Adaptive mesh refinement method, system, and device for physical information neural networks

Abstract

The present application relates to the technical field of the intersection of computational solid mechanics and deep learning, and particularly relates to an adaptive mesh refinement method, system and device of a physical information neural network, comprising: adopting a coarse mesh physical information neural network model to discretely establish a mapping relationship between node coordinates and node displacements of a mechanical problem, constructing a physical information function and a total loss function of the solid mechanics problem, training the coarse mesh physical information neural network model to obtain a total loss function value and a change rate of the solid mechanics problem; if the coarse mesh physical information neural network model meets a solving condition or the training times are greater than a preset number, outputting a displacement solution of the solid mechanics problem, otherwise selecting a grid element with a larger strain energy density value in the coarse mesh physical information neural network model for refinement, obtaining a refined mesh model, migrating parameters of the coarse mesh physical information neural network model to the refined mesh model, and repeatedly executing the above steps until a high-precision solution of the solid mechanics problem is output.

Description

Technical Field

[0001] This invention relates to the field of computational solid mechanics and deep learning, and in particular to an adaptive mesh refinement method, system and device for physical information neural networks. Background Technology

[0002] Numerical solutions to solid mechanics problems are a crucial component of engineering design and analysis. For a long time, traditional numerical methods such as the finite element method and the finite difference method have been the mainstream techniques in this field. These methods discretize a continuous physical domain into a finite number of elements or mesh nodes and solve the governing equations as a system of algebraic equations. Although these methods have matured considerably over the years, they still face some insurmountable challenges when dealing with complex engineering problems:

[0003] First, traditional numerical methods are computationally expensive. For problems with complex geometries or those requiring consideration of multiphysics coupling effects, the traditional mesh generation process is often very time-consuming, and solving large-scale algebraic equations requires a significant amount of computational resources.

[0004] Secondly, the quality of the mesh directly affects the accuracy of the calculation results. In order to obtain an accurate solution, a high-density mesh is often required, which will further increase the computational cost.

[0005] Furthermore, traditional numerical methods face convergence difficulties when solving nonlinear problems. Solid mechanics problems commonly exhibit complex phenomena such as material nonlinearity (e.g., elastoplastic deformation), geometric nonlinearity (e.g., large deformation), and contact nonlinearity. These nonlinear factors can significantly reduce the convergence speed of the iterative solution process, or even lead to non-convergence, severely impacting computational efficiency and reliability.

[0006] With the rapid development of artificial intelligence technology, physical information deep neural networks, as a disruptive numerical solution method, have attracted widespread attention in the field of solid mechanics. Physical information deep neural networks cleverly integrate the physical governing equations of solid mechanics (such as equilibrium equations, constitutive relations, and the law of conservation of energy) into the loss function of the neural network, thereby constructing a new paradigm of deep learning solution driven by both data and physics, providing new ideas for solving complex solid mechanics problems.

[0007] However, the training efficiency of physically-informed deep neural networks (PINs) remains a significant bottleneck when applied to practical solid mechanics problems. This is particularly true when dealing with structures exhibiting stress concentration (e.g., crack tips) or solutions involving high-order continuous displacement fields, where the convergence speed of PIN training decreases considerably. To alleviate this issue, existing techniques attempt to employ transfer learning strategies, using models pre-trained on similar problems to initialize the network parameters of PINs. However, in solid mechanics, due to the geometric complexity of engineering structures and the diversity of boundary conditions, the solution spaces differ significantly between different problems, often resulting in less than ideal transfer learning performance. Therefore, developing methods to effectively improve the training efficiency of PINs in solving complex solid mechanics problems remains a pressing challenge. Summary of the Invention

[0008] This invention provides an adaptive mesh refinement method, system, and device for physical information neural networks to address the problem of improving the training efficiency and solution accuracy of physical information deep neural networks in solving complex solid mechanics problems.

