Function gradient material optimization method based on physical information network and deep regression

Abstract

The invention discloses a functionally graded material structure optimization method based on a physical information network and deep regression. The method comprises the steps of 1, building a physical information network, calculating strain and stress through automatic differential, and building physical constraints in combination with a constitutive equation and an equilibrium equation; the deviation between the network prediction and the physical law is quantified through a mean square error to ensure that the prediction result conforms to the mechanics principle; 2, building a deep regression network, and optimizing and training the deep regression network through the mapping relation; an optimal expression is searched through gradient descent, and an analyzable volume fraction function model is finally generated and used for material distribution optimization; and step 3, adopting an alternating iteration strategy, finally outputting an expression of VA (x, y, z), verifying Young modulus distribution accuracy through Voigt homogenization, and realizing automatic optimization design of the functionally graded material. According to the method, the optimal distribution of each component of the functionally graded material structure is automatically obtained by adopting a physical information neural network method, so that the performance of the functionally graded structure is optimized.

Description

Technical Field

[0001] The present invention relates to the technical field of mechanical response analysis of functional gradient material structures in solid mechanics, and in particular to a functional gradient material structure optimization method based on physical information network and deep regression. Background Art

[0002] Functionally graded materials (FGMs) are a new type of composite material composed of two or more materials continuously compounded in a specific ratio. Their composition exhibits a continuous gradient along a specific direction. The development of these materials stems from the demand in modern aerospace for materials capable of long-term, stable operation in extreme environments of high temperature and pressure. Since the concept of FGMs was proposed, their manufacturing, design, and application have attracted widespread attention from the international academic and industrial communities, becoming a key research topic in mechanics, materials, and engineering.

[0003] In the preparation of functionally graded materials (FGMs), to meet practical needs, it is often necessary to optimize the design of different material components to achieve the best performance match. This process typically relies on finite element numerical simulation or experimental research, based on which appropriate optimization algorithms are selected. In recent years, the integration of data-driven solid mechanics analysis methods has become an important development direction in the modeling and research of FGMs. Data-driven methods (such as neural networks) can effectively process complex material models and predict the properties of FGMs, but these methods are highly dependent on high-quality data. Insufficient or low-quality data may limit the accuracy and reliability of the model. Therefore, although data-driven methods have strong fitting capabilities and advantages in nonlinear modeling, their limitation lies in the strict requirements on data quality and quantity. It is worth noting that existing FGM structural optimization design schemes are still in the exploratory stage, and many analysis strategies are highly dependent on expert experience, lacking versatility and operability. For inexperienced researchers, carrying out related work often faces great challenges.

[0004] Physics-informed neural networks (PINNs), an emerging numerical computational method, successfully solve complex physical fields by directly embedding physical laws into loss functions. With their significant advantages of low data dependency, meshing-free operation, and inverse problem-solving capabilities, PINNs offer a new approach for optimizing the structural design of functionally graded materials.

[0005] In summary, how to develop an efficient and accurate physical information neural network method based on advanced physical information neural network technology to solve the problem of insufficient physical field prediction accuracy in traditional data-driven has become one of the difficult problems in the structural optimization design of functional gradient materials. Summary of the Invention

[0006] In order to overcome the defects of the above-mentioned existing technologies, the purpose of the present invention is to provide a functional gradient material structure optimization method based on physical information network and deep regression, which adopts the physical information neural network method to automatically obtain the optimal distribution of each component of the functional gradient material structure, thereby realizing the optimization of the performance of the functional gradient structure.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is:

[0008] The functional gradient material structure optimization method based on physical information network and deep regression includes the following steps:

[0009] Step 1: Build a physical information network. The physical information network takes the Young's modulus and Poisson's ratio of the functional gradient material and the spatial coordinates (x, y, z) as input and outputs the displacement distribution (u, v, w) and volume fraction function V A (x, y, z); calculate strains and stresses through automatic differentiation, and construct physical constraints by combining constitutive equations with equilibrium equations; the optimization goal is to minimize the structural stress distribution. The loss function includes the equilibrium equation error, boundary condition error, and objective function error. The mean square error is used to quantify the deviation between the network prediction and the physical law to ensure that the prediction results conform to the principles of mechanics;

