Structural physical field solving method based on parameterized physical information neural network

Abstract

The invention discloses a structural physical field solving method based on a parameterized physical information neural network, and the method comprises the steps: taking a to-be-solved stress component as a rectangular thin plate, building a Cartesian coordinate system, and obtaining a space coordinate and a geometric parameter vector of the to-be-solved stress component; a geometric perception physical information neural network model with space coordinates and geometric parameter vectors as input and stress and deformation as output is constructed, the geometric perception physical information neural network model comprises a geometric perception neural network module and a physical constraint module based on total potential energy, the geometric perception neural network module encodes geometric information of a component and fuses the geometric information into a training process, and the physical constraint module is used for training the component. The input is converted into unconstrained scalar output, a physical constraint module based on the total potential energy converts the unconstrained scalar into a deflection field meeting physical laws and boundary conditions, and the total potential energy of a stress component to be solved is calculated to serve as a loss function of model training; and training the model by using a hybrid optimization strategy of Adam and LBFGS, and converging the model when the total potential energy reaches the minimum value to realize accurate mechanical property prediction.

Description

Technical Field

[0001] This invention relates to a method for solving structural physical fields based on parametric physical information neural networks, belonging to the field of structural mechanics stress and deformation solving technology. Background Technology

[0002] Computational structural mechanics is the theoretical cornerstone of modern engineering design and analysis, playing a crucial role in fields such as civil and structural engineering. Especially in the cutting-edge concept of digital twins, unprecedented demands are placed on computational methods, requiring virtual models to accurately and in real-time reproduce the entire lifecycle behavior of physical entities. Currently, the Finite Element Method (FEM), as a highly mature numerical tool, has been widely used in various commercial software (such as ANSYS and ABAUSQUS) due to its strong versatility and reliability. However, the success of FEM is rooted in its mesh-based discretization solution paradigm, which also constitutes its inherent limitations: whenever the geometry of the structure changes, mesh re-division and model reconstruction must be repeated. This process is not only tedious but, more importantly, its high time cost has become a key issue restricting the application of FEM in high-frequency real-time simulation scenarios such as digital twins.

[0003] In recent years, the continuous advancement of deep learning based on neural networks has enabled AI technology to be widely applied in engineering structures, such as damage identification, model optimization, and performance prediction. However, traditional pure data-driven models have two major drawbacks: first, they heavily rely on massive amounts of high-quality labeled data; and second, their "black box" nature leads to a lack of physical interpretability. Although some ex post-interpretation techniques (such as SHAP) can quantify the contribution of input features, they cannot intrinsically integrate prior physical knowledge into the model training process. To address this challenge, the Physics-Informed Neural Network (PINN) has emerged. PINN imbues deep learning models with physical laws by embedding control partial differential equations as regularization terms into the loss function of the neural network. A well-trained PINN can be considered an efficient surrogate model, achieving real-time prediction while ensuring physical consistency. Thanks to this, PINN has rapidly attracted widespread attention in academia and has been widely applied in various fields such as new material design, semiconductor performance optimization, surrogate models, sparse reconstruction of physical fields, thermodynamics, fluid mechanics, and solid mechanics.

[0004] Compared to FEM, PINN, as a meshless method, has three major theoretical advantages: (1) it can directly model continuous spatiotemporal domains, thus potentially avoiding the "curse of dimensionality" that plagues FEM in high-dimensional problems; (2) it can flexibly integrate physical laws with sparse data for joint driving; and (3) it has a natural framework advantage in solving inverse problems. These characteristics make PINN a promising technology for building digital twins, as it can not only capture the physical behavior of the system but also provide reliable predictions when data is sparse. However, when applied to structural mechanics problems characterized by high-order PDEs (such as plate and shell bending), the standard PINN framework encounters new challenges. The high-order differential terms included in the loss function lead to complex gradient calculations, unstable training processes, and even convergence failures. Although some improvement strategies have been proposed in the field, such as multi-network decomposition, adaptive activation functions, or adaptive sampling, these methods are essentially still within the framework of minimizing the PDE residuals point by point at coordination sites, thus their effect on improving training efficiency and accuracy is relatively limited.