[0009] The first aspect of this invention provides an adaptive mesh refinement method for a physical information neural network, comprising the following steps: determining whether there is a computational requirement for a solid mechanics problem involving a planar plate under tension; if there is a computational requirement for the solid mechanics problem, discretizing the mechanics problem using a coarse-grid physical information neural network model, establishing a mapping relationship between node coordinates and node displacements, constructing a physical information function and a total loss function for the solid mechanics control equations based on the mapping relationship between node coordinates and node displacements, obtaining the total loss function value and its rate of change for the solid mechanics problem by training the coarse-grid physical information neural network model; and determining whether the coarse-grid physical information neural network model meets preset solution conditions based on the total loss function value and its rate of change. The system checks whether the current number of training iterations of the coarse-grid physical information neural network model is greater than a preset number. If the coarse-grid physical information neural network model satisfies the preset solution conditions, or if the current number of training iterations is greater than the preset number, then the displacement solution of the solid mechanics problem is output. Otherwise, the strain energy density value of each grid cell in the coarse-grid physical information neural network model is calculated, and multiple target grid cells with larger strain energy density values ​​are selected for mesh refinement to obtain a refined grid model. The parameters of the coarse-grid physical information neural network model are then transferred to the deep neural network of the refined grid model to obtain the target neural network model. The above training-judgment-refinement-transfer loop is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem.

[0010] Optionally, the step of selecting multiple target mesh elements with larger strain energy density values ​​for mesh refinement, when the target mesh element is a triangular element, includes: if the target mesh element shares a boundary with three adjacent elements, then a first set of new nodes is added to the midpoint of each edge of the target mesh element, and the target mesh element is subdivided into multiple sub-elements; the first set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent elements, and each adjacent element is subdivided into multiple sub-elements; if the target mesh element shares a boundary with two adjacent elements, then... A second set of new nodes is added to the midpoint of each edge of the target mesh cell to be refined, and the target mesh cell to be refined is subdivided into multiple sub-cells. The second set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent cells, and each adjacent cell is subdivided into multiple sub-cells. If the target mesh cell to be refined shares a boundary with an adjacent cell, a third set of new nodes is added to the midpoint of each edge of the target mesh cell to be refined, and the target mesh cell to be refined is subdivided into multiple sub-cells. The third set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent cells, and the adjacent cells are subdivided into multiple sub-cells.

[0011] Optionally, determining whether the coarse-grid physical information neural network model satisfies the preset solution conditions based on the total loss function value and its rate of change includes: determining whether the total loss function value and its rate of change are less than a preset value; if the total loss function value and its rate of change are less than the preset value, then the neural network model based on the coarse-grid model is determined to satisfy the preset solution conditions; otherwise, the neural network model based on the coarse-grid model is determined not to satisfy the preset solution conditions.

[0012] Optionally, when selecting multiple target mesh elements with larger strain energy density values ​​for mesh refinement, the process includes: obtaining the current area of ​​the target mesh element and the mesh element with the largest area value among its adjacent elements; if the ratio of the current area to the largest area value is less than a preset ratio, then the mesh element with the largest area value is refined; otherwise, the target mesh element is refined.

[0013] Optionally, when refining the target mesh unit to be refined, the process includes: detecting the minimum area value of the target mesh unit to be refined; if the minimum area value of the target mesh unit to be refined is less than a preset area threshold, then refining the target mesh unit to be refined is stopped.

[0014] Optionally, after initializing the deep neural network of the refined mesh model using the parameters of the deep neural network model trained with the coarse mesh model to obtain the target neural network model, the process includes: determining whether the target neural network model satisfies the preset solution conditions, or determining whether the current training count of the target neural network model is greater than a preset count; if the target neural network model satisfies the preset solution conditions, or the current training count of the target neural network model is greater than the preset count, then outputting the displacement solution of the solid mechanics problem; otherwise, continuing to refine the mesh of the target neural network model until the target neural network model after mesh refinement satisfies the preset solution conditions or the current training count of the target neural network model after mesh refinement is greater than the preset count.