[0010] Step 2: Build a deep regression network and optimize the training of the deep regression network through mapping relationships; the deep regression network maps the coordinates (x, y, z) into an explicit expression of the volume fraction function through a binary tree structure, and the symbol library contains mathematical operators and elementary functions; during training, the V predicted by the physical information network is used A (x, y, z) is the label, the fitting quality reward R (based on the normalized root mean square error) is calculated, and the network is optimized to improve the R value; the optimal expression is searched through gradient descent, and finally an analyzable volume fraction function model is generated for material distribution optimization;

[0011] Step 3: Use an alternating iterative strategy to first train the physical information network to reduce Loss, and then train the deep regression network to improve R, until both meet the threshold (such as Loss < 10 -4 , R>99%); Joint optimization ensures that the volume fraction function is both physically consistent and mathematically explicit. The final output is V A The expression of (x, y, z) is used to verify the accuracy of Young's modulus distribution through Voigt homogenization, realizing the automated optimization design of functional gradient materials.

[0012] The step 1 is specifically as follows:

[0013] Build a physical information network W net The input is the functional gradient material parameters (Young's modulus and Poisson's ratio) and the structural coordinate position point (x, y, z), and the output is the displacement distribution u(x, y, z), v(x, y, z), w(x, y, z) and volume fraction function V A (x,y,z), that is

[0014] {u pred (x,y,z),v pred (x,y,z),w pred (x,y,z),V Apred (x,y,z)}=W net (x,y,z;ω,b) (1)

[0015] where u pred (x,y,z),v pred (x,y,z),w pred (x,y,z) and V Apred (x, y, z) are the displacement distribution and volume fraction function predicted based on the coordinate position (x, y, z), respectively, W net represents a fully connected neural network, ω represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network.

[0016] Based on the Voigt homogenization theory, the continuous spatial distribution of the two-phase material (component A and component B) is regulated by the volume fraction output by the above physical information network, and the equivalent Young's modulus and Poisson's ratio of the functionally graded structure are predicted to be

[0017]

[0018] Among them E A 、E B is the Young's modulus of component A and component B, μ A 、μ B is the Poisson's ratio of component A and component B, V A (x,y,z) and V B (x, y, z) are the volume fraction expressions of the functionally gradient material components A and B at (x, y, z), satisfying V A (x,y,z)+V B (x,y,z)=1.

[0019] Based on the displacement distribution output by the physical information network and combined with the automatic differentiation calculation capability of the neural network, the strain of the functional gradient structure is obtained.

[0020]

[0021] where ε x , ε y , ε z and γ xy , γ yz , γ zx are normal strain and shear strain, respectively.

[0022] The constitutive equation of the functionally graded structure is:

[0023]

[0024] θ=ε x (x,y,z)+ε y (x,y,z)+ε z (x,y,z)

[0025] where σ x , σ y , σ z and τ xy , τ yz , τ zx are normal stress and shear stress respectively;

[0026] The equilibrium equation of the functionally graded structure is:

[0027]

[0028] where f x 、f y and f z It is physical strength;

[0029] Substituting the constitutive equation (4) into the equilibrium equation (5), we can obtain the control equation of displacement control. Combined with the boundary conditions, we can obtain the stress distribution σ x , σ y , σ z , τ yz , τ zx , τ xy For the elastic response problem of functionally graded structures, the optimization objective is defined as

[0030]

[0031] in

[0032]

[0033] Combining constitutive equations with equilibrium equations to construct physical constraints:

[0034] Constraint 1: Control equations: ensure that the model follows the basic physical equations of functionally graded materials;

[0035] Constraint 2 boundary conditions: ensure that the predicted stress field meets the boundary conditions of the functionally graded plate with holes;

[0036] Constraint 3 optimization objective: Ensure that the optimization objective requirements are calculated under the predicted volume fraction function. Through the combined effect of these three constraints, accurate prediction of the functionally gradient plate with holes under tensile load can be achieved.

[0037] The total loss function Loss of the physical information network, including the mean square error Loss of the balance equation f ; Boundary condition mean square error Loss b And the objective function mean square error Loss M ,Right now

[0038] Loss=Loss f +Loss b +Loss M (8)

[0039] The mean square error Loss of the functional gradient structure balance equation f

[0040]

[0041] The mean square error of the boundary conditions of the functional gradient structure is Loss b , objective function mean square error Loss M

[0042]

[0043] where N p is the number of sampling points of displacement data points (u, v, w), (x i ,y i ,z i ) is the test sample point, N m is the number of sampling points of the objective function.

