Topological optimization method and system based on physical information radial basis deep neural network

Abstract

The invention provides a topological optimization method and system based on a physical information radial basis deep neural network. The method comprises the following steps: S1, initializing parameters and a network model; s2, calculating a volume fraction; s3, training a network to predict a displacement field; s4, calculating a total potential energy loss function of the system; s5, judging network convergence; s6, calculating an optimization target value; s7, calculating a Lagrange multiplier and a velocity field; s8, calculating an expansion coefficient and an updating level set function; and S9, optimizing cyclic convergence judgment. Wherein the network model used for displacement prediction uses a physical information radial basis depth neural network (PIRDN), and a first hidden layer of the PIRDN is designed to be a self-adaptive radial basis function layer, so that the calculation precision and efficiency of a displacement field are improved; the total potential energy of the system is calculated through Gaussian integral, and training of the PIRDN model is achieved by minimizing the total potential energy of the system; and calculating unit strain energy according to the displacement field predicted by the PIRDN, and further calculating a level set function evolution velocity field to realize updating of the structure topology.

Description

Technical Field

[0001] This invention relates to a structural topology optimization design method, specifically, to a topology optimization method and system based on a physical information radial basis deep neural network, which can be used for topology optimization design of deep-sea pressure-resistant structures. Background Technology

[0002] In traditional engineering structures and materials design, topology optimization can find the optimal material distribution within a given design region based on given optimization objectives and constraints, thereby achieving optimized design of high-performance structures. As an efficient design tool, topology optimization is widely used in aerospace, civil engineering, and shipbuilding engineering.

[0003] However, traditional finite element-based topology optimization methods require iterative finite element analysis (FEA) to solve for structural responses. For large-scale structural designs, especially three-dimensional engineering structures, this consumes significant computational resources. Computational efficiency severely limits the application of topology optimization methods in practical engineering structural optimization design. With the in-depth development of machine learning technology, especially the rapid rise of the physical information-driven paradigm in recent years, new opportunities have emerged for the further development of topology optimization methods. As a classic network structure form of the physical information-driven paradigm, Physical Information Neural Networks (PINNs) can encode the physical laws of the problem to be solved into a loss function, and the problem can be solved through network training. Currently, the extension of PINNs in topology optimization methods has made initial progress, but it still faces problems such as high network training costs and low computational accuracy.

[0004] Therefore, given the challenges of combining PINNs with topology optimization, there is an urgent need to develop an efficient and more accurate topology optimization method. This invention utilizes a topology optimization method based on physically-informed radial basis deep neural networks to perform efficient and high-precision topology optimization design for engineering structures such as deep-sea pressure-resistant structures. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the purpose of this invention is to provide a topology optimization method and system based on a physical information radial basis deep neural network.

[0006] A topology optimization method based on a physically-informed radial basis deep neural network, according to the present invention, includes: Step S1: Initialize the Physical Information Radial Basis Deep Neural Network (PIRDN) model; Step S2: Discretize the initial pressure-resistant structural design domain and calculate the initial level set function accordingly; Step S3: Calculate the volume fraction of the current pressure-resistant structural unit based on the level set function. and volume fractionv ; Step S4: Obtain discrete spatial coordinates based on the discrete design domain X and obtain discrete space coordinates X Input the PIRDN model to obtain the displacement field u ; Step S5: Based on displacement field u Calculate the total potential energy loss function Train the PIRDN model and determine whether the preset training convergence condition is met. If not, backpropagate to update the PIRDN model and repeat steps S4 to S5 until the preset training convergence condition is met. Step S6: Based on displacement field u Calculation of pressure-resistant structural material parameters: element strain energy of the current pressure-resistant structure. and target softness value J ; Step S7: Based on the unit volume fraction of the current pressure-resistant structure and volume fraction v Calculate Lagrange multipliers ; Element strain energy based on current pressure-resistant structures and Lagrange multipliers Calculate the normal velocity field ; Step S8: Based on the normal velocity field Update expansion coefficients According to the expansion coefficient Update level set function Based on volume fraction v and target softness value J Determine whether the preset topology optimization convergence condition is met. If the preset condition is not met, repeat steps S3 to S8 until the preset condition is met, and output the optimized pressure-resistant structure topology shape.