[0015] A second aspect of the present invention provides an adaptive mesh refinement system for a physical information neural network, comprising: a first judgment module, configured to determine whether there is a computational requirement for a solid mechanics problem involving a tensile process of a planar plate; a calculation module, configured to, if there is a computational requirement for the solid mechanics problem, discretize the mechanics problem using a coarse-grid physical information neural network model, establish a mapping relationship between node coordinates and node displacements, construct a physical information function and a total loss function for the solid mechanics control equations based on the mapping relationship between node coordinates and node displacements, and obtain the total loss function value and its rate of change for the solid mechanics problem by training the coarse-grid physical information neural network model; and a second judgment module, configured to determine whether the coarse-grid physical information neural network model satisfies a preset solution based on the total loss function value and its rate of change. The system determines whether the current number of training iterations of the coarse-grid physical information neural network model is greater than a preset number. The mesh refinement module is used to output the displacement solution of the solid mechanics problem if the coarse-grid physical information neural network model satisfies the preset solution conditions or the current number of training iterations is greater than the preset number. Otherwise, it calculates the strain energy density value of each grid cell in the coarse-grid physical information neural network model, selects multiple target grid cells with larger strain energy density values ​​for mesh refinement, obtains a refined mesh model, and uses the parameters of the coarse-grid physical information neural network model to transfer to the deep neural network of the refined mesh model to obtain the target neural network model. This training-determination-refinement-transfer loop is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem.

[0016] Optionally, when the target mesh cell to be refined is a triangular cell, the mesh refinement module is further configured to: if the target mesh cell to be refined shares a boundary with three adjacent cells, add a first set of new nodes to the midpoint of each edge of the target mesh cell to be refined, subdivide the target mesh cell to be refined into multiple sub-cells, connect the first set of new nodes to the opposite vertices of the corresponding edges in the adjacent cells, and subdivide each adjacent cell into multiple sub-cells; if the target mesh cell to be refined shares a boundary with two adjacent cells, then in the target mesh cell to be refined... A second set of new nodes is added to the midpoint of each edge of the target mesh unit, and the target mesh unit to be refined is subdivided into multiple sub-units. The second set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent units, and each adjacent unit is subdivided into multiple sub-units. If the target mesh unit to be refined shares a boundary with an adjacent unit, a third set of new nodes is added to the midpoint of each edge of the target mesh unit to be refined, and the target mesh unit to be refined is subdivided into multiple sub-units. The third set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent units, and the adjacent units are subdivided into multiple sub-units.

[0017] Optionally, the second judgment module is used to: determine whether the total loss function value and its rate of change are less than a preset value; if the total loss function value and its rate of change are less than the preset value, then determine that the neural network model based on the coarse grid model satisfies the preset solution condition; otherwise, determine that the neural network model based on the coarse grid model does not satisfy the preset solution condition.

[0018] Optionally, when selecting multiple target mesh elements with larger strain energy density values ​​for mesh refinement, the mesh refinement module is further configured to: obtain the current area of ​​the target mesh element to be refined and the mesh element with the largest area value among the adjacent elements; if the ratio of the current area to the largest area value is less than a preset ratio, then refine the mesh element with the largest area value; otherwise, refine the target mesh element to be refined.

[0019] Optionally, when refining the target mesh unit to be refined, the mesh refinement module is further configured to: detect the minimum area value of the target mesh unit to be refined; if the minimum area value of the target mesh unit to be refined is less than a preset area threshold, then stop refining the target mesh unit to be refined.

[0020] Optionally, after initializing the deep neural network of the refined mesh model using the parameters of the deep neural network model trained with the coarse mesh model to obtain the target neural network model, the mesh refinement module is further configured to: determine whether the target neural network model satisfies the preset solution conditions, or determine whether the current training count of the target neural network model is greater than the preset count; if the target neural network model satisfies the preset solution conditions, or the current training count of the target neural network model is greater than the preset count, then output the displacement solution of the solid mechanics problem; otherwise, continue to refine the mesh of the target neural network model until the target neural network model after mesh refinement satisfies the preset solution conditions or the current training count of the target neural network model after mesh refinement is greater than the preset count.