[0044] The step 2 is specifically as follows:

[0045] Construct a deep regression network to establish the function V of the functional gradient structure coordinate point (x, y, z) and the volume fraction function of component A A *(x, y, z); in this deep regression network, the symbolic expression is an explicit mathematical expression instantiated by a binary tree data structure and is abstracted into a binary tree structure, where each node represents a symbol in a symbol library, which contains input variables such as the coordinates (x, y, z) of the functional gradient structure, constants (such as m, n) or operators (such as +, -, ×, / , trigonometric functions such as sin, cos, tan, logarithmic function log, exponential function exp, etc.), and the final output of the network is the predicted volume fraction function V A * Explicit expressions of (x,y);

[0046] When evaluating each candidate solution V A * When (x, y, z), the deep regression network calculates the fit quality reward R = 1 / (1 + NRMSE) as the following form, where NRMSE is the root mean square error normalized by the deviation from the target S:

[0047]

[0048] Where S is the variance of the volume fraction data set calculated from the target volume fraction, V A (x,y,z) is represented by the physical information network W net The predicted volume fraction function, V A * (x,y,z) is the volume fraction function predicted by the deep regression network.

[0049] The step three is specifically as follows:

[0050] When the loss function Loss of the physical information network and the deep regression network fitting quality reward R both meet the set conditions, the network training process will terminate;

[0051] For the physical information network and deep regression network, two independent thresholds are set: when the loss function of the physical information network is less than a predetermined threshold (for example, 10 -4 ), and the fitting quality reward R exceeds a certain set threshold (e.g. 99%), the training process will be terminated.

[0052] Taking into account the differences in training efficiency and convergence between physical information networks and deep regression networks, an alternating iterative training strategy is adopted. First, the physical information network is optimized to reduce its loss function Loss, and then the deep regression network is trained to improve the fitting quality reward R. The joint optimization of the two is achieved through an iterative algorithm.

[0053] In this process, the final output of the deep regression network is the volume fraction function V of the functionally gradient material. A* Explicit expression of (x,y,z), where the volume fraction V A * The parameters (x, y, z) are jointly determined by a deep regression network. The Young's modulus distribution is further calculated using the Voigt homogenization scheme. To verify the accuracy of the obtained Young's modulus distribution, it can be compared with the results of numerical calculation methods such as finite element analysis.

[0054] Through the above steps, the final output of the deep regression network is the volume fraction function V of the functionally gradient material A * The explicit expression of (x, y, z) reflects the distribution of each component of the functionally gradient material at different positions. At the same time, the volume fraction function V A * The model parameters in (x, y, z) are simultaneously determined by a deep regression network, realizing the optimized design of functionally graded material structures.

[0055] Beneficial effects of the present invention:

[0056] The present invention implements a method for optimizing the structure of functional gradient materials based on a physical information network and a deep regression network. This method automatically establishes the relationship between the structure and performance of functional gradient materials by combining a physical information neural network with a data-driven deep regression network. The model is obtained entirely through the joint training of the physical information network and the deep regression network. The volume fraction function and Young's modulus distribution in the optimized design results may be significantly different from those of traditional material design methods. This optimization method can intelligently discover new design rules for functional gradient materials, and can accurately predict the properties of complex materials that cannot be covered by traditional optimization methods, providing a more accurate structural optimization solution.

[0057] In addition, during the design process of functional gradient materials, engineering and technical personnel usually need to repeatedly verify multiple design schemes in order to find the optimal material structure that meets specific needs as much as possible. Each verification process often requires relying on a large amount of experimental data, test experience and appropriate optimization algorithms to adjust the model and calibrate the parameters. The functional gradient material structure optimization design method based on physical information network and deep regression network provided by the present invention does not need to rely on the continuous updating of a large amount of experimental data and manual parameter adjustment, thereby greatly reducing the problem of design inapplicability caused by lack of experience or insufficient testing. At the same time, it eliminates the tedious process of repeated experimental verification for engineering and technical personnel, significantly simplifies the workflow of functional gradient material optimization design, improves the accuracy and efficiency of material design, and reduces calculation and experimental costs. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1Schematic diagram of metal (component A)-ceramic (component B) two-phase functionally gradient material and common functionally gradient rectangular thin plate with holes.