[0005] As an important variant of PINN, the Deep Energy Method (DEM) offers a new approach. DEM cleverly returns to the first principle of physics—the principle of minimum potential energy variation—using the total potential energy functional of the system as the loss function of the neural network. By minimizing the global energy rather than the local residuals, DEM avoids directly calculating higher-order derivatives in the loss function, thus significantly improving the stability and efficiency in solving higher-order PDE problems. This method has demonstrated its superiority in solving problems related to heat conduction, stress and strain in circular tubes, contact mechanics, stress and deformation in beams and plates, deformation of hyperelastic materials, and elastoplasticity, and is considered a powerful alternative technique for data-driven physical modeling.

[0006] However, both traditional PINN based on residuals and DEM based on variational principles share a common and critical limitation in current research paradigms: the trained models are typically only applicable to solving problems with a specific geometric configuration. Once the geometry changes, the model needs to be retrained for the new configuration. From this perspective, existing PINN / DEM methods essentially reproduce the pain point faced by FEM—the inability to efficiently handle geometric parameterization problems, which greatly limits their potential as a general-purpose computational tool. Summary of the Invention

[0007] The technical problem to be solved by this invention is to provide a method for solving the physical field of a structure based on a parametric physical information neural network. The geometric features of the structure are used as input parameters of the neural network, enabling the model to perceive geometric changes. This transforms PINN from a traditional single-unit problem solver into an efficient parametric surrogate model, enabling real-time and accurate prediction of the mechanical properties of structures with arbitrary geometric shapes in the parameter domain.

[0008] To solve the above-mentioned technical problems, the present invention adopts the following technical solution: The structure physics field solution method based on parametric physical information neural networks includes the following steps: Step 1: Treat the stress-bearing member to be solved as a rectangular thin plate, establish a Cartesian coordinate system, and obtain the spatial coordinates and geometric parameter vectors of the stress-bearing member to be solved; Step 2: Construct a geometrically perceptive physical information neural network model with spatial coordinates and geometric parameter vectors as inputs and the stress and deformation of the stress-bearing component to be solved as outputs. The geometrically perceptive physical information neural network model includes a geometrically perceptive neural network module and a physical constraint module based on total potential energy. The geometrically perceptive neural network module is used to encode the geometric information of the stress-bearing component to be solved and incorporate it into the training process. It also converts the input into an unconstrained scalar and outputs it to the physical constraint module based on total potential energy. The physical constraint module based on total potential energy is used to convert the unconstrained scalar into a deflection field that satisfies physical laws and boundary conditions, and calculates the total potential energy of the stress-bearing component to be solved as the loss function for model training. Step 3: The geometric perception physical information neural network model is trained using a hybrid optimization strategy of Adam and LBFGS. When the total potential energy reaches the minimum value, the model converges and outputs the stress and deformation of the stressed component to be solved.

[0009] Compared with the prior art, the present invention, employing the above technical solution, has the following technical effects: 1. This invention proposes a novel training paradigm for physical information neural networks—the Geometry-Aware Physics-Informed Neural Network (GA-ePINN). The core idea of ​​this network is to use the geometric features of the structure, with fixed boundary conditions, as variable input parameters for the neural network. By training on a specified parameter domain, GA-ePINN can learn the intrinsic mapping relationship between physical fields (such as displacement and stress) and geometric parameters. Ultimately, the trained single model can serve as a general surrogate solver, enabling real-time and accurate prediction of the mechanical properties of structures with arbitrary geometries within the parameter domain.

[0010] 2. This invention transforms the problem of minimizing local PDE residuals into minimizing global energy functionals. This method not only avoids the computation of complex higher-order derivatives, but also outperforms the traditional PINN method based on strong forms in terms of training efficiency, convergence stability, and solution accuracy.