[0007] Preferably, the PIRDN model in step S1 includes: an input layer, multiple hidden layers, and an output layer; The multiple hidden layers include: the first hidden layer is an adaptive radial basis function layer, and the activation function of the other hidden layers is tanh.

[0008] Preferably, step S4 includes: The forward propagation process of discrete spatial coordinates X in the PIRDN model is as follows:

[0009] in, Represents the radial basis function layer. This represents a linear layer, where n is the number of fully connected layers. This represents the activation function tanh. and These represent the network training weights and biases, respectively. Indicates the connection between different layers of a neural network; The activation function of the radial basis function layer has the following form:

[0010] in, Represents the Gaussian function, shape parameter a and b By controlling the shape of the Gaussian function, PIRDN can be trained in the network to achieve automatic updates, enabling it to learn more complex physical features. c This represents the center point of the Gaussian function, which remains fixed during network training.

[0011] Preferably, step S5 includes:

[0012]

[0013] in, The characteristic load amplitude introduced; The characteristic length; For the Heaviside function, Indicates internal strain energy. As external potential energy, This indicates an external force load.

[0014] Preferably, the preset training convergence conditions in step S5 include:

[0015] Where epochs represents the maximum number of training epochs in the network settings. Indicates the current training step. tol To improve network convergence tolerance; and These are all the average values ​​of the loss function. The table is an integer representing the average over the selected window length. For the first in the selected window range i The potential energy loss function value.

[0016] Preferably, the preset topology optimization convergence conditions in step S8 include:

[0017] in, Due to volume constraints, and They represent the first k Volume fraction and compliance target value for each iteration step. and These represent the volume constraint convergence tolerance and the objective value convergence tolerance, respectively. Indicates selecting the first One iteration step to The target value of the iteration step.

[0018] A topology optimization system based on a physically-informed radial basis deep neural network, according to the present invention, includes: Module M1: Initializes the physical information radial basis deep neural network (PIRDN) model; Module M2: Discretizes the initial pressure-resistant structural design domain and calculates the initial level set function accordingly; Module M3: Calculates the volume fraction of the current pressure-resistant structural unit based on the level set function. and volume fraction v ; Module M4: Obtaining Discrete Spatial Coordinates Based on Discrete Design Domain X and obtain discrete space coordinates X Input the PIRDN model to obtain the displacement field u ; Module M5: Based on displacement field u Calculate the total potential energy loss function The PIRDN model is trained, and it is determined whether the preset training convergence condition is met. If not, the PIRDN model is updated by backpropagation, and modules M4 to M5 are triggered repeatedly until the preset training convergence condition is met. Module M6: Based on displacement field u Calculation of pressure-resistant structural material parameters: element strain energy of the current pressure-resistant structure. and target softness value J ; Module M7: Based on the unit volume fraction of the current pressure-resistant structure and volume fraction v Calculate Lagrange multipliers ; Element strain energy based on current pressure-resistant structures and Lagrange multipliers Calculate the normal velocity field ; Module M8: Based on normal velocity field Update expansion coefficients According to the expansion coefficient Update level set function Based on volume fraction v and target softness value JDetermine whether the preset topology optimization convergence condition is met. If the preset condition is not met, repeatedly trigger modules M3 to M8 until the preset condition is met, and output the optimized pressure-resistant structure topology shape.

[0019] Preferably, the PIRDN model in module M1 includes: one input layer, multiple hidden layers, and one output layer; The multiple hidden layers include: the first hidden layer is an adaptive radial basis function layer, and the activation function of the other hidden layers is tanh.

[0020] Preferably, the module M4 includes: The forward propagation process of discrete spatial coordinates X in the PIRDN model is as follows:

[0021] in, Represents the radial basis function layer. This represents a linear layer, where n is the number of fully connected layers. This represents the activation function tanh. and These represent the network training weights and biases, respectively. Indicates the connection between different layers of a neural network; The activation function of the radial basis function layer has the following form:

[0022] in, Represents the Gaussian function, shape parameter a and b By controlling the shape of the Gaussian function, PIRDN can be trained in the network to achieve automatic updates, enabling it to learn more complex physical features. c This represents the center point of the Gaussian function, which remains fixed during network training.

