dimensional feature space, and low-spectral information is learned in the frequency domain using weighted learning. Simultaneously, local linear transformations and nonlinear activation functions are combined to achieve the fusion and update of global and local information. Finally, the high-dimensional feature space is projected back into the target space through an output mapping layer, yielding multi-scale shape function components.

[0021] With the above structure, both the model input and output are represented in the form of a three-dimensional spatial field, thus avoiding the spatial information loss problem caused by vectorization processing and more completely preserving the geometric and topological relationships between material distribution and structural response. In addition, the Fourier neural operator significantly reduces computational complexity through the Fast Fourier Transform, enabling the model to maintain high training efficiency and good generalization ability in large-scale three-dimensional problems.

[0022] To address the computational burden caused by high-dimensional output in three-dimensional second-order coarse-scale units, this invention further reduces the dimensionality of multi-scale shape functions. For uniformly discrete... A coarse-scale unit, whose complete output dimension can reach ,when The output dimension has already approached 240,000, and with the increasing density of the grid, the dimensions of both input and output are growing rapidly. To reduce model complexity, this invention directly predicts only four multi-scale shape function components (…). and The remaining multi-scale shape function components are reconstructed through spatial rotation. The fine-scale material distribution is then reconstructed using a coordinate system. Rotate space to a new coordinate system The new coordinate system is predicted by the trained neural network. The four multi-scale shape function components below and Rotate them back to their original coordinate system The four multi-scale shape function components in the original coordinate system are obtained. and By repeating the above operations with different spatial rotation methods, all the multiscale shape function components can be obtained, which together form a complete multiscale shape function.

[0023] Finally, regarding training data construction, this invention proposes a sample generation method based on the Cahn-Hilliard phase-field equations. By solving the phase separation evolution equations within a three-dimensional uniform computational domain, a material distribution with continuity and physical consistency is obtained. By randomly perturbing the initial conditions and model parameters, diverse material distribution samples can be generated and divided into multiple uniform subdomains as coarse-scale unit training samples. Simultaneously, data augmentation techniques such as spatial rotation are introduced to improve the diversity and robustness of the training samples. Compared to random "mosaic" material distributions, the data generated by this method better reflects the true evolution of materials during topology optimization, thereby significantly improving the model's generalization performance.

[0024] This invention trains Fourier neural operators using PyTorch and implements large-scale topology optimization design using MATLAB software. Figure 3The specific implementation process of the problem-independent machine learning large-scale topology optimization method based on Fourier neural operators proposed in this invention will be shown in the following pseudocode form: 1) Define parameters (design domain, Young's modulus, Poisson's ratio, penalty coefficient, displacement / external force boundary conditions, filter radius, volume fraction, and discretization method); 2) Initialize the stiffness matrix, filtering coefficient matrix, and element density; 3) Begin the topology optimization iterative loop; 3.1) Based on the discrete finite element model and the material distribution of fine elements, select the second-order coarse-scale elements that need to predict multi-scale shape functions, and perform several spatial rotations on the corresponding material distribution to form the input of the neural network. 3.2) Call the trained neural network based on Fourier neural operators, take the material distribution of fine-scale units before and after each spatial rotation as input, and predict the corresponding four multi-scale shape function components. 3.3) Rotate the predicted multi-scale shape function components after each spatial rotation back to their original positions, and combine them with the multi-scale shape function components before the spatial rotation to form a complete multi-scale shape function, and then correct it according to the rigid body displacement constraint. 3.4) Calculate the stiffness matrix and load vector of all coarse elements and assemble them; 3.5) Apply displacement / external force boundary conditions; 3.6) Solve for the displacement field of the coarse-scale nodes and map it back to the displacement field of the fine-scale nodes; 3.7) Calculate the sensitivity of fine-scale elements; 3.8) Update the material distribution of fine-scale elements using optimization criteria; 4) Return to step 3) until the maximum number of iterations is reached.

[0025] Example: Combination Figures 4-5 The accuracy, reliability, and superior performance of the problem-independent machine learning large-scale topology optimization design method based on Fourier neural operators proposed in this invention are further described in detail.