[0011] 3. The method of this invention has been validated on the classic and challenging problem of fourth-order Kirchhoff plates. Results show that in the aspect ratio parameterization case, GA-ePINN achieves a computational efficiency 10 times that of FEM while maintaining an error of less than 1%. Furthermore, this framework has been successfully extended to the parameterization of internal circular hole boundaries, with an average solution error of only (1.32 ± 0.61)%. Compared to FEM, GA-ePINN achieves high accuracy while reducing the total computation time for solving parameterized curves or surfaces by an order of magnitude, representing a breakthrough in efficiency. More importantly, it provides the ability to continuously predict parameters, which is difficult to achieve with numerical methods such as the finite element method.

[0012] 4. This invention is the first to systematically integrate the concept of geometric parameterization into the energy-based PINN framework for structural mechanics analysis, which can significantly accelerate the development of structural optimization analysis and digital twin systems. The paradigm pioneered by GA-ePINN can provide a powerful and computationally efficient new approach for parametric mechanics problems in the engineering field, and is expected to play an important role in fields such as digital twins, structural optimization, and real-time simulation, with broad application prospects. Attached Figure Description

[0013] Figure 1 This is a flowchart of the structural physics field solution method based on parametric physical information neural network of the present invention; Figure 2 This is the solution to the maximum deflection problem in Example 1; Figure 3 This is the result of evaluating GA-ePINN based on the FEM dataset in Example 1; Figure 4 The results and error contour plots of GA-ePINN and FEM solutions in Example 1 are shown. Figures 5(a)-5(e) These are the complete solution results for 41 cases using Example 1 GA-ePINN; Figure 6 This is a schematic diagram of the circular fixed support area, i.e., the center of the circle, moving along a preset trajectory in Embodiment 2; Figure 7 This is the solution result of GA-ePINN when the position of the circular fixed support region changes, i.e., when the center of the circle moves, in Example 2; Figure 8This is the solution result of GA-ePINN when the position of the circular fixed region remains unchanged and the radius changes from small to large in Example 2; Figure 9 This is the result of evaluating GA-ePINN based on the FEM dataset in Example 2; Figure 10 The results and error cloud diagrams of GA-ePINN and FEM solutions in Example 2 are shown. Detailed Implementation

[0014] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0015] This invention proposes a structural physics field solution method based on a parameterized physical information neural network. First, it introduces the concepts of the minimum potential energy principle and the deep energy method, which are the core physical principles for optimizing the model. Then, specifically for Kirchhoff plates, it derives their total potential energy functional, laying the groundwork for determining the input and loss functions of GA-ePINN. Finally, based on the derivation results, it constructs and trains the corresponding GA-ePINN to perform real-time and accurate mechanical performance prediction of structures with arbitrary geometric shapes in the parameter domain.

[0016] 1. The principle of minimum potential energy and the deep energy method In elastic statics, there are three equivalent classical approaches to describe the equilibrium state of a system: local description, work description, and energy description. Local description directly derives the system's control parameters (PDEs) by analyzing the force balance of infinitesimal elements; work description utilizes the principle of virtual work; and energy description is based on variational principles, corresponding to the principle of minimum potential energy.

[0017] Traditional PINN is based on local description, summing the PDE residuals at a series of coordination points in the solution domain and minimizing them as a loss function. While this "pointwise equation satisfaction" strategy is intuitive, it often faces challenges of training instability when dealing with structural mechanics problems characterized by high-order differential equations due to the complexity of gradient calculation.

[0018] In contrast, DEM is rooted in a more macroscopic and global energy description. The principle of minimum potential energy states that in a conservative force system, for all possible deformation fields (i.e., virtual displacement fields) that satisfy displacement boundary conditions, the true equilibrium state of the system is the only state that minimizes its total potential energy.

