Topological optimization method of embedded physical neural network
By embedding a topology optimization method with a physical neural network, using PINN to construct a displacement and density network, and combining the energy minimization loss function and dynamic sampling, the problems of high computational resources and time requirements of traditional topology optimization methods are solved, achieving more efficient and accurate structural design.
Patent Information
- Application Number
- CN202510525910.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-04-24
- Publication Date
- 2025-09-05
AI Technical Summary
Traditional topology optimization methods rely on finite element analysis and complex numerical calculations, resulting in high computing resource and time requirements, many design variables and cumbersome iterative processes, making it difficult to maintain flexibility and innovation in the optimization of large-scale or complex structures.
A topology optimization method embedded in a physical neural network is adopted. The displacement and density networks are constructed using PINN. The energy minimization loss function and dynamic sampling strategy are combined to embed physical constraints. The displacement field and material distribution are determined by minimizing the total potential energy functional.
It improves the efficiency and accuracy of the topology optimization process, enables the exploration of a wider design space while maintaining physical constraints, achieves better design solutions, reduces dependence on data and computing resources, and improves the durability and reliability of the structure.
Smart Images
Figure CN120600173A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to a topology optimization method, in particular to a dynamically configured physical information neural network (PINN) topology optimization method, which is used in the fields of structural design and material optimization. Background Art
[0002] In traditional engineering and design fields, topology optimization is a powerful technique used to find the optimal material distribution under given loads and constraints. This method is particularly important in fields such as aerospace, automotive manufacturing, and mechanical engineering because it can significantly improve structural efficiency, reduce material usage, lower costs, and improve performance.
[0003] However, traditional topology optimization methods typically rely on finite element analysis (FEA) and complex numerical calculations, which require significant computational resources and time. Furthermore, these methods often require a large number of design variables and iterative processes, making the optimization process cumbersome and time-consuming. In many cases, these methods also require discretization of the design space, which limits design flexibility and innovation.
[0004] The development of deep learning technology, particularly the emergence of physical-informed neural networks (PINNs), has opened up new possibilities for addressing these challenges. PINNs can learn complex physical laws directly from data, without explicitly solving differential equations. This provides a new, data-driven approach to topology optimization. However, applying deep learning to topology optimization still faces several challenges, including effectively embedding physical constraints, handling high-dimensional design spaces, and improving the efficiency and accuracy of the optimization process.
[0005] The background technology of the present invention is based on these challenges and aims to develop a new topology optimization method. Summary of the Invention
[0006] Purpose of the invention: The purpose of the present invention is to provide a topology optimization method embedded in a physical neural network in order to overcome the defects of the prior art. Two networks are proposed to improve topology optimization. Through this topology optimization method embedded in a physical neural network, the efficiency of the optimization process is improved, and a better design solution is achieved while maintaining physical constraints.
[0007] This method leverages deep learning techniques, specifically PINN, to address the limitations of traditional methods. This approach not only improves the efficiency of the optimization process but also explores a wider range of design spaces while maintaining physical constraints, leading to more optimal design solutions.
[0008] Technical solution: The topology optimization method embedded in a physical neural network includes the following steps:
[0009] S 1: Construct a dynamically configured PINN, design and implement two interconnected neural networks: a displacement network for calculating the structural displacement field and a density network for estimating the material density distribution;
[0010] S2: defining a loss function to connect the two networks via a loss function based on energy minimization, which is derived from the variational principle of the governing equations to connect the two neural networks;
[0011] S3: Embed deep learning within the physical constraints of the problem to eliminate the need for large-scale data and analytical sensitivity analysis;
[0012] S4: Produces high-resolution designs for 2D and 3D optimization problems by minimizing compliance through energy-based objectives while enforcing volume fraction constraints;
[0013] S5: Dynamic sampling strategy, which selectively samples checkpoints based on pseudo-density to reduce the number of input checkpoints and lower the training cost;
[0014] S6: Calculate the strain energy of the element by Gaussian integration to decouple the mapping of the material at the calibration point;
[0015] S7: Displacement analysis based on the principle of minimum potential energy. Construct a total potential energy functional of a physical system, including strain potential energy and external work, and determine the displacement field by minimizing the total potential energy functional.
[0016] S8: Calculate the second-order variation of the total potential energy functional to determine the equilibrium state of the system;
[0017] S9: Determine the displacement field of the system by minimizing the total potential energy functional so that the stress field corresponding to the displacement field satisfies the equilibrium equations and boundary conditions within the domain.