[0026] This embodiment is used to verify the computational accuracy and numerical efficiency of the proposed problem-independent machine learning topology optimization design method based on Fourier neural operators. For example... Figure 4As shown, the example is a 2×1×1 cantilever beam structure with its left end fully fixed and a uniformly distributed downward load applied to the lower edge of the right end, with a total resultant force of 1. Considering the symmetry of the structural geometry and load, only half of the structure is calculated to reduce the computational scale. Topology optimization is performed using the variable density method, with 60 iterations, a volume fraction of 10%, and a filter radius five times the size of the fine-scale element. The computational domain is discretized using two coarse-scale meshes: 40×10×20 and 80×20×40, corresponding to 8 million and 64 million fine-scale elements respectively, to verify the applicability of the method under different computational scales. Simultaneously, the optimization results of linear coarse-scale elements and second-order coarse-scale elements are compared for each mesh scale.

[0027] like Figure 5 As shown, with mesh refinement, the ability to represent the details of the optimized structure gradually increases. The optimization results obtained using linear coarse-scale elements exhibit obvious discontinuities, and the structural boundaries show significant jagged features, indicating that linear coarse-scale elements are difficult to accurately describe the complex deformation modes within the elements. This phenomenon improves somewhat with an increase in the number of coarse-scale elements, but some degree of boundary unsmoothness still exists. In contrast, second-order coarse-scale elements, due to the introduction of more degrees of freedom and higher-order deformation modes, produce smoother overall optimization results, significantly improve the continuity of structural boundaries, and with mesh refinement, the structural details become clearer and the physical consistency is stronger.

[0028] Furthermore, although the computation time increases due to the increased degrees of freedom, the second-order coarse-scale element is significantly superior to the linear coarse-scale element in terms of displacement field accuracy. A comparison of structural compliance results shows that, in some cases, the linear coarse-scale element may yield slightly lower compliance values, but the corresponding displacement field accuracy is relatively poor, exhibiting some numerical deviation. While the second-order coarse-scale element has a slightly higher computational cost, it provides more reliable structural response predictions, thus making the topology optimization results more credible and engineering-applicable.

[0029] In summary, the above embodiments verify the accuracy, effectiveness, and applicability of the problem-independent machine learning large-scale topology optimization design method based on Fourier neural operators proposed in this invention. They demonstrate significant advantages in terms of optimization result quality, computational efficiency, and the scale of problems that can be handled, proving the necessity of this invention. Numerical results show that by constructing second-order coarse-scale elements, this invention can effectively improve the jagged boundary problem caused by linear coarse-scale elements, improving the smoothness and continuity of the optimized structure. Simultaneously, the PIML method based on Fourier neural operators can efficiently replace repetitive local calculations in extended multi-scale finite element methods, significantly reducing the computational cost in large-scale three-dimensional topology optimization. Therefore, the method proposed in this invention combines high precision, high efficiency, and large-scale problem handling capabilities, and has certain engineering application prospects.

[0030] The embodiments of the present invention are given for illustrative and descriptive purposes only, and are not intended to be exhaustive or to limit the invention to the forms disclosed. Many modifications and variations will be apparent to those skilled in the art. The embodiments were chosen and described to better illustrate the principles and practical application of the invention, and to enable those skilled in the art to understand the invention and design various embodiments with various modifications suitable for a particular purpose.