[0023] Preferably, the module M5 includes:

[0024]

[0025] in, The characteristic load amplitude introduced; The characteristic length; For the Heaviside function, Indicates internal strain energy. As external potential energy, Indicates external force load; The preset training convergence conditions in module M5 include:

[0026] Where epochs represents the maximum number of training epochs in the network settings. Indicates the current training step. tol To improve network convergence tolerance; and These are all the average values ​​of the loss function. The table is an integer representing the average over the selected window length. For the first in the selected window range i One potential energy loss function value; The preset topology optimization convergence conditions in module M8 include:

[0027] in, Due to volume constraints, and They represent the first k Volume fraction and compliance target value for each iteration step. and These represent the volume constraint convergence tolerance and the objective value convergence tolerance, respectively. Indicates selecting the first One iteration step to The target value of the iteration step.

[0028] Compared with the prior art, the present invention has the following beneficial effects: 1. This invention designs an adaptive radial basis function layer in the hidden layer, which significantly improves the training efficiency and computational accuracy of PINNs; 2. This invention solves the structural displacement field based on a radial basis deep neural network of physical information, achieving efficient and high-precision structural topology optimization design; 3. The present invention is based on a physical information radial basis deep neural network topology optimization method, which can realize the topology optimization design of complex engineering structures such as deep-sea cylindrical shell pressure-resistant structures.

[0029] 4. The topology optimization method and system based on physical information radial basis deep neural networks provided by the present invention are based on the addition of an adaptive radial basis function layer to the classic PINNs. The shape of the radial basis function is adjusted through network training to enhance the network's ability to learn complex physical features. This effectively improves the computational efficiency and accuracy of the PINN-based topology optimization method, and enables efficient solution of topology optimization problems. 5. This invention achieves efficient and high-precision solutions to 2D and 3D topology optimization problems, and is also well applicable to complex engineering structures such as deep-sea pressure-resistant structures. Attached Figure Description

[0030] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings: Figure 1 This is a flowchart of a topology optimization method based on a radial basis deep neural network with physical information.

[0031] Figure 2 This is a diagram of a radial basis deep neural network structure for physical information.

[0032] Figure 3 A schematic diagram is provided for the optimization calculation example of the cantilever beam.

[0033] Figure 4 This is a comparison diagram of the two-dimensional topology optimization results of the present invention and the PLSM method.

[0034] Figure 5 This is a comparison diagram of the three-dimensional topology optimization results of the present invention and the PLSM method.

[0035] Figure 6 A schematic diagram is provided for the calculation example of a three-dimensional deep-sea pressure-resistant structure.

[0036] Figure 7 A schematic diagram of the optimized structure for a three-dimensional deep-sea pressure-resistant structure. Detailed Implementation

[0037] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the protection scope of the present invention.

[0038] Example 1 According to the present invention, a topology optimization method based on a physical information radial basis deep neural network is provided, such as... Figure 1 As shown, it includes the following steps: Step S1: Initialize the Physical Information Radial Basis Deep Neural Network (PIRDN) model; Specifically, such as Figure 2 As shown, the PIRDN model consists of an input layer, multiple hidden layers, and an output layer. The first hidden layer is an adaptive radial basis function layer, and the activation functions of the other hidden layers are tanh.

[0039] Step S2: Define the discrete design domain and calculate the initial level set function accordingly; Specifically, step S2 includes: defining initial parameters such as material parameters, design domain geometry parameters, and optimization algorithm parameters; initializing the PIRDN network model and setting hyperparameters for network training; setting the initial level set function and calculating the expansion coefficients of the level set function interpolated using radial basis functions based on the geometry of the initial design domain.

[0040] Step S3: Calculate the unit volume fraction based on the level set function and volume fraction v ; Step S4: Obtain discrete spatial coordinates based on the discrete design domain X and obtain discrete space coordinates X Input the PIRDN model to obtain the displacement field u ; Specifically, spatial coordinates X The forward propagation process in the PIRDN model is as follows:

[0041] in, Represents the radial basis function layer. Indicates a linear layer. n The number of fully connected layers. This represents the activation function tanh. and These represent the network training weights and biases, respectively. This represents the connection operator between different layers of a neural network. Taking the Gaussian function as an example, the activation function of the radial basis function layer has the following form:

[0042] in Represents the Gaussian function, shape parameter a and b By controlling the shape of the Gaussian function and training the network to achieve automatic updates, PIRDN can learn more complex physical features. c This represents the center point of the Gaussian function, which remains fixed during network training.