[0059] Figure 2 Schematic diagram of the process flow for the optimization design of functionally graded rectangular thin plates with holes based on deep regression and physical information networks.

[0060] Figure 3 Schematic diagram of the mechanical response of a functionally gradient rectangular plate with holes based on the optimized design. (a) Distribution of displacement u(x,y), (b) distribution of displacement v(x,y), (c) distribution of stress σ λ (x,y) distribution. DETAILED DESCRIPTION

[0061] The present invention will be described in further detail below with reference to the accompanying drawings.

[0062] In the optimization design of functionally graded structures, the optimal safety index is usually used as the optimization target in solving the optimization problem, and then the optimal material distribution is inferred from the known physical field information to meet the given performance indicators. Here, the optimization design of a two-dimensional functionally graded rectangular thin plate with holes is analyzed. It is usually assumed that the functionally graded material is a mixture of a high-pressure-resistant metal material and a high-temperature-resistant ceramic material. An appropriate volume fraction function is obtained from the metal-ceramic (i.e., Ti alloy-ZrO2) functionally graded material, and the optimization design problem of the functionally graded material is studied in combination with the homogenization scheme, such as Figure 1 (a) is shown. A (x,y) and V B (x, y) represent the volume fractions of component A (Ti alloy) and component B (ZrO2), respectively. The Young's modulus can be described by the Voigt homogenization scheme and the volume fraction assumption, where V A (x,y) and V B (x,y) satisfies V A (x,y)+V B (x,y)=1. The purpose of the optimization design is to find the optimal volume fraction function V of the functionally gradient rectangular thin plate with holes about the position coordinate (x,y) A (x,y), minimizing the mechanical response (mechanical properties) of the functionally graded rectangular plate with holes.

[0063] To complete this process, it is necessary to estimate the distribution of material properties such as Young's modulus based on a homogenization scheme of volume fractions. The stress distribution is then obtained so that the stress distribution meets the optimization goal.

[0064] The present invention will be further described in detail below with reference to the embodiments.

[0065] The present invention realizes an artificial intelligence neural network automated modeling scheme for the optimization design of functional gradient materials based on a physical information network and deep regression functional gradient material structure optimization design method.

[0066] The following steps describe the present invention in detail. Figure 2 As shown:

[0067] Step 1: Build a physical information network to predict the coordinate position (x, y) and displacement distribution u(x, y), v(x, y) and volume fraction function V of the functionally gradient rectangular thin plate with holes A The mapping relationship of (x,y).

[0068] First, establish a fully connected neural network W net (w,b) predicted displacement distribution u(x,y), v(x,y) and volume fraction function V A (x, y), where the neural network input is the coordinate position point (x, y) and material parameters, and the output is the displacement distribution u(x, y), v(x, y) and volume fraction function V A (x,y), as attached Figure 2 shown

[0069] {u pred (x,y),v pred (x,y),V Apred (x,y)}=W net (x,y;ω,b) (1)

[0070] Among them, u pred 、v pred and V Apred are the displacement distribution and volume fraction predicted based on the coordinate position (x, y), W net represents a fully connected neural network, ω represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network.

[0071] Based on elasticity theory, calculate different volume fraction functions V A The corresponding Young's modulus under the (x,y) expression. Taking the Voigt homogenization scheme as an example, different volume fraction functions V A The calculation formulas for Young's modulus and Poisson's ratio under the (x,y) expression are as follows

[0072]

[0073] Among them E A and E B is the Young's modulus of component A and component B, μ A and μ B is the Young's modulus of component A and component B, VA (x,y) and V B (x, y) are the volume fraction expressions of functionally graded material components A and B, V A (x,y) and V B (x,y) satisfies V A (x,y)+V B (x,y)=1.

[0074] In addition, other homogenization schemes can also be considered, such as the Reuss scheme

[0075]

[0076] Combined with the automatic differentiation computing capability of neural networks, the strain of functionally graded thin plates is calculated.

