Maximum stress intensity prediction method and system based on physical information neural network

Abstract

The invention discloses a maximum stress intensity prediction method based on a physical information neural network, and the method comprises the steps: collecting stress analysis data of different structures under various load conditions, carrying out the normalization of the data, and mapping all data to a specific interval; a neural network structure is constructed, and the optimal configuration is determined by comparing the performance of different structure models through experiments; determining a control equation related to the maximum stress intensity, and converting the control equation into a residual form; a data fitting item and a physical information constraint item are constructed, a loss function is defined as the sum of the two items, the data fitting item is calculated by adopting a mean square error (MSE), and the physical information constraint item is based on a control equation residual error; and optimizing the loss function by using a mathematical optimization algorithm, and predicting the maximum stress intensity value of the composite material. According to the method, stress analysis data of different structures under various load working conditions are accurately simulated, so that the prediction accuracy and efficiency are improved.

Description

Technical Field

[0001] The present invention belongs to the cross - technical field of computational mechanics and artificial intelligence, and particularly relates to a maximum stress intensity prediction method and system based on the physics - informed neural network PINNs. Background Technique

[0002] In the engineering field, accurately predicting the maximum stress intensity of materials and structures under different working conditions is crucial. Although traditional numerical calculation methods (such as the finite element method) are widely used, they have limitations. Fine mesh division is required, and the computational cost increases sharply with the complexity of the problem, and the processing efficiency for complex geometries and nonlinear problems is low. At the same time, when traditional neural networks are used for stress intensity prediction, since physical knowledge is not incorporated, a large number of samples are required for training, the model interpretability is poor, and the generalization ability is limited. It is not easy to determine a suitable neural network structure. A simple structure is difficult to learn complex physical relationships, and a complex structure is prone to overfitting, thus affecting the performance and efficiency of the model. It is difficult to accurately incorporate physical information into the model. There are challenges in transforming the constraint form and combining it with training. If incorporated improperly, the model prediction results lack physical interpretability. During the training process, the computational amount is large, the convergence is slow, gradient problems are likely to occur, resulting in training instability, and it will also increase the development cost, affecting the final performance and efficiency of the model.

[0003] In summary, the stress distribution of materials under complex boundary conditions and load conditions is complex. Traditional calculation methods are likely to make the prediction results deviate from the actual situation, thus affecting engineering safety and economy. Studying the maximum stress intensity prediction method of the physics - informed neural network under different material conditions is an urgent problem to be solved at present, with broad application prospects and important research value. Summary of the Invention

[0004] Object of the Invention: The object of the present invention is to provide a maximum stress intensity prediction method and system based on the physics - informed neural network, so as to accurately simulate the stress analysis data of different structures under various load conditions and improve the prediction accuracy and efficiency.

[0005] Technical Solution: A maximum stress intensity prediction method based on the physics - informed neural network of the present invention includes the following steps:

[0006] Step 1: Collect the stress analysis data of different structures under various load conditions, normalize the data, map all the data to a specific interval, and obtain the processed data;

[0007] Step 2: Construct a neural network structure. The input layer is the data after collection and processing to determine the number of neurons; the output layer is a single neuron that outputs a value representing the maximum stress intensity. Determine the optimal configuration by experimentally comparing the performance of different structural models; determine the governing equation related to the maximum stress intensity and transform the governing equation into a residual form;

[0008] Step 3: Construct a data fitting term and a physical information constraint term, define the loss function as the sum of the two terms. The data fitting term is calculated using the mean square error (MSE), and the physical information constraint term is based on the residual of the governing equation;

[0009] Step 4: Use a mathematical optimization algorithm to optimize the loss function and predict the maximum stress intensity value of the composite material.

[0010] Furthermore, in Step 1, the data includes geometric dimensions, material parameters, the magnitude and direction of the applied load, and the simulated maximum stress intensity value.

[0011] Furthermore, Step 2 specifically includes the following steps:

[0012] Step 2.1: Determine the input layer, hidden layer, and output layer of the neural network, construct an L-layer deep feedforward neural network, let the input be layer 0, and the L-th layer represent the output. This network contains L - 1 hidden layers, and each hidden layer consists of a linear mapping and a corresponding non-linear transformation, which can be formally expressed as follows:

[0013]

[0014] where, \(z\) l represents the input of the \(l\)-th layer, \(\sigma\) l represents the activation function, and \(a\) l represents the input corresponding to the activation function. The weight matrix \(\omega\) 1 and the bias vector \(b\) l are free parameters that need to be learned during the training.