[0018] Furthermore, the displacement network is constructed: the goal is to map the spatial coordinates (x, y) to their corresponding displacement components The output of the shifted NN is expressed as:
[0019]
[0020] Where x, y represent spatial coordinates, θ disp is the learnable parameter of the displacement NN, output and Represents the displacement field component at the input coordinates, and the spatial coordinates are described layer by layer through the forward propagation of the network as follows:
[0021] h j =α(W j h j-1 +b j)j=1,...,N,
[0022] Among them, N represents the number of layers in the fully connected network, h0 is the input layer, represents the spatial coordinates, and h N represents the output layer, represents the displacement component, W i and b i are the weight and bias of layer i respectively, α is an activation function that introduces nonlinearity, enabling the network to model more complex relationships between input and output, and the sine representation network SIREN is used as the activation function:
[0023]
[0024] Furthermore, the density network uses Fourier eigenmaps to define design variables to handle complex design configurations.
[0025] The density neural network uses the Adam optimizer to minimize the loss function.
[0026] The formula of density neural network can be expressed as follows:
[0027] Fourier eigenmaps enable neural networks consisting of multiple hidden layers to express high-frequency geometry in a low-dimensional domain and are expressed as:
[0028]
[0029] Among them, the matrix is from N(0,σ 2 ). The dimension m is the size of the Fourier feature map.
[0030] Furthermore, the method trains two neural networks simultaneously, and reduces the overall training time by periodically updating the density network:
[0031] For an object in static equilibrium, with no external forces, the potential energy of the system is:
[0032]
[0033] Loss Function in Shift Neural Network It can be defined as the potential energy of the system:
[0034] The solution to the elasticity problem provided by the displacement neural network model can be expressed as:
[0035] In this work, the Adam optimizer is used to train the model, and the learning rate is another parameter that the user can change to achieve the best performance of the framework. This optimization process is repeated in an iterative manner until the design meets the following stopping criteria:
[0036]
[0037] Furthermore, the method updates the density distribution by minimizing potential energy, which is equivalent to minimizing compliance or maximizing structural stiffness. The displacement network is shown to minimize the total potential energy as its loss function, which is equivalent to minimizing compliance or maximizing structural stiffness. To update the density distribution, a specialized density network is used, whose design ensures that volume constraints are explicitly considered in its loss function. In this way, the density network helps maintain the desired material distribution during the optimization process while respecting volume constraints.
[0038] Furthermore, the dynamic sampling strategy solves the problem of automatic adjustment of tracking sampling rate and the problem of how to quickly and accurately find services that need to adjust tracking sampling rate through two data structures: sampling strategy tree and execution trajectory graph.
[0039] Furthermore, the total potential energy functional consists of strain potential energy density and external force potential energy density, where the strain potential energy density is expressed as The external potential energy density is expressed as Strain potential energy density is related to the strain ∈, which is a function of the displacement u:
[0040] The strain potential energy density function is defined as:
[0041] Here is the strain potential energy density function, which depends on the strain and material density ρ, where λ and μ are Lame constants, which are related to the Young's modulus E(ρ) and Poisson's ratio v of the material.
[0042] The Lamé constant can be calculated as:
[0043] The strain potential energy density is further described by the equation How it depends on the material properties and the design variable ρ, and how these properties are related to the displacement through the strain ∈.
[0044] Furthermore, the method also includes calculating the second-order variation of the work done by the external force to determine the impact of the external force on the equilibrium state of the system.
[0045] External potential energy:
[0046] Ignore the effect of B:
[0047] The calculation of the total potential energy includes the internal strain energy and external energy
[0048]
[0049] Finally, the compliance of the structure Related to the total potential energy, it can be expressed as:
[0050]
[0051] The topology optimization problem can then be reformulated as minimizing the total potential energy:
[0052]
[0053] This shows that the second-order variation of the work done by the external force is zero, which means that in the equilibrium state, the effect of the work done by the external force on the equilibrium state of the system is determined by the second-order variation of the strain energy. By calculating these variations, the effect of the external force on the equilibrium state of the system can be determined.
[0054] Compared with the prior art, the present invention has the following advantages:
[0055] 1. Traditional topology optimization methods typically rely on finite element analysis (FEA) and complex numerical calculations, requiring a large amount of computing resources and time. This paper proposes a displacement network using a sinusoidal representation network (SIREN) activation function, which is particularly suitable for problems that require the calculation of high-order derivatives. The SIREN activation function, through a combination of sine and cosine functions, can accurately calculate the high-order derivatives of the neural network weights with respect to the output without adding additional computational complexity. This is particularly important for structural design involving complex mechanical behavior because it can provide more accurate predictions of displacement and stress distribution, thereby improving the reliability and performance of the design.