Claims

1. A problem-independent machine learning method for large-scale topology optimization design based on Fourier neural operators, characterized in that, The steps are as follows: Step 1: Construct multi-scale shape functions: Second-order coarse-scale elements are constructed based on the extended multi-scale finite element framework, and m is represented by the multi-scale shape function matrix N for scaling down. m The physical field of m fine-scale elements, wherein the multi-scale shape function matrix N is obtained through the following finite element analysis: The second-order coarse-scale element consists of 20 nodes, each containing degrees of freedom in the x, y, and z directions. Each node's degree of freedom corresponds to a multi-scale shape function component, and all multi-scale shape function components together form a complete multi-scale shape function matrix N. The calculation process for the multi-scale shape function component corresponding to any degree of freedom of a coarse-scale element node is as follows: the stiffness matrix is ​​calculated using the material distribution of the fine-scale element, and a unit displacement boundary condition is applied to the corresponding degree of freedom. Zero displacement boundary conditions are applied to the degrees of freedom of the remaining coarse-scale element nodes. At the same time, the fine-scale element degrees of freedom corresponding to the boundaries of the coarse-scale element are obtained by quadratic Lagrange interpolation. The corresponding multi-scale shape function component is then obtained through finite element analysis. Step 2: Construct the training dataset: The training dataset includes fine-scale element material distributions generated based on the Cahn-Hilliard phase-field equations, and four multi-scale shape function components of second-order coarse elements. The selected multi-scale shape function components satisfy the following criteria: 1) At any vertex of a coarse-scale unit, select a multi-scale shape function component; 2) Select three multi-scale shape function components at the midpoint of any coarse-scale unit; After selecting four multi-scale shape function components, they are obtained by finite element analysis in step 1; Step 3: Construct a neural network based on Fourier neural operators: The fine-scale unit material distribution generated in step 2 based on the Cahn-Hilliard phase-field equation is used as the input of the neural network, and the four multi-scale shape function components of the second-order coarse unit in step 2 are used as the output of the neural network. The neural network is trained to learn the mapping relationship between the fine-scale unit material distribution and the four multi-scale shape function components in the frequency domain space. The neural network based on Fourier neural operators consists of three parts: an input mapping layer, a Fourier feature update layer, and an output mapping layer. First, the input mapping layer projects the fine-scale unit material distribution onto a high-dimensional feature space. Then, the Fourier layer performs a fast Fourier transform on the high-dimensional feature space and performs weighted learning of low-spectral information in the frequency domain. At the same time, it combines local linear transformation and nonlinear activation functions to achieve the fusion and update of global and local information. Finally, the output mapping layer projects the transformed high-dimensional feature space back into the target space to obtain multi-scale shape function components. Step 4: Preprocessing for topology optimization design: The objective condition is to minimize the structural flexibility; the parameters for topology optimization design are defined as follows: design domain, discretization method, Young's modulus, Poisson's ratio, penalty coefficient, displacement / external force boundary conditions, filter radius, and volume fraction; the design domain is discretized into coarse-scale elements and fine-scale elements; the stiffness matrix, filter coefficient matrix, and element density are initialized. Step 5: Iterative solution of topology optimization design: The specific implementation process will be demonstrated through the following steps: 5.1) Based on the material distribution of the discrete fine-scale units, select the second-order coarse-scale units that need to solve the multi-scale shape function, and perform several spatial rotations on the corresponding material distributions to form the input of the neural network. 5.2) Call the neural network based on Fourier neural operators that has been trained in step 2, and use the fine-scale unit material distribution before and after each spatial rotation in step 5.1) as input to predict the corresponding four multi-scale shape function components. 5.3) Rotate the multiscale shape function components predicted in step 5.2) after each spatial rotation back to their original positions, and together with the multiscale shape function components before the spatial rotation, form a complete multiscale shape function matrix. 5.4) Correct the complete multi-scale shape function matrix in step 5.3) according to the rigid body displacement constraints; 5.5) Calculate the stiffness matrix and load vector of all second-order coarse-scale elements based on the corrected complete multi-scale shape function matrix in step 5.4), and assemble them. 5.6) Apply displacement / external force boundary conditions; 5.7) Solve for the displacement field of coarse-scale nodes; 5.8) Based on the corrected complete multi-scale shape function matrix in step 5.4), the displacement field of the coarse-scale node is mapped back to the displacement field of the fine-scale node. 5.8) Calculate the sensitivity of fine-scale elements; 5.9) Update the material distribution of fine-scale elements using optimization criteria; 5.10), return to step 5.1), until the maximum number of iterations is reached.

2. The method for large-scale topology optimization design based on problem-independent machine learning using Fourier neural operators according to claim 1, characterized in that, In step 2, by randomly generating initial perturbation conditions and model parameters, the Cahn-Hilliard phase-field equation is solved in a three-dimensional uniform computational domain to obtain a phase-field distribution with continuity and physical consistency. This distribution is then divided into multiple uniform subdomains, and the phase-field distribution of the subdomains is used as the distribution of fine-scale unit materials. Simultaneously, spatial rotation is performed to increase the diversity of neural network inputs.

3. The method for large-scale topology optimization design based on problem-independent machine learning using Fourier neural operators according to claim 1, characterized in that, The spatial rotation in steps 5.1), 5.2), and 5.3) is as follows: To reduce the complexity of the neural network, only the four multi-scale shape function components mentioned in step 2 are selected as the output of the neural network. When the selected four multi-scale shape function components are... and hour: The distribution of fine-scale materials is determined by the coordinate system. Rotate space to a new coordinate system The new coordinate system is obtained by predicting using the neural network trained in step 3. The four multi-scale shape function components below and Rotate them back to their original coordinate system The four multi-scale shape function components in the original coordinate system are obtained. and By repeating the above operations with different spatial rotation methods, all multi-scale shape function components can be obtained, which together form a complete multi-scale shape function matrix.

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