[0019] Therefore, solving the equilibrium problem of a mechanical system is essentially equivalent to a variational problem: finding a specific displacement function. This makes the total potential energy functional of the system The minimum value is obtained. Total potential energy functional. From the system's strain energy and external potential energy constitute: , Real equilibrium displacement field This is the minimum point of the functional: , in, It is defined in the set of kinematically permissible displacement fields Any virtual displacement field in, i.e. The system's displacement boundary conditions must be satisfied. The deep energy method cleverly utilizes the powerful function approximation capabilities of deep neural networks to explore this infinite-dimensional function space. Specifically, construct a parameter as Deep neural networks It is used to approximate the real displacement field. By designing the output format of the network architecture, we can ensure... The displacement boundary conditions are consistently satisfied. In this way, the output of the neural network... It's equivalent to a virtual displacement field. .

[0020] Neural network Substituting the total potential energy functional, the original variational problem can be transformed into a parameter optimization problem. The total potential energy functional of the system Directly defined as the loss function of a neural network : , Then, algorithms such as gradient descent can be used to iteratively update the network parameters. This is done to minimize the loss function. When the loss function converges, the resulting neural network... This provides a high-precision approximate solution for the displacement field in the true equilibrium state of the system.

[0021] Therefore, the core idea of ​​the PINN (DEM) framework based on the principle of minimum potential energy is to transform a complex problem of solving PDEs into an unconstrained optimization problem with a clearly defined objective function and clear physical meaning. The advantage of this method is that it avoids the direct processing of higher-order differential operators, thus providing a more stable and efficient path for solving structural mechanics problems.

[0022] 2. The total potential energy functional of the Kirchhoff plate Kirchhoff-Love plate theory, used to determine the stress and deformation of thin plates under forces and moments, is an extension of Euler-Bernoulli beam theory and has wide applications in engineering and mechanics, holding significant importance for engineering practice. It is governed by a fourth-order differential equation, a high-order equation that is very challenging to solve using traditional PINN methods. However, by applying the minimum potential energy principle, the solution can be reduced to a second-order derivative, significantly improving efficiency. This paper takes a Kirchhoff plate as an example, first deriving its total potential energy functional, and then introducing how to combine GA-ePINN for parametric modeling and solution.

[0023] Consider a rectangular thin plate with length . Width is Thickness is Establish a Cartesian coordinate system, where... The plane coincides with the neutral plane of the plate. ), The axis is perpendicular to the plate surface. ).

[0024] According to the core assumption of Kirchhoff's theory (i.e., the "plane section assumption"), any point within the plate... Three-dimensional displacement field The lateral deflection of the neutral plane can be completely determined. express: , Under the assumption of small deformation in linear elasticity, the non-zero strain components are: , in, For positive strain, Let be the shear strain. For an isotropic linear elastic material, its constitutive relation is: , in, The elasticity matrix of the material: , It is Young's modulus. It is Poisson's ratio.

[0025] The strain energy density per unit volume It can be represented as: , In the formula, the strain vector The total strain energy of the system It is the strain energy density over the entire volume of the plate. Points on: , strain vector and elasticity matrix Substitute into the above formula and adjust the thickness. Integrating in direction The strain energy can be obtained. It concerns mid-surface deflection. Functional: , in, This refers to the bending stiffness of the plate. Simultaneously, the load... The work done (i.e., the potential energy generated by external forces). for: , According to the principle of minimum potential energy, the total potential energy of the system is... Therefore, for a domain defined in a rectangle... The total potential energy functional of the plate is the sum of strain energy and external force potential energy.

[0026] When a hole exists within the plate, the expression for the total potential energy functional remains unchanged, but its integration domain changes. Let the hole be a circle, defined by its center coordinates... and radius definition.

[0027] At this point, the integral must be in a multi-connected domain. The above is performed; this area is the original rectangular region. The portion after removing the circular hole area: , Therefore, the total potential energy functional of the perforated plate It can be represented as: .