[0056] 2. Regarding how to effectively embed physical constraints, the present invention proposes to combine physical prior knowledge to reduce data requirements, and utilize the constraint information provided by physical equations to reduce dependence on large amounts of training data. In many practical applications, collecting large amounts of accurate data is expensive or even infeasible. By embedding physical constraints, such as conservation laws or mechanical principles, the model can be effectively trained and predicted when data is scarce, which is especially important in small sample learning scenarios. This advantage enables the model to adapt to new physical systems more quickly, and reduces dependence on experimental data, reducing cost and time. At the same time, it is proposed to improve the interpretability of the model, and based on physical equations, its prediction results have stronger interpretability. Compared with traditional black-box deep learning models, the output of PINN can correspond to physical laws, making the model's prediction process and results easier to understand and verify.
[0057] 3. In order to improve the efficiency and accuracy of the optimization process, the present invention proposes efficient calculation of strain energy, precise determination of equilibrium state, and precise satisfaction of stress field. By calculating the strain energy of elements through Gaussian integral, the present invention can efficiently decouple the mapping of materials at the checkpoints. This integral-based method can more accurately capture the local behavior of materials while reducing computational complexity. This enables faster results when dealing with large-scale or complex structural optimization problems while maintaining the accuracy of the results. At the same time, by calculating the second-order variation of the total potential energy functional, the present invention can accurately determine the equilibrium state of the system. This step is crucial for identifying the stability and safety of the structure. Second-order variation analysis provides sensitive information about the stability of the system, allowing designers to predict and avoid potential instability problems and ensure the reliability of the structural design. By minimizing the total potential energy functional, the displacement field of the system is determined, ensuring that the stress field corresponding to the displacement field satisfies the equilibrium equations and boundary conditions within the domain. This energy-based method can provide more accurate stress distribution predictions, which helps to design structures that meet functional requirements and have sufficient strength and stiffness. By precisely controlling the stress distribution, the present invention helps to improve the durability and reliability of the structure.
[0058] 4. Innovation in structural design is achieved by combining dynamically configured PINN (Physical Information Neural Network) topology optimization technology; involving two interconnected displacement networks and density networks; connecting these two networks through a loss function based on energy minimization; embedding deep learning within physical constraints to reduce reliance on large-scale data and analytical sensitivity analysis.
[0059] 5. The displacement network uses a sinusoidal representation network (SIREN) activation function to improve the accuracy of calculating high-order derivatives, while the density network uses Fourier eigenmaps to define design variables to handle complex design configurations. Minimizing potential energy is equivalent to minimizing compliance or maximizing structural stiffness to update the density distribution while complying with volume constraints. It is suitable for topological optimization problems with high resolution, multiple loads, and displacement constraints, and can determine the impact of external forces on the equilibrium state of the system by calculating the second-order variation of the work done by external forces. Through this topological optimization method, it can be reformulated as a problem of minimizing total potential energy, thereby achieving automation and optimization of structural design. BRIEF DESCRIPTION OF THE DRAWINGS
[0060] Figure 1 The steps of the topology optimization method embedded in physical neural network;
[0061] Figure 2 It is the flow chart of displacement and density physics network;
[0062] Figure 3 This is a schematic diagram of this case;
[0063] Figure 4 This is the topology optimization result diagram;
[0064] Figure 5 This is a comparison chart of the optimization results of the present invention and the SIMP method. DETAILED DESCRIPTION
[0065] This paper proposes several methods to improve topology optimization, including a dynamic configuration PINN method, a loss function method based on energy minimization, a deep learning-embedded physical constraint method, an energy-targeted compliance method based on minimization, a dynamic sampling method, a Gaussian integral calculation element method, and a displacement analysis method based on the minimum potential energy principle. This method improves topology optimization methods by embedding them into a physical neural network to optimize material structures, achieving strength and stiffness in the material design, and improving the durability and reliability of the material structure.
[0066] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. Component models, material names, connection structures, control methods, algorithms, and other features not explicitly described in this technical solution are considered common technical features disclosed in the prior art.