[0015] Step 2.2: The process of determining the governing equation related to the maximum stress intensity:

[0016] In a two-dimensional plane stress problem, the equilibrium equations are:

[0017]

[0018] where, \(\sigma\) xx , \(\sigma\) yy are the normal stress components in the \(x\) and \(y\) directions respectively, \(\sigma\) xy =\(\sigma\) yx is the shear stress component, \(f\) x , \(f\) y are the body force components in the \(x\) and \(y\) directions respectively.

[0019] The geometric equations are:

[0020]

[0021] The constitutive equations are:

[0022]

[0023] Among them, the parameter E is the elastic modulus for plane stress problems, and μ is the Poisson's ratio for plane stress problems.

[0024] Finally, transform the above control equations into residual form:

[0025] The stress components predicted by the neural network and Substitute into the equation to obtain the residuals:

[0026]

[0027] And the corresponding residuals after transforming the geometric equations and constitutive equations, denoted as r 3 , r 4 , …, r 8 ;

[0028] Apply the stress boundary conditions and displacement boundary conditions to ensure the compatibility of boundary displacements and stresses.

[0029] Furthermore, the stress boundary conditions:

[0030]

[0031] The displacement boundary conditions:

[0032]

[0033] Among them, respectively refer to the surface forces in the x and y directions, respectively refer to the displacements in the x and y directions.

[0034] Furthermore, step 3 specifically includes the following steps:

[0035] Step 3.1: Construct the data fitting term and the physical information constraint term:

[0036]

[0037] Among them, N is the number of samples, is the predicted value of the i-th sample, is the actual value. N r is the number of collocation points, r 1 i , r2 i , …, r 8 i are the control equation residuals at each allocation point.

[0038] Step 3.2: Define the loss function as the sum of two terms:

[0039] Loss = MSE data + λMSE r

[0040] Furthermore, in step 4, the optimization of the loss function using the mathematical optimization algorithm is specifically as follows: construct a loss function composed of a data fitting term and a physical information constraint term, balance the importance of the two by setting the weight coefficient λ, and use the Adam algorithm based on the adaptive learning rate to predict the maximum stress intensity.

[0041] Furthermore, use different activation functions (such as ReLU function, Sigmoid function, Tanh function, and Softplus function), and compare their effects on the model performance. Observe the convergence speed of the model under different activation functions, and whether problems such as gradient disappearance or explosion occur. Consider the influence of multiple factors of the material on the prediction results and make necessary corrections.

[0042] The present invention also discloses a maximum stress intensity prediction system based on a physics-informed neural network for implementing the method as described in claim 1, including:

[0043] Data input module: Through this module, input the stress analysis-related data of different structures under various load conditions, covering the basic information of geometric dimensions, material parameters, the magnitude and direction of the applied load;

[0044] Model construction and loading module: According to the input data, construct a neural network structure, compare the performance of the model under different activation functions, determine the control equation and transform it into the residual form, construct a physical information constraint module and define a loss function for prediction;

[0045] Numerical calculation module: Use an efficient numerical calculation method to solve the PINNs model, and use the mean square error index to evaluate the model performance and measure the accuracy and reliability of the model prediction;

[0046] Result analysis module: Conduct an in-depth analysis of the calculated stress distribution, identify the distribution under different working conditions, and judge the generalization ability and stability of the model.

[0047] The present invention also discloses a computer device, including a memory, a processor, and a computer program stored on the memory, and the processor executes the computer program to implement the steps of the method of the present invention.

[0048] The present invention also discloses a computer-readable storage medium, on which a computer program / instructions are stored, and when the computer program / instructions are executed by a processor, the steps of the method of the present invention are implemented.

[0049] The present invention also discloses a computer program product, including a computer program / instructions, and when the computer program / instructions are executed by a processor, the steps of the method of the present invention are implemented.