[0043] By inputting the spatial coordinates after discretization of the design domain X PIRDN can predict displacement fields. u ,Right now u It can also be represented as spatial coordinates X and training parameters Functions:

[0044] Step S5: Based on displacement field u Calculate the total potential energy loss function Train the PIRDN model and determine whether the preset training convergence condition is met. If not, backpropagate to update the PIRDN model and repeat steps S4 to S5 until the preset training convergence condition is met. Specifically, step S5 includes: PIRDN uses the total system potential energy as a loss function, calculates the loss function using Gaussian integrals to drive network training, and the total system potential energy loss function... as follows:

[0045] in, and These are the system strain energy and the work done by external forces, respectively. For the Heaviside function This indicates an external force load. and Both represent integral symbols. This represents the internal strain energy; This is the double dot product of stress and strain. To improve the stability of the network training process, a characteristic load amplitude is introduced. and feature length For the loss function Perform dimensionless processing:

[0046] Furthermore, step S5 includes: setting network convergence conditions; when the network meets the convergence conditions, the displacement field can be output. u The convergence conditions for network training are as follows:

[0047] Where epochs represents the maximum number of training epochs set for the network. Indicates the current training step. tol To reduce network convergence tolerance, and These are all the average values ​​of the loss function. The table is an integer representing the average over the selected window length. For the first in the selected window range i The potential energy loss function value. Setting network convergence conditions can effectively avoid unnecessary computational overhead.

[0048] Step S6: Based on displacement field u Calculate the strain energy of the unit and target softness value J ; Step S7: Based on the unit volume fraction and volume fraction vCalculate Lagrange multipliers Based on unit strain energy and Lagrange multipliers Calculate the normal velocity field ; Step S8: Based on the normal velocity field Update expansion coefficients According to the expansion coefficient Update level set function Based on volume fraction v and target softness value J Determine whether the preset topology optimization convergence condition is met. If the preset condition is not met, repeat steps S3 to S8 until the preset condition is met.

[0049] Specifically, step S8 includes: setting convergence conditions for the optimization iteration to obtain the final optimized design structure. The convergence conditions for the optimization iteration are as follows:

[0050] in Due to volume constraints, and They represent the first k Volume fraction and compliance target value for each iteration step. and These represent the volume constraint convergence tolerance and the objective value convergence tolerance, respectively. Indicates selecting the first One iteration step to The target value for each iteration step.

[0051] The present invention also provides a topology optimization system based on a physical information radial basis deep neural network. The topology optimization system based on the physical information radial basis deep neural network can be implemented by executing the process steps of the topology optimization method based on the physical information radial basis deep neural network. That is, those skilled in the art can understand the topology optimization method based on the physical information radial basis deep neural network as a preferred embodiment of the topology optimization system based on the physical information radial basis deep neural network.

[0052] Example 2 Example 2 is a preferred example of Example 1. A topology optimization method based on a physically-informed radial basis deep neural network, according to the present invention, includes the following steps: Step 1: Initialize the parameters and PIRDN model, discrete design domain, and define the initial level set function.

[0053] Initialization parameters mainly include: material parameters, design domain geometric parameters, and optimization algorithm parameters. Material parameters include Young's modulus E (initialized to...). MPa) and Poisson's ratio (Initialized to 0.3). Initial design domain geometric parameters are as follows: Figure 3 As shown, the design domain mainly includes its length, width, and height. The length and height of the two-dimensional cantilever beam design domain are initialized to 12m and 4m respectively, with an initial void diameter of 0.8m. The length, width, and height of the three-dimensional cantilever beam design domain are initialized to 2m, 1m, and 1m respectively. The initial design domain and boundary conditions of the three-dimensional cylindrical pressure shell are as follows... Figure 6 As shown, the length, thickness, and outer diameter of the cylindrical shell are 8m, 0.6m, and 3m, respectively, and the environmental pressure is 100MPa. The optimization results are as follows. Figure 7 As shown. The optimization algorithm parameters include: the support domain radius dsp (initialized to a length of 3 units), and the optimization iteration parameters. (Initialized to 5) and (Initialized to 30).