[0077]

[0078] Further combined with the constitutive equation, the stress prediction value of the functionally graded thin plate is calculated

[0079]

[0080] The equilibrium equation of the functionally graded structure is:

[0081]

[0082] where f x 、f y and f z It is physical strength;

[0083] Substituting the constitutive equation (4) into the equilibrium equation (5), we can obtain the control equation of displacement control. Combined with the boundary conditions, we can obtain the stress distribution σ x , σ y , σ z , τ yz , τ zx , τ xy For the elastic response problem of functionally graded structures, the optimization objective is defined as

[0084]

[0085] in

[0086]

[0087] The constraints include three aspects, which work together in the training process of the physical information neural network. Constraint 1 (control equation): ensure that the model follows the basic physical equations of functional gradient materials; Constraint 2: (boundary conditions) ensure that the predicted stress field meets the boundary conditions of the functional gradient plate with holes; Constraint 3: (optimization target) ensures that the optimization target requirements are calculated under the predicted volume fraction function. Through the joint action of these three constraints, accurate prediction of functional gradient plates with holes under tensile loads can be achieved. That is, the total loss function Loss of the physical information network includes the following parts: the mean square error Lossf of the equilibrium equation; the mean square error of the boundary conditions and Loss b3 And the objective function mean square error Loss M ,Right now

[0088] Loss=Loss f +Loss b1 +Loss b2 +Loss b3 +Loss M (8)

[0089] The mean square error Lossf of the functionally graded thin plate equilibrium equation is

[0090]

[0091] Mean Square Error of Stress Boundary Conditions for Functionally Graded Plates for

[0092]

[0093] Mean Square Error of Displacement Boundary Conditions for Functionally Graded Thin Plates for

[0094]

[0095] Mean square error Loss of circular hole stress boundary condition b3 for

[0096]

[0097] Where l and n are the sine and direction cosine of the external normal direction at the circular hole, that is,

[0098] Objective function mean square error Loss M for

[0099]

[0100] where N p is the number of sampling points of the displacement data point (u,v). (x i ,yi ) is the test sample point, and is the sample point on the boundary, N m is the number of sampling points of the objective function, and q is the boundary sample point The true stress value on .

[0101] Step 2: Build a deep regression network to generate a function expression for optimizing the volume fraction of functionally graded thin plates with holes;

[0102] Construct a deep regression network to analyze the volume fraction function V represented by the coordinate position point (x, y) of the functional gradient thin plate A * (x,y) expression, as shown in the following Figure 2 As shown in Figure 1. This network performs computations by representing symbolic expressions as a binary tree, where each node corresponds to a symbol from a symbol library. The input variables for the nodes include the coordinates (x, y) of the functional gradient sheet, constants (such as m, n), and operators (such as +, -, ×, / , sin, cos, tan, log, exp, etc.). In this representation, the input variables and constants are considered terminal nodes; they have no child nodes. Symbols that accept single-argument operations (such as cos and exp) are called unary symbols (with one child node), while symbols that accept two-argument operations (such as +, -, ×, =, etc.) are binary symbols. By analyzing each node in depth and processing from left to right, the expression can be converted into a one-dimensional list. In this representation, operators precede their corresponding operands, reducing the need for parentheses. Using prefix notation and treating symbols (i.e., tokens) as categories, any expression can be converted into a sequence of category vectors. This sequence is generated by a recurrent neural network (RNN) and significantly reduces the search space by zeroing out the probabilities of certain tokens based on the context encoded in the generated expression tree. In addition, a priori constraints are used to limit the size of expressions (maximum length is 20 tokens), and to restrict the nesting of trigonometric functions in expressions to no more than two levels (for example, expressions such as sin(f.t+cos(x / x0+tan(□))) are prohibited, but sin(f.t+cos(□))) is allowed), as well as to avoid self-nesting of exponential and logarithmic operations (for example, e e□ ) and invalid unary inverse operations (e.g., prohibiting e log(□) ). Ultimately, the output of the deep regression network is an explicit expression of the predicted volume fraction function.

[0103] When evaluating each candidate solution V A * When (x, y), the fitting quality reward R = 1 / (1 + NRMSE) is calculated in the deep regression network, where NRMSE is the root mean square error normalized by the target S deviation:

[0104]

[0105] Where S is the variance of the volume fraction data set calculated by the target volume fraction function, V A (x,y) is represented by the physical information network W net The predicted Young's modulus, V A * (x,y) is the volume fraction function expression predicted by the deep regression network.