[0028] 3. Geometric Perception Physical Information Neural Network (GA-ePINN) like Figure 1 As shown, the GA-ePINN framework consists of two main modules: a geometry-aware neural network agent module and a physical constraint module based on total potential energy. The former is responsible for approximating the solution function and learning the complex mapping from spatial coordinates and geometric parameters to the physical field; the latter is responsible for transforming the minimum potential energy principle into an optimizable loss function and driving the training of the entire model through automatic differentiation and backpropagation.

[0029] In this framework, the geometry-aware neural network proxy module is treated as a general function approximator, and its structure mainly includes an input layer, a hidden layer, and an output layer.

[0030] (1) Input layer: The input layer is the key to realizing geometry perception. It is a multi-dimensional vector composed of two types of information, namely, necessary spatial coordinates. and optional geometric parameter vectors The choice of geometric parameters depends on the specific form of the problem to be solved. In the aspect ratio parameterization case, the input is a three-dimensional vector. In the case of parameterizing the inner circular boundary, the input is a five-dimensional vector. When co-parameterization is performed, the input is expanded to a six-dimensional vector. .

[0031] (2) Hidden Layers: A fully connected feedforward network with multiple hidden layers is used. The network capacity (number of layers and neurons) should match the complexity of the problem: higher-dimensional parameter inputs and denser training points usually require deeper and wider networks to ensure the model's expressive power. Based on the complexity of different cases, configurations of "3 hidden layers × 20 neurons", "3 hidden layers × 30 neurons", and "4 hidden layers × 50 neurons" were tested. All hidden layers use the hyperbolic tangent function (Tanh) as the activation function. These hidden layers are responsible for mapping the low-dimensional information of the input to a high-dimensional feature space, thereby learning the deflection function. It has a complex nonlinear form.

[0032] (3) Output layer: The output of the network is a scalar. It represents the current network parameters. A potential, unconstrained solution function under (weights and biases).

[0033] The physical constraint module based on total potential energy is responsible for processing the unconstrained output of the neural network. Transformed into a real deflection field that satisfies physical laws and boundary conditions. And calculate its total potential energy as the loss function for training.

[0034] (1) Apply hard boundary constraints: the original output of the neural network The boundary conditions of the problem cannot be automatically satisfied. To solve this problem, a "hard constraint" method is adopted, which involves designing an algebraic function related to geometry. The function is always zero on its defined boundaries. The final deflection field satisfies the boundary conditions. Constructed using the following formula: , For the plate whose inner and outer boundaries are both fixed in this invention, the boundary function is... Can be derived from the outer boundary function and inner boundary function Composed of: , Multiplying the two together and squaring them, we can simultaneously satisfy the fixed support condition where both displacement and rotation are zero: .

[0035] (2) Calculate the total potential energy: obtain the physically effective deflection field. Then, the total potential energy of the system can be calculated. Among them, the internal potential energy of the plate The calculation depends on deflection Second-order partial derivatives and By utilizing the built-in automatic differentiation feature of the deep learning framework Pytroch, it is possible to... These derivative terms are efficiently obtained from the computational graph, and then substituted into the strain energy functional formula to obtain the total potential energy. .

[0036] (3) Loss function and optimization: The total potential energy of the final system Defined as the loss function for the entire model, this loss function is minimized using a hybrid optimization strategy combining Adam and LBFGS. The Adam optimizer quickly explores the parameter space in the early stages of training, while LBFGS, as a second-order optimization method, achieves higher convergence accuracy in the later stages. The gradient of the loss function is propagated back into the network via backpropagation, iteratively updating the parameters. The model converges when the total potential energy of the system reaches its minimum value.