[0054] Initialize the PIRDN model. Figure 3 The PIRDN structures corresponding to the two cantilever beam design domains in the diagram are [2,30×10,400,200,100,2] and [3,12×6×6,800,400,200,100,3], respectively. Figure 6 The PIRDN network structure corresponding to the cylindrical shell in the example is [3, 16×28×6, 800, 400, 200, 100, 3]. The square brackets represent the number of layers from the input layer to the output layer and the number of neurons in each layer. The 30×10 indicates that the radial basis functions have 30 centers in the length and 10 centers in the height directions. The learning rate is initialized to 0.001, the activation function of the first hidden layer is a Gaussian function, and the activation functions of the other hidden layers are tanh. The total number of training epochs is 8000.

[0055] Discrete design domains: The discrete element set for the two-cantilever beam is nx=240, ny=80; the discrete element set for the three-dimensional cantilever beam is nx=40, ny=20, nz=20; the discrete element set for the three-dimensional cylindrical pressure shell is nx=40, ny=60, nz=20. Boundary conditions include displacement boundary conditions and load boundary conditions. Both the two-dimensional and three-dimensional design domains have a fixed left boundary. A static load of 1000 N is set, with the application location referenced... Figure 3 .

[0056] Initial level set function definition. By appropriately setting the positions of the holes in the design domain, the initial level set function can be defined as a signed distance function. It has the following expression: ; in, For radial basis functions, compactly supported radial basis functions are selected in this embodiment. The expansion factor can be calculated. .

[0057] Step 2: Calculate the unit volume fraction based on the level set function, and take the average of the unit volume fractions to obtain the volume fraction of the design domain.

[0058] Step 3: Obtain node coordinates from the discrete design domain X ,Will X The input features are used as input features to the PIRDN model to obtain the output displacement u as follows: ; The forward propagation process of spatial coordinate X in the PIRDN model is as follows:

[0059] in, Represents the radial basis function layer. This represents a linear layer, where n is the number of fully connected layers. This represents the activation function tanh. and These represent the network training weights and biases, respectively. This represents the connection operator between different layers of a neural network. Taking the Gaussian function as an example, the activation function of the radial basis function layer has the following form:

[0060] in, Represents the Gaussian function, shape parameter a and b Control the shape of the Gaussian function. c This represents the center point of the Gaussian function, which remains fixed during network training.

[0061] Step 4: Calculate the total potential energy loss function The total potential energy loss function is defined as follows:

[0062] in, and These represent the system strain energy and the work done by the external force, respectively. To improve the stability of the network training process, characteristic load amplitudes are introduced. and feature length For the loss function Perform dimensionless processing:

[0063] Among them, for characteristic load amplitude exist Figure 3 In the two-dimensional and three-dimensional cantilever beam problems shown, the N value is taken as 1000 N, and the characteristic length is... The dimensions are taken as 12m and 2m respectively, and the three-dimensional cylindrical pressure shell is taken as... For an environmental pressure of 100 MPa, take for .

[0064] Step 5: Set the network training convergence conditions as follows:

[0065] Where epochs represents the maximum number of training epochs set for the network. Indicates the current training step. tol To improve network convergence tolerance, setting convergence conditions can effectively avoid unnecessary computational overhead. In this embodiment, the number of epochs is 8000. tol for , The value is 10. During network training, if the convergence condition is met or the maximum number of training rounds is exceeded, network training terminates, and the output shift is set. u Proceed to step six. Otherwise, continue the training loop until the network convergence condition is met.

[0066] Step 6: Calculate the element strain energy and target softness value J The element strain energy can be calculated from the displacement field output after PIRDN training using Gaussian integrals. The target compliance value can then be obtained by summing the element strain energies. J .

[0067] Step 7: Calculate the Lagrange multipliers and normal velocity field Lagrange multipliers The updated formula is as follows:

[0068] in, and To optimize the iteration parameters, take , . , and These represent the initial volume fraction, the volume fraction at the k-th iteration step, and the volume constraint value, respectively, representing the volume constraints for the two-dimensional and three-dimensional problems in the embodiments. All are 0.5. The normal velocity field is calculated according to the following formula:

[0069] in, This represents the strain energy term, which can be calculated based on the element strain energy.