[0067] Example 1
[0068] Please combine Figure 1 , this embodiment adopts the following technical solution, including the following steps:
[0069] Step 1: Build a neural network model, i.e., dynamically configure the PINN. Design and implement two interconnected neural networks: a displacement network and a density network. The displacement network is responsible for calculating the displacement field of the structure, while the density network is used to estimate the density distribution of the material. Construction of the displacement network (NN architecture): The goal is to map spatial coordinates (x, y) to their corresponding displacement components. The output of the shift NN can be expressed as:
[0070]
[0071] Where x, y represent spatial coordinates, θ disp is the learnable parameter of the displacement NN, output and Represents the displacement field component at the input coordinates, and the spatial coordinates are described layer by layer through the forward propagation of the network as follows:
[0072] h j =α(W j h j-1 +b j )j=1,...,N,
[0073] Where N represents the number of layers in the fully connected network, h0 is the input layer (representing spatial coordinates), hN is the output layer (representing displacement components), Wi and b i are the weight and bias of layer i, respectively. α is an activation function that introduces nonlinearity, enabling the network to model more complex relationships between input and output. We use the Sinusoidal Representation Network (SIREN) as the activation function.
[0074]
[0075] The architecture of a density neural network. The input consists of spatial coordinates x, y, which are first processed through a Fourier projection layer. The data then passes through a series of hidden layers, including linear transformations, batch normalization, and leaky ReLU activation functions, and the final output of the network is the corresponding density field ρ.
[0076]
[0077] Fourier eigenmaps enable neural networks consisting of multiple hidden layers to express high-frequency geometry in a low-dimensional domain and are expressed as:
[0078]
[0079] Among them, the matrix is from N(0,σ 2 ). The dimension m is the size of the Fourier feature map.
[0080] Step 2: Define a loss function and connect the networks based on the energy minimization loss function. This function is derived from the variational principle of the governing equations and connects the two neural networks. The method trains two neural networks simultaneously and reduces the overall training time by periodically updating the density network:
[0081] For an object in static equilibrium, with no external forces, the potential energy of the system is:
[0082]
[0083] Loss Function in Shift Neural Network It can be defined as the potential energy of the system: The solution to the elasticity problem provided by the displacement neural network model can be expressed as: In this work, the Adam optimizer is used to train the model, and the learning rate is another parameter that the user can change to achieve the best performance of the framework. This optimization process is repeated in an iterative manner until the design meets the following stopping criteria:
[0084]
[0085] Step 3: Embed physical constraints. This embeds the deep learning model within the constraints of the physical problem, eliminating the need for large datasets and sensitivity analysis. Prior knowledge from physics is directly incorporated into the model during training. These physical constraints appear as additional terms in the loss function, ensuring that the model adheres to physical laws during training.
[0086] Step 4: Objective minimization based on constraints, compliance is achieved by minimizing the energy-based objective, while volume fraction constraints are enforced to produce high-resolution designs for 2D and 3D optimization problems. The method updates the density distribution by minimizing the potential energy, which is equivalent to minimizing compliance or maximizing structural stiffness. It is explained that the displacement network uses minimizing the total potential energy as its loss function, which is equivalent to minimizing compliance or maximizing structural stiffness. In order to update the density distribution, a specialized density network is used. The design of this network ensures that the volume constraint is explicitly considered in its loss function. In this way, the density network helps maintain the desired material distribution during the optimization process while complying with the volume constraint.
[0087] Step 5: Dynamic Sampling Strategy: The two data structures, sampling strategy tree and execution trajectory graph, solve the problem of automatically adjusting the tracking sampling rate and how to quickly and accurately find the services that need to adjust the tracking sampling rate.
[0088] Step 6: Gaussian integral calculation, using Gaussian integral to calculate the strain energy of the element, decoupling the mapping of the checkpoint material. The total potential energy functional consists of strain potential energy density and external force potential energy density, where the strain potential energy density is expressed as The external potential energy density is expressed as Strain potential energy density is related to the strain ∈, which is a function of the displacement u:
[0089] The strain potential energy density function is defined as: Here is the strain potential energy density function, which depends on the strain and material density ρ, where λ and μ are Lame constants, which are related to the Young's modulus E(ρ) and Poisson's ratio v of the material.
[0090] The Lamé constant can be calculated as: The strain potential energy density is further described by the equation How it depends on the material properties and the design variable ρ, and how these properties are related to the displacement through the strain ∈.
[0091] Step 7: Construct a total potential energy functional. Based on the principle of minimum potential energy, perform displacement analysis to construct a total potential energy functional for the physical system, including strain potential energy and external force work. Simultaneously, determine the displacement field by minimizing the total potential energy functional. The method also includes calculating the second-order variation of the work done by the external force to determine its effect on the equilibrium state of the system.