[0037] 4. Adaptive Sampling Strategy In cases with holes, the sampling process generates a large number of non-physical collocations located inside the holes. Calculating the potential energy at these points is meaningless. Therefore, an accept-reject sampling algorithm is used to generate a valid set of collocations. The process is as follows: (1) Generate candidate point: Generate a candidate point in the complete parameterized hyperrectangular space. ; (2) Apply geometric constraints: Examine the candidate points and determine their physical coordinates. Whether it is located in the same sample of defect parameters Outside of the defined opening. Acceptance conditions are: , (3) Screening: If a candidate point meets the above conditions, it is accepted as a valid match point; otherwise, the point is discarded. (4) Iteration: Repeat the above steps until the expected number of valid points are collected.

[0038] This method ensures that the Monte Carlo integral of the total potential energy is performed only within the valid physical domain. This is performed to ensure the correctness of the loss function calculation.

[0039] The effectiveness of the proposed GA-ePINN framework is verified through two numerical examples. Example 1 parameterizes the aspect ratio of the structure, allowing a single-trained model to solve for the deformation and internal forces of structures with arbitrary aspect ratios within the training domain. Example 2 parameterizes the position and size of the internal circular holes; similarly, a single-trained model can solve for the deformation and internal forces of any structure in the parameter domain. All examples are quantitatively evaluated using high-precision finite element method simulation results.

[0040] Example 1 - Aspect Ratio Parameterization GA- The core advantage of ePINN lies in its ability to quickly predict the deformation and internal forces of all geometries within the parameter domain after a single training iteration. This can be illustrated by discussing a classic plate and shell problem: Figure 2 As shown, the structure is a rectangular thin plate with fixed supports on all four sides. When its width... Given a value of 1, find: the length. When it changes continuously within a certain range, its maximum deflection The evolutionary pattern.

[0041] For traditional numerical methods such as the finite element method, solving such problems requires an independent and repetitive "modeling-simulation" process for a series of discrete aspect ratios. This process is not only cumbersome, but the computational cost is also proportional to the number of sampling points.

[0042] Unlike traditional methods, GA-ePINN achieves "train once, predict any subject" by constructing a unified parameterized surrogate model. Figure 2 This demonstrates the predictions of GA-ePINN. With aspect ratio The nonlinear relationship curve shows a high degree of agreement with the discrete simulation points of the FEM. The simulation results reveal a clear physical law: as the plate becomes slender, the bending behavior in its central region gradually tends to resemble that of a one-dimensional beam, with the maximum deflection at... It then converges to an asymptotic value.

[0043] Table 1 provides a quantitative comparison of the efficiency of the two methods in solving this parameterization problem. To obtain... Figure 3 The original model required 41 independent simulations for 41 FEM data points, taking a total of 255 seconds. In contrast, GA-ePINN only requires one training iteration (25 seconds) to generate a complete continuous curve, improving computational efficiency by 10 times. More importantly, the trained model can generate a complete continuous curve within the parameter range. It enables instantaneous prediction of arbitrary aspect ratios, achieving a continuous solution capability across parameter dimensions that is difficult to achieve with traditional numerical methods.

[0044] Table 1 Comparison of FEM and GA-ePINN Method Number of solutions Solving for total time Discrete or Continuous FEM 41 255s Discrete GA-ePINN 1 25s continuous To comprehensively evaluate the accuracy of GA-ePINN, the above 41 sets of FEM simulation results with different aspect ratios were used as benchmark solutions to systematically verify the solution results of the GA-ePINN model.

[0045] Macroscopic statistical results (Table 2) show that the GA-ePINN predictions are in high agreement with the FEM benchmark across all 41 test samples. Deflection field The average R² value is as high as 0.99985, and the standard deviation is only 0.00012, proving that the model is highly stable under different geometric parameters. For the bending moment field derived from the second derivative of displacement... The R² values ​​were all above 0.995, verifying the reliability of the model in predicting higher-order physical quantities.