[0106] Step 3: Neural network training to find the optimal volume fraction function expression;

[0107] Joint training is performed based on the physical information network in step 1 and the deep regression network in step 2;

[0108] When the loss function Loss of the physical information network and the reward R of the fitting quality of the deep regression network meet the conditions at the same time, stop network training;

[0109] In order to optimize the training process of physical information network and deep regression network, this method sets different thresholds for these two networks. Specifically, when the loss function of physical information network is less than a certain threshold (for example, 10 -4 ), and the fitting quality reward R of the deep regression network exceeds a preset threshold (for example, 99%) at the same time, the training process will stop. This setting is intended to ensure that the training of the network can be terminated after reaching a certain optimization accuracy, thereby avoiding overtraining and waste of computing resources. Taking into account the differences in training efficiency and convergence effect between physical information networks and deep regression networks, this method adopts an iterative alternating training strategy. First, the network reduces the loss function Loss by training the physical information network to ensure that it approaches the optimal solution. After the physical information network is fully trained, the deep regression network is trained to improve the fitting quality reward R and optimize its expressive power and generalization performance. Through this alternating training method, the strong constraint characteristics of the physical information network driven by physical laws can be fully utilized, while relying on the advantages of the deep regression network in data-driven optimization to gradually improve the performance and convergence effect of the overall network. This iterative training process gradually achieves more accurate fitting effects and lower losses by alternately optimizing the two networks in each round of training, thereby improving the accuracy and stability of the overall system while ensuring computational efficiency. Ultimately, through iterative optimization, the networks can converge to the ideal performance level, achieving efficient joint training of physical information networks and deep regression networks.

[0110] Through the above steps, the final output of the deep regression network is the volume fraction function V of the functional gradient material obtained by this method. A* (x, y) expression, that is, the volume fraction expression that can reflect the optimal design of functionally gradient materials is generated, and the volume fraction function V A * The expression parameters of the (x,y) expression are simultaneously determined by a deep regression network. The obtained volume fraction function expression can be compared with data sets from methods such as finite element methods to verify it.

[0111] In summary, the method proposed in the present invention combines deep regression with the training of physical information networks to directly obtain the explicit expression of the volume fraction function for the optimal design of functionally graded materials and simultaneously determine the model parameters in the volume fraction.

[0112] The physical information network in step 1 obtains the coordinate position point (x, y) and volume fraction function V A The (x, y) expression mapping relationship is used for the optimization training of the deep regression network in step 2; the deep regression network in step 3 ultimately generates a volume fraction expression that can reflect the optimization design of functionally graded materials;

[0113] Example:

[0114] The invention discloses an application of the volume fraction expression for optimizing the design of functionally gradient materials to deformation analysis of functionally gradient rectangular thin plates containing holes.

[0115] The uniaxial tension of a functionally gradient rectangular thin plate with holes was studied. Figure 1 (b) shown.

[0116] Through the optimization design scheme of the functional gradient thin plate with holes of the present invention, the functional gradient rectangular thin plate with holes is optimized. Figure 1 As shown in (b), the size of the functionally gradient rectangular thin plate is 300mm×100mm, and the radius of the circular hole is 10mm. The upper and lower boundaries of the functionally gradient rectangular thin plate are free, a fixed constraint is applied to the left boundary (displacement is 0), and the right end is subjected to a uniform tensile load q. Assume that the mechanical parameters of the Ti alloy-ZrO2 functionally gradient material are E A =122.7GPa, E B =132.2GPa, μ A =0.29, μ B =0.33, uniaxial tensile load q = 100 MPa. The physical information network training uses Tanh as the activation function. The network structure contains 20 hidden layers, 20 neurons in each layer, and the learning rate is 1×10 -4 , the number of training times is 2×10 4 In the training of the deep regression network, ReLU is used as the activation function. The network structure contains 10 hidden layers, each with 64 neurons, and the number of training times is 1×10 4, input variable coordinate points, constants, and operators for the functionally gradient thin plate. The expression is limited to a maximum length of 20 tokens, the number of nested layers of trigonometric functions does not exceed two, and the number of nested layers of exponential and logarithmic operations is also limited. Based on deep regression and physical information network training, the explicit expression of the volume fraction function of the functionally gradient thin plate is obtained as

[0117]

[0118] At the same time, the parameter in the volume fraction function expression is determined to be k=0.8, where R is the radius of the circular hole.