[0092] External potential energy:
[0093] Ignore the effect of B:
[0094] The calculation of the total potential energy includes the internal strain energy and external energy
[0095]
[0096] Finally, the compliance of the structure Related to the total potential energy, it can be expressed as:
[0097]
[0098] The topology optimization problem can then be reformulated as minimizing the total potential energy:
[0099]
[0100] This shows that the second-order variation of the work done by the external force is zero, which means that in the equilibrium state, the effect of the work done by the external force on the equilibrium state of the system is determined by the second-order variation of the strain energy. By calculating these variations, the effect of the external force on the equilibrium state of the system can be determined.
[0101] Application Example 1
[0102] The learning rate of the case is initially 10e-3, and the Young's modulus E=10 is fixed. The case diagram is as follows Figure 3 shown.
[0103] This case 1) Figure 3 As shown, the object v f = 0.5, and the left side is fixed, a force F is applied to the midpoint of its right side, and topology optimization is performed. Using the present invention, a topology optimization method embedded in a physical neural network is obtained, and the topology optimization results are as follows Figure 3 As shown, its flexibility value is 2.8467
[0104] In this case 2) Figure 3 As shown, the object v f = 0.7, and the left side is fixed, and a force F is applied to the bottom of the right side to perform topological optimization. Using the present invention, a topological optimization method embedded in a physical neural network is used to obtain the topological optimization results as follows Figure 3 As shown, its flexibility value is 1.2494
[0105] In this case 3) Figure 3 As shown, the object v f =0.45, and the left and right sides are fixed, and an average force F is applied on them to perform topological optimization. Using the present invention, a topological optimization method embedded in a physical neural network is used to obtain the topological optimization results as follows Figure 3 As shown, its flexibility value is 374.5602
[0106] This case 4) Figure 3 As shown, the object v f = 0.3, and the two sides of the bottom are fixed, and a force F is applied downward at the midpoint below it to perform topological optimization. Using the present invention, a topological optimization method embedded in a physical neural network is used to obtain the topological optimization results as follows Figure 4 As shown, its flexibility value is 1.4583
[0107] This implementation demonstrates that this topology optimization method, embedded in a physical neural network, significantly enhances topology optimization, enabling better optimization of material structures. This not only improves the efficiency of the optimization process but also allows for the exploration of a wider design space while maintaining physical constraints, leading to more optimal design solutions. This provides an innovative approach to topology optimization, addressing the growing complexity and performance requirements of modern engineering design.
[0108] At the same time, the optimization of the case using the topology optimization method embedded in the physical neural network of the present invention is compared with the case optimized by the conventional SIMP method. It can be seen that the flexibility value obtained by the present method is smaller and the optimization result is better (such as Figure 5 )
[0109] - Case 1 Case 2 Case 3 The flexibility value of this topology optimization method 2.8467 1.2494 374.5602 Flexibility value of SIMP method 5.0871 1.9994 406.4573
[0110] It can be seen from the table that the structural flexibility values obtained by this topology optimization method are smaller than those obtained by the SIMP method.
[0111] The above description is merely a preferred embodiment of the present invention and does not limit the present invention in any way. Any person skilled in the art who, without departing from the scope of the present invention, makes any equivalent substitution, modification, or other changes to the technical solution and technical content disclosed in the present invention shall be deemed to be within the scope of the present invention and still fall within the scope of protection of the present invention.
Claims
1. A topological method for embedding physical knowledge into a neural network, characterized in that: The following steps are involved: S1: Construct a dynamically configured PINN, design and implement two interconnected neural networks: a displacement network for calculating the structural displacement field and a density network for estimating the material density distribution; S2: defining a loss function to connect the two networks via a loss function based on energy minimization, which is derived from the variational principle of the governing equations to connect the two neural networks; S3: Embed deep learning within the physical constraints of the problem to eliminate the need for large-scale data and analytical sensitivity analysis; S4: Produces high-resolution designs for 2D and 3D optimization problems by minimizing compliance through energy-based objectives while enforcing volume fraction constraints; S5: Dynamic sampling strategy, which selectively samples checkpoints based on pseudo-density to reduce the number of input checkpoints and lower the training cost; S6: Calculate the strain energy of the element by Gaussian integration to decouple the mapping of the material at the calibration point; S7: Displacement analysis based on the principle of minimum potential energy. Construct a total potential energy functional of a physical system, including strain potential energy and external work, and determine the displacement field by minimizing the total potential energy functional. S8: Calculate the second-order variation of the total potential energy functional to determine the equilibrium state of the system; S9: Determine the displacement field of the system by minimizing the total potential energy functional so that the stress field corresponding to the displacement field satisfies the equilibrium equations and boundary conditions within the domain.