[0050] Beneficial effects: Compared with the prior art, the present invention has the following remarkable advantages:

[0051] The present invention introduces the theory of Physics-Informed Neural Networks (PINNs). By constructing a neural network structure, comparing the performance of models under different activation functions, determining the governing equations and transforming them into residual forms, constructing a physics-informed constraint module and defining a loss function for prediction, it accurately simulates the stress analysis data of different structures under various load conditions, so as to more accurately predict the stress distribution and changes of materials during the stress-bearing process, providing a scientific basis for engineering fields such as architecture, machinery, and aerospace.

[0052] As a new deep learning method, Physics-Informed Neural Networks (PINNs) introduce physical laws or physical prior knowledge into the loss function of neural networks in the form of regularization terms, greatly reducing the complexity of the hypothesis space and showing certain advantages in solving problems such as partial differential equations. In the field of stress analysis, PINNs can effectively utilize a small number of samples to learn physical laws and train an effective model, giving full play to its advantages and overcoming the deficiencies of traditional methods. Based on this, the present invention constructs a PINNs model system for predicting the maximum stress intensity to improve the accuracy and efficiency of prediction, providing a new technological transformation for fields such as engineering structure design and material property evaluation, and promoting breakthroughs and developments in related fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a schematic diagram of the method and system flow for predicting the maximum stress intensity of different structures of the present invention;

[0054] Figure 2 It is a schematic diagram of a deep feedforward neural network and the calculation details of one neuron therein;

[0055] Figure 3 It is a framework for solving problems by the PINNs framework;

[0056] Figure 4 It is a force diagram of a two-dimensional plane stress pure bending beam structure and a linear distributed load diagram;

[0057] Figure 5 It is the stress distribution of pure bending deformation simulated by the PINN method;

[0058] Figure 6 The influence of different activation functions on the convergence characteristics of the loss function.

[0059] Figure 7 The relative error precision curve of the stress calculation results in the plane domain. Specific implementation mode

[0060] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0061] The maximum stress intensity prediction method and system based on the physics-informed neural network (PINNs) of the present invention, as Figure 1 shown, includes the following steps:

[0062] Step 101: Collect data on stress analysis of different structures under various load conditions, including geometric dimensions, material parameters, magnitudes and directions of applied loads, and the simulated maximum stress intensity values. Normalize this data to map all data to a specific interval.

[0063] Step 102: Determine the structure of the neural network, and determine the optimal configuration by experimentally comparing the performance of different structure models; determine the control equations related to the maximum stress intensity, and transform these equations into residual forms.

[0064] Specifically, determine the input layer, hidden layer, and output layer of the neural network. Considering an L-layer deep feedforward neural network, let the input be the 0th layer and the Lth layer represent the output. Then this network contains L - 1 hidden layers, and each hidden layer consists of a linear mapping and a corresponding non-linear transformation, which is formally expressed as follows:

[0065]

[0066] Then, the process of determining the control equations related to the maximum stress intensity:

[0067] In the two-dimensional plane stress problem, the equilibrium equation is:

[0068]

[0069] The geometric equation is:

[0070]

[0071] The constitutive equation is:

[0072]

[0073] Finally, transform the above control equations into residual forms:

[0074] The stress components predicted by the neural network and Substitute into the equation to obtain the residuals:

[0075]

[0076] and the corresponding residuals after transformation of the geometric equation and the constitutive equation, denoted as r 3 , r 4 , …, r 8 etc.

[0077] Transform the physical control equation related to the maximum stress intensity into the residual form and construct the physical information constraint term. To ensure the compatibility of the boundary displacement and stress, apply the boundary conditions:

[0078] Stress boundary conditions:

[0079]

[0080] Displacement boundary conditions:

[0081]

[0082] where respectively refer to the surface forces in the x and y directions, respectively refer to the displacements in the x and y directions.

[0083] Step 103: Construct the data fitting term and the physical information constraint term, and define the loss function as the sum of the two terms. The data fitting term is calculated using the mean square error (MSE), and the physical information constraint term is based on the residual of the control equation:

[0084] Construct the data fitting term and the physical information constraint term:

[0085]

[0086] Define the loss function as the sum of the two terms:

[0087] Loss = MSE data + λMSE r

[0088] The framework for solving the problem through the PINNs framework is shown by Figure 3 as follows.

[0089] Step 104: Use a mathematical optimization algorithm to optimize the loss function and predict the maximum stress intensity value of the composite material.