[0119] Based on the volume fraction obtained by network model training, the displacement and stress components can be further calculated. The results are shown in the attached figure. Figure 3 The volume fraction function expression of the functionally gradient material obtained through network model training can be further combined with finite element analysis to calculate the stress and strain distribution of the material under different loads, thereby conducting quantitative analysis of the elastic deformation mechanics of the material under different loads.

[0120] The proposed method combines deep regression with physical information network training to directly obtain an explicit expression for the optimal volume fraction function in functionally gradient materials (FGMs), while also simultaneously determining the model parameters within the expression. Specifically, the volume fraction function expression for FGM optimization is derived entirely through neural network training, significantly reducing computational costs. This provides a more scientific and systematic solution for the optimization design of FGMs and offers technical support for their engineering applications.

Claims

1. A functional gradient material structure optimization method based on physical information network and deep regression, characterized by: The following steps are included: Step 1: Build a physical information network. The physical information network takes the Young's modulus and Poisson's ratio of the functional gradient material and the spatial coordinates (x, y, z) as input and outputs the displacement distribution (u, v, w) and volume fraction function V A (x, y, z); calculate strains and stresses through automatic differentiation, and construct physical constraints by combining constitutive equations with equilibrium equations; quantify the deviation between network predictions and physical laws through mean square error to ensure that the prediction results conform to mechanical principles; Step 2: Build a deep regression network and optimize the training of the deep regression network through mapping relationship; the deep regression network maps the coordinates (x, y, z) into an explicit expression of the volume fraction function through a binary tree structure, and uses the V predicted by the physical information network to calculate the volume fraction. A (x, y, z) is the label, the fitting quality reward (R) is calculated, and the network is optimized to improve the R value; the optimal expression is searched through gradient descent, and finally an analyzable volume fraction function model is generated for material distribution optimization; Step 3: Using an alternating iterative strategy, first train the physical information network to reduce Loss, then train the deep regression network to improve R, until both meet the threshold; finally output V A (x, y, z) expression to achieve automated optimization design of functionally graded materials.

2. The functional gradient material structure optimization method based on physical information network and deep regression according to claim 1 is characterized in that: The step 1 is specifically as follows: Build a physical information network W net The input is the functional gradient material parameters (Young's modulus and Poisson's ratio) and the structural coordinate position point (x, y, z), and the output is the displacement distribution u(x, y, z), v(x, y, z), w(x, y, z) and volume fraction function V A (x,y,z), that is {u pred (x,y,z),v pred (x,y,z),w pred (x,y,z),V Apred (x,y,z)}=W net (x,y,z;ω,b) (1) where u pred (x,y,z),v pred (x,y,z),w pred (x,y,z) and V Apred (x, y, z) are the displacement distribution and volume fraction function predicted based on the coordinate position (x, y, z), respectively, W net represents a fully connected neural network, ω represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network.

3. The functional gradient material structure optimization method based on physical information network and deep regression according to claim 2 is characterized in that: Based on the Voigt homogenization theory, the continuous spatial distribution of the two-phase material (component A and component B) is regulated by the volume fraction output by the above physical information network, and the equivalent Young's modulus and Poisson's ratio of the functionally graded structure are predicted to be Among them E A 、E B is the Young's modulus of component A and component B, μ A 、μ B is the Poisson's ratio of component A and component B, V A (x,y,z) and V B (x, y, z) are the volume fraction expressions of the functionally gradient material components A and B at (x, y, z), satisfying V A (x,y,z)+V B (x,y,z)=1; Based on the displacement distribution output by the physical information network and combined with the automatic differentiation calculation capability of the neural network, the strain of the functional gradient structure is obtained. where ε x , ε y , ε z and γ xy , γ yz , γ zx are normal strain and shear strain, respectively.