[0070] Step 8: Update the expansion coefficients and level set function Expansion coefficient Update using the following formula:

[0071] in, The time step is represented by 0.25 and 0.1 for the two-dimensional and three-dimensional problems, respectively, in the embodiment. Let represent an invertible matrix associated with the interpolation basis function. V The velocity field representing the evolution of the level set function is as follows:

[0072] in, It is a coefficient. Figure 3 The two-dimensional and three-dimensional problems are taken as 1.5 and 0.5, respectively. The level set function can be updated according to the expansion coefficient, as follows:

[0073] Step 9: Set the convergence conditions for the optimization iteration to obtain the final optimized design structure. The optimization iteration convergence conditions used in this embodiment are as follows:

[0074] in Due to volume constraints, and Let these represent the volume fraction and compliance target value at the k-th iteration step, respectively. and Let these represent the volume constraint convergence tolerance and the objective value convergence tolerance, respectively. , , , Indicates selecting the first One iteration step to The target value of the iteration step.

[0075] Figure 4 and Figure 5 The topology optimization problems of cantilever beams in two and three dimensions were solved respectively. The three-dimensional problem is more difficult to solve, which verifies that the method of the present invention has a wide range of applicability.

[0076] Those skilled in the art will understand that, in addition to implementing the system, apparatus, and their modules provided by this invention in purely computer-readable program code, the same program can be implemented in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers by logically programming the method steps. Therefore, the system, apparatus, and their modules provided by this invention can be considered a hardware component, and the modules included therein for implementing various programs can also be considered structures within the hardware component; alternatively, modules for implementing various functions can be considered both software programs implementing the method and structures within the hardware component.

[0077] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A topology optimization method based on a physical information radial basis deep neural network, characterized by, Comprising: Step S1: initializing a physical information radial basis deep neural network (PIRDN) model; Step S2: discretizing an initial pressure-resistant structure design domain and calculating an initial level set function according to the same; Step S3: Calculate the current pressure-resistant structure unit volume fraction based on the level set function and volume fraction v ; Step S4: obtaining discrete space coordinates based on the discrete design domain X and obtaining the discrete space coordinates X inputting the PIRDN model to obtain the displacement field u ; Step S5: based on displacement field u Computing total potential energy loss function The PIRDN model is trained, and it is judged whether the preset training convergence condition is met. If the preset training convergence condition is not met, the PIRDN model is updated through back propagation, and steps S4 to S5 are repeatedly triggered until the preset training convergence condition is met. Step S6: calculating the displacement field based on the strain energy u , and the target flexibility value ; and the target flexibility value J ; Step S7: Calculate the unit volume fraction of the current pressure-resistant structure and the volume fraction v Calculate the Lagrange multiplier ; based on the unit strain energy of the current pressure-resistant structure and the Lagrange multiplier Calculate the normal velocity field ; Step S8: updating the normal velocity field based on the normal velocity field updating the expansion coefficient ; updating the expansion coefficient according to the expansion coefficient updating the level set function ; updating the level set function based on the volume fraction v and the target compliance value J determining whether a preset topological optimization convergence condition is met, if the preset condition is not met, repeating triggering steps S3 to S8 until the preset condition is met, and outputting the optimized pressure-resistant structure topological shape.

2. The topology optimization method based on physical information radial basis deep neural network according to claim 1, wherein, The PIRDN model in the step S1 comprises: an input layer, multiple hidden layers, and an output layer. The multiple hidden layers comprise: a first layer of the hidden layers is an adaptive radial basis function layer, and an activation function of other hidden layers is tanh.

3. The topology optimization method based on physical information radial basis deep neural network according to claim 1, wherein, The step S4 comprises: A forward propagation process of the discrete spatial coordinates X in the PIRDN model is as follows: wherein, represents a radial basis function layer, represents a linear layer, n is the number of fully connected layers, represents an activation function tanh, and represent network training weights and biases, respectively, represents a connector between different layers of the neural network; An activation function of the radial basis function layer has the following form: wherein, denotes a Gaussian function, the shape parameter a and b controls the shape of the Gaussian function, trained in the network to achieve automatic update, enabling PIRDN to learn more complex physical features; c denotes the center point of the Gaussian function, which remains fixed during network training.