[0090] The Adam algorithm with an adaptive learning rate is used to predict the maximum stress intensity. This algorithm utilizes the first - moment estimate (mean) and second - moment estimate (uncentered variance) of the gradient to dynamically adjust the learning rate for each parameter. The first - moment estimate can be understood as the average of the gradient, which is used to accelerate convergence and reduce oscillations; the second - moment estimate is used to measure the magnitude of the gradient change, so as to adjust the learning rate more precisely. The specific steps are as follows:

[0091] During the training process, for each parameter θ, it maintains two variables, namely the first - moment estimate m t and the second - moment estimate v t . At each iteration t, first calculate the current gradient g t , and then update the first - moment estimate and second - moment estimate according to the following formulas:

[0092] m t =β 1 *m t-1 +(1 - β 1 )*g t

[0093] and

[0094]

[0095] where β 1 and β 2 are the decay rates of the first - moment and second - moment respectively, usually taking values close to 1, such as β 1 =0.9, β 2 =0.999. To correct the bias, it is also necessary to perform bias correction on m t and v t :

[0096]

[0097] Finally, update the parameter θ according to the corrected first - moment and second - moment:

[0098]

[0099] Here, α is the initial learning rate, and ε is a very small constant used to prevent the denominator from being zero, usually taking 1e - 8.

[0100] The following further describes the present invention in detail by combining the embodiments with the prediction process of the maximum stress of different structures under various load conditions.

[0101] Verify the feasibility and effectiveness of the model under complex boundary conditions and complex load conditions, and calculate the two - dimensional plane stress pure - bending beam problem. The structural force diagram takes Figure 4 (a) as an example. The material properties are given as follows:

[0102] A simply supported beam with an elastic modulus \(E = 1000\ Pa\) and a Poisson's ratio \(\mu=0.3\). The length of the beam is \(1\ m\) and the width is \(0.1\ m\). A bending moment \(M = \frac{1}{12}\ N\cdot m\) is applied on both sides of the beam. For the convenience of realizing the bending moment, a linearly distributed force \(F(y)=1000y\) is applied on both sides of the beam. The distributed load is applied to equivalently represent the bending moments at both ends and simulate the structural stress, as shown in Figure 4 (b).

[0103] The stress boundary conditions are:

[0104]

[0105] The arc stress boundary conditions are:

[0106]

[0107] A total of 10,000 internal particles and 2,000 boundary particles are randomly selected to discretize the computational domain and the boundary. Four hidden layers are used, with 60 neurons in each hidden layer. The activation function used is the Tanh function. Figure 5 It shows that the pure bending deformation simulated by the PINN method is better, and the random selection of particles also ensures that there is no situation where the stress concentration area cannot be captured by the grid in terms of stress. A better training process in the neural network can solve the problem of shear locking and has a high accuracy.

[0108] Figure 6 It shows the influence of different activation functions on the convergence characteristics of the loss function. A suitable activation function can improve the training efficiency and generalization ability of the model. For the above embodiments, using the Tanh activation function can achieve the best accuracy. The results obtained by the Tanh activation function are significantly better than the other three activation functions, and it also performs well in terms of the loss reduction speed.

[0109] According to the stress distribution obtained by the physics-informed neural network (PINNs), a mathematical optimization algorithm is used to search for the stress peak region, so as to predict the maximum stress intensity of different structures under various load conditions. Considering the uncertainties of geometric dimensions, material parameters, applied load magnitudes and directions, etc., the mean square error is used to evaluate the uncertainty range of the prediction results. It helps to understand the generalization ability and prediction accuracy, optimize the performance of the model, and provide more valuable references for engineering decisions.

[0110] The above is only a preferred embodiment of the present invention and is not intended to limit the protection scope of the present invention.

Claims

1. A maximum stress intensity prediction method based on physical information neural network, characterized in that: The following steps are involved: Step 1: Collect stress analysis data of different structures under various load conditions, normalize the data, map all data to a specific interval, and obtain processed data; Step 2: Construct a neural network structure. The input layer is the collected and processed data to determine the number of neurons. The output layer is a value representing the maximum stress intensity output by a single neuron. The optimal configuration is determined by comparing the performance of different structural models through experiments. The control equation related to the maximum stress intensity is determined and converted into a residual form. Step 3: Construct data fitting terms and physical information constraint terms, define the loss function as the sum of the two terms, the data fitting term is calculated using the mean square error (MSE), and the physical information constraint term is based on the residual of the control equation; Step 4: Use a mathematical optimization algorithm to optimize the loss function and predict the maximum stress intensity value of the composite material.