4. The functional gradient material structure optimization method based on physical information network and deep regression according to claim 3 is characterized in that: The constitutive equation of the functionally graded structure is: where σ x , σ y , σ z and τ xy , τ yz , τ zx are normal stress and shear stress respectively; The equilibrium equation of the functionally graded structure is: where f x 、f y and f z It is physical strength; Substituting the constitutive equation (4) into the equilibrium equation (5), we can obtain the control equation of displacement control. Combined with the boundary conditions, we can obtain the stress distribution σ x , σ y , σ z , τ yz , τ zx , τ xy For the elastic response problem of functionally graded structures, the optimization objective is defined as in 5. The functional gradient material structure optimization method based on physical information network and deep regression according to claim 4 is characterized in that: Combining constitutive equations with equilibrium equations to construct physical constraints: Constraint 1: Control equations: ensure that the model follows the basic physical equations of functionally graded materials; Constraint 2 boundary conditions: ensure that the predicted stress field meets the boundary conditions of the functionally graded plate with holes; Constraint 3 optimization objective: Ensure that the optimization objective requirements are calculated under the predicted volume fraction function. Through the combined effect of these three constraints, accurate prediction of the functionally gradient plate with holes under tensile load can be achieved.

6. The functional gradient material structure optimization method based on physical information network and deep regression according to claim 5 is characterized in that: The total loss function Loss of the physical information network, including the mean square error Loss of the balance equation f ; Boundary condition mean square error Loss b And the objective function mean square error Loss M ,Right now Loss=Loss f +Loss b +Loss M (8) The mean square error Loss of the functional gradient structure balance equation f The mean square error of the boundary conditions of the functional gradient structure is Loss b , objective function mean square error Loss M where N p is the number of sampling points of displacement data points (u, v, w), (x i ,y i ,z i ) is the test sample point, N m is the number of sampling points of the objective function.

7. The functional gradient material structure optimization method based on physical information network and deep regression according to claim 6 is characterized in that: The step 2 is specifically as follows: Construct a deep regression network to establish the function V of the functional gradient structure coordinate point (x, y, z) and the volume fraction function of component A A * (x, y, z); in this deep regression network, the symbolic expression is an explicit mathematical expression instantiated by a binary tree data structure, which is abstracted into a binary tree structure, where each node represents a symbol in a symbol library. The symbol library contains input variables, such as the coordinates (x, y, z) of the functional gradient structure, constants (m, n) or operators (+, -, ×, / , trigonometric functions such as sin, cos, tan, logarithmic function log, exponential function exp, etc.). The final output of the network is the predicted volume fraction function V A * Explicit expressions of (x,y); When evaluating each candidate solution V A * When (x, y, z), the deep regression network calculates the fit quality reward R = 1 / (1 + NRMSE) as the following form, where NRMSE is the root mean square error normalized by the deviation from the target S: Where S is the variance of the volume fraction data set calculated from the target volume fraction, V A (x,y,z) is represented by the physical information network W net The predicted volume fraction function, V A * (x,y,z) is the volume fraction function predicted by the deep regression network.

8. The functional gradient material structure optimization method based on physical information network and deep regression according to claim 7 is characterized in that: The step three is specifically as follows: When the loss function Loss of the physical information network and the deep regression network fitting quality reward R both meet the set conditions, the network training process will terminate; For the physical information network and deep regression network, two independent thresholds are set: when the loss function of the physical information network is less than a predetermined threshold (for example, 10 -4 ), and when the fitting quality reward R exceeds a certain set threshold (for example, 99%), the training process will be terminated; Taking into account the differences in training efficiency and convergence between physical information networks and deep regression networks, an alternating iterative training strategy is adopted. First, the physical information network is optimized to reduce its loss function Loss, and then the deep regression network is trained to improve the fitting quality reward R. The joint optimization of the two is achieved through an iterative algorithm.

9. The functional gradient material structure optimization method based on physical information network and deep regression according to claim 8, characterized in that: In this process, the final output of the deep regression network is the volume fraction function V of the functionally gradient material. A * Explicit expression of (x,y,z), where the volume fraction V A * The parameters in (x, y, z) are jointly determined by a deep regression network; the Young's modulus distribution is further calculated using the Voigt homogenization scheme; in order to compare it with the results of numerical calculation methods such as finite element analysis; Through the above steps, the final output of the deep regression network is the volume fraction function V of the functionally gradient material A * The explicit expression of (x, y, z) reflects the distribution of each component of the functionally gradient material at different positions. At the same time, the volume fraction function V A * The model parameters in (x, y, z) are simultaneously determined by a deep regression network, realizing the optimized design of functionally graded material structures.

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