4. The topology optimization method based on physical information radial basis deep neural network according to claim 1, wherein, The step S5 comprises: wherein, is the introduced characteristic load amplitude; is the characteristic length; is the Heaviside function, denotes the internal strain energy, is the external force potential energy, denotes the external force load.

5. The topology optimization method based on physical information radial basis deep neural network according to claim 1, wherein, The preset training convergence condition in the step S5 comprises: where epochs represents the maximum number of training rounds set for the network, represents the current training step, tol is the network convergence tolerance; and are the average values of the loss function, table is an integer representing the average in the selected window length, is the i potential energy loss function value in the selected window range.

6. The topology optimization method based on physical information radial basis deep neural network according to claim 1, wherein, The preset topological optimization convergence condition in the step S8 comprises: wherein, is a volume constraint, and denote the volume fraction and the compliance target value of the k th iteration step, respectively, and denote the volume constraint convergence tolerance and the target value convergence tolerance, respectively; denotes the target value of the th iteration step to the th iteration step.

7. A topology optimization system based on physical information radial basis deep neural network, characterized in that, Comprising: Module M1: initializing a physical information radial basis deep neural network (PIRDN) model; Module M2: discretizing an initial pressure-resistant structure design domain and calculating an initial level set function according to the same; Module M3: Calculate current pressure-resistant structure unit volume fraction based on level set function and volume fraction v ; Module M4: obtaining discrete spatial coordinates based on the discrete design domain X and obtaining the discrete spatial coordinates X inputting the PIRDN model to obtain the displacement field u ; Module M5: based on displacement field u Computing total potential energy loss function The PIRDN model is trained, and it is judged whether the preset training convergence condition is met. If not, the PIRDN model is updated by back propagation, and the modules M4 to M5 are repeatedly triggered until the preset training convergence condition is met. Module M6: Based on displacement field u , and target flexibility value of the current pressure-resistant structure J ; Module M7: Calculate the fraction of the unit volume of the current pressure-resistant structure and the volume fraction v Calculate the Lagrange multiplier ; Based on current pressure-resistant structure unit strain energy And lagrange multipliers Computing normal velocity field ; Module M8: based on normal velocity field updating the expansion coefficient ; according to the expansion coefficient updating the level set function ; based on the volume fraction v and the target compliance value J determine whether the preset topological optimization convergence condition is met, if the preset condition is not met, repeat triggering modules M3 to M8 until the preset condition is met, and output the optimized pressure-resistant structure topological shape.

8. The topology optimization system based on physical information radial basis deep neural network according to claim 7, wherein, The PIRDN model in the module M1 comprises: an input layer, multiple hidden layers, and an output layer. The multiple hidden layers comprise: a first layer of the hidden layers is an adaptive radial basis function layer, and an activation function of other hidden layers is tanh.

9. The topology optimization system based on physical information radial basis deep neural network according to claim 7, wherein, The module M4 comprises: A forward propagation process of the discrete spatial coordinates X in the PIRDN model is as follows: wherein, represents a radial basis function layer, represents a linear layer, n is the number of fully connected layers, represents an activation function tanh, and respectively represent network training weights and biases, represents a connector between different layers of the neural network; An activation function of the radial basis function layer has the following form: wherein, represents a Gaussian function, a shape parameter a and b controls the shape of the Gaussian function, trained in the network to achieve automatic update, enabling PIRDN to learn more complex physical features; c represents the center point of the Gaussian function, which remains fixed during network training.

10. The topology optimization system based on physical information radial basis deep neural network according to claim 7, wherein, The module M5 comprises: wherein, is the introduced characteristic load amplitude; is the characteristic length; is the Heaviside function, denotes the internal strain energy, is the external force potential energy, denotes the external force load; The preset training convergence condition in the module M5 comprises: wherein epochs represents the maximum number of training rounds set for the network, represents the current training step, tol is the network convergence tolerance; and are the average values of the loss function, table is an integer representing the averaging over a selected window length, is the i potential energy loss function value in the selected window range. The preset topological optimization convergence condition in the module M8 comprises: wherein, is a volume constraint, and denote the volume fraction and the compliance target value of the k th iteration step, respectively, and denote the volume constraint convergence tolerance and the target value convergence tolerance, respectively; denotes the target value of the th iteration step to the th iteration step.

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