2. The maximum stress intensity prediction method based on physical information neural network according to claim 1 is characterized in that: In step 1, the data include geometric dimensions, material parameters, magnitude and direction of applied loads, and the maximum simulated stress intensity value.

3. The maximum stress intensity prediction method based on physical information neural network according to claim 1 is characterized in that: Step 2 specifically includes the following steps: Step 2.1: Determine the input layer, hidden layer, and output layer of the neural network, and construct an L-layer deep feedforward neural network. Let the input be layer 0, and the L-th layer represent the output. This network contains L-1 hidden layers, and each hidden layer consists of a linear mapping and a corresponding nonlinear transformation. The formal expression is as follows: Among them, z l represents the input of the lth layer, σ l represents the activation function, a l Represents the input of the corresponding activation function, the weight matrix ω 1 and the bias vector b l are free parameters that need to be learned during retraining; Step 2.2: Procedure for determining the governing equations associated with the maximum stress intensity: In a two-dimensional plane stress problem, the equilibrium equation is: Among them, σ xx , σ yy are the normal stress components in the x and y directions, σ xy =σ yx is the shear stress component, f x , f y are the body force components in the x and y directions respectively; The geometric equation is: The constitutive equation is: Among them, the parameter E is the elastic modulus of the plane stress problem, and μ is the Poisson's ratio of the plane stress problem; Finally, the above control equations are transformed into residual form: Stress components predicted using neural networks and Substituting into the equation, we get the residual: and the corresponding residuals after transformation of the geometric equation and the constitutive equation, denoted as r3, r4,…, r8; Stress boundary conditions and displacement boundary conditions are applied to ensure the compatibility of boundary displacements and stresses.

4. The maximum stress intensity prediction method based on physical information neural network according to claim 3 is characterized in that: The stress boundary conditions are: The displacement boundary condition: in, are the surface forces in the x and y directions respectively, Refers to the displacement in the x and y directions respectively.

5. The maximum stress intensity prediction method based on physical information neural network according to claim 1 is characterized in that: Step 3 specifically includes the following steps: Step 3.1: Construct data fitting terms and physical information constraints: Where N is the number of samples, is the predicted value of the i-th sample, is the actual value, N r is the number of points, is the residual of the control equation at each collocation point; Step 3.2: Define the loss function as the sum of two terms: Loss=MSE data +λMSE r 6. The maximum stress intensity prediction method based on physical information neural network according to claim 1 is characterized in that: In step 4, the use of a mathematical optimization algorithm to optimize the loss function is specifically as follows: constructing a loss function consisting of a data fitting term and a physical information constraint term, balancing the importance of the two by setting a weight coefficient λ, and using an Adam algorithm based on an adaptive learning rate to predict the maximum stress intensity.

7. A maximum stress intensity prediction system based on physical information neural network, used to implement the method according to claim 1, characterized in that: include: Data input module: This module is used to input stress analysis related data of different structures under various load conditions, covering basic information such as geometric dimensions, material parameters, and applied load magnitude and direction; Model building and loading module: Based on the input data, build the neural network structure, compare the performance of the model under different activation functions, determine the control equation and convert it into residual form, build the physical information constraint module and define the loss function for prediction; Numerical calculation module: Use efficient numerical calculation methods to solve the PINNs model, use the mean square error indicator to evaluate the model performance, and measure the accuracy and reliability of the model prediction; Result analysis module: conducts in-depth analysis on the calculated stress distribution, identifies the distribution under different working conditions, and determines the generalization ability and stability of the model.

8. A computer device comprising a memory, a processor and a computer program stored in the memory, characterized in that: The processor executes the computer program to implement the steps of the method of claim 1.

9. A computer-readable storage medium having a computer program / instruction stored thereon, characterized in that: When the computer program / instructions are executed by a processor, the steps of the method according to claim 1 are implemented.

10. A computer program product comprising a computer program / instructions, characterized in that When the computer program / instructions are executed by a processor, the steps of the method according to claim 1 are implemented.

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