[0046] Table 2. Average accuracy (41 samples, different aspect ratios)

[0047] Figure 4 by For example, a visual error analysis was performed on a single computational example, where, It is the bending moment in the x-direction. It is the bending moment in the y-direction. It is the bending moment in the tangential direction. This represents the maximum principal bending moment. It can be seen that the prediction error for the deflection field is extremely small (0.71%). For the more challenging bending moment field, the prediction error is mainly distributed in regions with large bending moment gradients or extreme values. Nevertheless, the relative L2 errors of each bending moment remain at an acceptablely low level (4-7%). This observation also provides insights for future model optimization: drawing on the idea of ​​adaptive mesh refinement in finite element methods, an adaptive sampling strategy can be introduced to dynamically increase the density of coordination sites in these key regions during training, aiming to further improve the model's prediction accuracy for the internal force field. Figures 5(a)-5(e) This provides the complete solution results for 41 cases using GA-ePINN.

[0048] Example 2 - Internal Boundary Parameterization In computational mechanics, dealing with varying internal boundaries (such as pores and inclusions) is more challenging than dealing with external boundaries. For any traditional mesh-based approach, any variation in the location, size, or topology of the internal boundaries inevitably requires complex mesh refactoring, making continuous parametric analysis exceptionally difficult and costly. This case study aims to demonstrate that the framework can elegantly address this challenge by using the geometric description of the internal boundaries as input to GA-ePINN.

[0049] 1. Changes in location like Figure 6 As shown, a radius is set. The circular internal fixed region is moved along a preset sine trajectory. Figure 7 The solution results show that GA-ePINN can predict the deflection and internal force field distribution of the entire plate in real time and continuously when the internal constraint is at any position on the path, clearly demonstrating how the physical field evolves smoothly with the change of the position of the internal constraint.

[0050] 2. Changes in area The circular fixed support area is fixed at the center of the plate, and its radius is... A continuous change from small to large. For example... Figure 8 As shown, the model accurately captures the key physical phenomenon: as the internal fixed support area expands, the overall effective stiffness of the plate increases significantly, resulting in a significant reduction in the maximum deflection.

[0051] To quantitatively evaluate model performance, this invention constructs a validation dataset containing 196 independent FEM simulation results, such as... Figure 9 As shown, these samples cover different hole locations and radii. Table 3 shows that the macroscopic statistical results further validate the high accuracy of the model. For the deflection field... The average R² value reached 0.99961, the relative L2 error was less than 1.4%, and the standard deviation was only 0.61%, indicating that the model prediction has high stability.

[0052] Table 3. Accuracy assessment results (196 samples)

[0053] Figure 10 A specific example is provided ( Error analysis was performed with a radius of 0.1 and a center at (0.75, 0.5), revealing the distribution characteristics of the error. Consistent with expectations, for the bending moment field calculated from the second derivative of displacement, the error is mainly concentrated in the stress concentration region at the edge of the hole. Especially... In the field, the high stress gradient region near the hole is the main source of error.

[0054] Based on the same inventive concept, embodiments of this application provide a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the aforementioned method for solving structural physics fields based on a parameterized physical information neural network.

[0055] Based on the same inventive concept, embodiments of this application provide a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the aforementioned method for solving structural physics fields based on a parameterized physical information neural network.

[0056] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0057] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0058] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0059] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0060] The above embodiments are merely illustrative of the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solutions based on the technical concept proposed in this invention shall fall within the scope of protection of this invention.