2. The topological method for embedding physical knowledge neural network according to claim 1 is characterized in that: In step S1: Construction of displacement network: The target maps the spatial coordinates (x, y) to their corresponding displacement components The output of the shifted NN is expressed as: Where x, y represent spatial coordinates, θ disp is the learnable parameter of the displacement NN, output and Represents the displacement field component at the input coordinates, and the spatial coordinates are described layer by layer through the forward propagation of the network as follows: h j =α(W j h j-1 +b j )j=1,...,N, Among them, N represents the number of layers in the fully connected network, h0 is the input layer, represents the spatial coordinates, and h N represents the output layer, represents the displacement component, W i and b i are the weight and bias of layer i respectively, α is an activation function that introduces nonlinearity, enabling the network to model more complex relationships between input and output, and the sine representation network SIREN is used as the activation function:
3. The topological method for embedding physical knowledge neural network according to claim 2 is characterized in that: In step S1: the density network uses Fourier eigenmaps to define design variables to handle complex design configurations, and the density neural network uses Adam optimizer to minimize the loss function. The formula of density neural network is as follows: Fourier eigenmaps enable neural networks consisting of multiple hidden layers to express high-frequency geometry in a low-dimensional domain and are expressed as: Among them, the matrix is from N(0,σ 2 ), and the dimension m is the size of the Fourier feature map.
4. The topological method for embedding physical knowledge neural network according to claim 3 is characterized in that: In step S2, two neural networks are trained simultaneously, and the overall training time is reduced by periodically updating the density network: For an object in static equilibrium, without external forces, the potential energy of the system is: Loss Function in Shift Neural Network It can be defined as the potential energy of the system: The solution to the elasticity problem provided by the displacement neural network model can be expressed as: The Adam optimizer is used to train the model. The learning rate is another parameter that the user can change to achieve the best performance of the framework. This optimization process is repeated in an iterative manner until the design meets the following stopping criteria:
5. The topological method for embedding physical knowledge neural network according to claim 4 is characterized in that: In the step S4: the density distribution is updated by minimizing the potential energy, which is equivalent to minimizing the compliance or maximizing the structural stiffness, indicating that the displacement network uses minimizing the total potential energy as its loss function, which is equivalent to minimizing the compliance or maximizing the structural stiffness. In order to update the density distribution, a dedicated density network is used. The design of this network ensures that the volume constraint is explicitly considered in its loss function. In this way, the density network helps maintain the desired material distribution during the optimization process while complying with the volume constraint.
6. The topological method for embedding physical knowledge into a neural network according to claim 5, characterized in that: In step S5, the dynamic sampling strategy solves the problem of automatic adjustment of the tracking sampling rate and the problem of how to quickly and accurately find the service that needs to adjust the tracking sampling rate through two data structures: the sampling strategy tree and the execution trajectory graph.
7. The topological method for embedding physical knowledge neural network according to claim 6, characterized in that: In step S6: the total potential energy functional is composed of strain potential energy density and external force potential energy density, wherein the strain potential energy density is expressed as The external potential energy density is expressed as Strain potential energy density is related to the strain ∈, which is a function of the displacement u: The strain potential energy density function is defined as: Here is the strain potential energy density function, which depends on the strain and material density ρ, where λ and μ are Lame constants, which are related to the Young's modulus E(ρ) and Poisson's ratio v of the material. The Lamé constant can be calculated as: The strain potential energy density is further described by the equation How it depends on the material properties and the design variable ρ, and how these properties are related to the displacement through the strain ∈.
8. The topological method for embedding physical knowledge into a neural network according to claim 7, characterized in that: In step S7: the second-order variation of the work done by the external force is calculated to determine the influence of the external force on the equilibrium state of the system, and the potential energy of the external force is: Ignore the effect of B: The calculation of the total potential energy includes the internal strain energy and external energy Finally, the compliance of the structure Related to the total potential energy, it can be expressed as: The topology optimization problem can be reformulated as minimizing the total potential energy: The second-order variation of the work done by the external force is zero, which means that in the equilibrium state, the influence of the work done by the external force on the equilibrium state of the system is determined by the second-order variation of the strain energy. By calculating these variations, the influence of the external force on the equilibrium state of the system can be determined.