Claims

1. A method for solving structured physical fields based on parametric physical information neural networks, characterized in that, Includes the following steps: Step 1: Treat the stress-bearing member to be solved as a rectangular thin plate, establish a Cartesian coordinate system, and obtain the spatial coordinates and geometric parameter vectors of the stress-bearing member to be solved; Step 2: Construct a geometrically perceptive physical information neural network model with spatial coordinates and geometric parameter vectors as inputs and the stress and deformation of the stress-bearing component to be solved as outputs. The geometrically perceptive physical information neural network model includes a geometrically perceptive neural network module and a physical constraint module based on total potential energy. The geometrically perceptive neural network module is used to encode the geometric information of the stress-bearing component to be solved and incorporate it into the training process. It also converts the input into an unconstrained scalar and outputs it to the physical constraint module based on total potential energy. The physical constraint module based on total potential energy is used to convert the unconstrained scalar into a deflection field that satisfies physical laws and boundary conditions, and calculates the total potential energy of the stress-bearing component to be solved as the loss function for model training. Step 3: The geometric perception physical information neural network model is trained using a hybrid optimization strategy of Adam and LBFGS. When the total potential energy reaches the minimum value, the model converges and outputs the stress and deformation of the stressed component to be solved.

2. The method for solving structured physics fields based on parametric physical information neural networks according to claim 1, characterized in that, In step 1, the stress-bearing component to be solved is considered as a rectangular thin plate, and the length of the stress-bearing component to be solved is... Width is Thickness is Establish a Cartesian coordinate system. The plane coincides with the neutral plane of the thin plate. The axis is perpendicular to the neutral plane. , The geometric parameter vector includes the aspect ratio of the thin plate. When there is a circular hole in the thin plate, the geometric parameter vector also includes the center coordinates and radius of the circular hole.

3. The method for solving structured physics fields based on parametric physical information neural networks according to claim 2, characterized in that, In step 2, when the aspect ratio of the stressed component changes, the input to the model is a three-dimensional vector. When the aspect ratio of the load-bearing component to be solved remains constant and circular holes exist, the input to the model is a five-dimensional vector. When the aspect ratio of the stressed component to be solved changes and circular holes exist, the input of the model is a six-dimensional vector. , To determine the spatial coordinates of the stressed component, The coordinates of the center of the circular hole are: The radius of the circular hole.

4. The method for solving structured physical fields based on parametric physical information neural networks according to claim 3, characterized in that, In step 2, the geometric perception neural network module includes an input layer, a fully connected feedforward network, and an output layer. The input of the input layer is the input of the model. The fully connected feedforward network includes multiple hidden layers, each of which uses the hyperbolic tangent function as the activation function. The output layer outputs an unconstrained scalar.

5. The method for solving structured physical fields based on parametric physical information neural networks according to claim 4, characterized in that, In step 2, the total potential energy of the stressed component to be solved is calculated using the following formula: ， in, This represents the total potential energy functional of the stressed component to be solved. For strain energy, As external potential energy, The transverse deflection field of the neutral surface; ， ， in, For the bending stiffness of thin plates, , For Young's modulus, Poisson's ratio, Spatial coordinates Load at the location, and For deflection field The second-order partial derivative; When there are circular holes in a thin plate, the total potential energy of the stressed member to be solved is expressed as: ， ， in, Let represent the total potential energy functional of the stressed component containing a circular hole. Represents the original rectangular region Multiple connected domains after removing the circular hole region.

6. The method for solving structured physics fields based on parametric physical information neural networks according to claim 5, characterized in that, In step 2, when transforming the unconstrained scalar into a deflection field that satisfies physical laws and boundary conditions, the applied hard boundary constraints are as follows: Design an algebraic function related to geometry. If the function is always zero on its defined boundary, then the deflection field satisfies the boundary conditions. Constructed using the following formula: ， Since the plate has fixed supports at both its inner and outer boundaries, the boundary function... From the outer boundary function and inner boundary function Composed of: ， ， The fixed support condition that simultaneously satisfies zero displacement and zero rotation is: ， in, Represents an unconstrained scalar. For model parameters, This is a vector of geometric parameters.

7. A computer device comprising a memory, a processor, and a computer program stored in the memory and capable of running on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the structural physics field solution method based on a parameterized physical information neural network as described in any one of claims 1 to 6.

8. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of the structural physics field solution method based on a parameterized physical information neural network as described in any one of claims 1 to 6.

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