[0021] Optionally, a third aspect of the present invention provides an electronic device, including: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the adaptive mesh refinement method of the physical information neural network as described in the above embodiments.

[0022] A fourth aspect of the present invention provides a computer program product having a computer program stored thereon, which is executed by a processor to implement the adaptive mesh refinement method for a physical information neural network as described in the above embodiments.

[0023] In the above embodiments, if there is a computational requirement for a solid mechanics problem, a coarse-grid physical information neural network model is used to discretize the mechanical problem, establish a mapping relationship between node coordinates and node displacements, and construct the physical information function and total loss function of the solid mechanics governing equations based on the mapping relationship between node coordinates and node displacements. By training the coarse-grid physical information neural network model, the total loss function value and its rate of change of the solid mechanics problem are obtained. Based on the total loss function value and its rate of change, it is determined whether the coarse-grid physical information neural network model meets the preset solution conditions, or whether the current training number of the coarse-grid physical information neural network model is greater than the preset number. If the coarse-grid physical information neural network model meets the preset solution conditions, or the current training number is greater than the preset number, the displacement solution of the solid mechanics problem is output. Otherwise, the strain energy density value of each grid element of the coarse-grid physical information neural network model is calculated, and multiple target grid elements with larger strain energy density values ​​are selected for mesh refinement to obtain a refined grid model. The parameters of the coarse-grid physical information neural network model are then transferred to the deep neural network of the refined grid model to obtain the target neural network model. The above training-judgment-refinement-transfer loop is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem. This solves the problem of how to improve the training efficiency and solution accuracy of deep neural networks for physical information in solving complex solid mechanics problems, significantly improving overall training efficiency, reducing computational costs, and ensuring the stability and reliability of computational results.

[0024] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0025] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of the embodiments taken in conjunction with the accompanying drawings, wherein:

[0026] Figure 1 A flowchart of an adaptive mesh refinement method for a physical information neural network according to an embodiment of the present invention;

[0027] Figure 2 A schematic diagram illustrating the discrete calculation of a solid mechanics problem in element form according to an embodiment of the present invention;

[0028] Figure 3 This is a flowchart illustrating the calculation of physical information using a deep neural network according to an embodiment of the present invention.

[0029] Figure 4 This is a schematic diagram of an adaptive mesh update method according to an embodiment of the present invention;

[0030] Figure 5 This is a schematic diagram illustrating adaptive mesh updating for a specific solid mechanics problem according to an embodiment of the present invention.

[0031] Figure 6 This is an optimized schematic diagram illustrating the selection of refined target units according to an embodiment of the present invention;

[0032] Figure 7 This is a schematic diagram of adaptive mesh refinement of a physical information neural network according to an embodiment of the present invention;

[0033] Figure 8 This is a schematic diagram of an adaptive mesh refinement system for a physical information neural network according to an embodiment of the present invention;

[0034] Figure 9 This is a schematic diagram of the structure of an electronic device according to an embodiment of the present invention. Detailed Implementation

[0035] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0036] The adaptive mesh refinement method, system, and device of the physical information neural network according to embodiments of the present invention are described below with reference to the accompanying drawings. Addressing the issue mentioned in the background art regarding how to improve the training efficiency and solution accuracy of physical information deep neural networks in solving complex solid mechanics problems, the present invention provides an adaptive mesh refinement method for physical information neural networks. In this method, if there is a computational requirement for a solid mechanics problem, a coarse-mesh physical information neural network model is used to discretize the mechanics problem, establishing a mapping relationship between node coordinates and node displacements. Based on this mapping relationship, a physical information function and a total loss function for the solid mechanics governing equations are constructed. By training the coarse-mesh physical information neural network model, the total loss function value and its rate of change for the solid mechanics problem are obtained. Based on the total loss function value and its rate of change, it is determined whether the coarse-mesh physical information neural network model satisfies... The algorithm presets solution conditions or checks if the current training iterations of the coarse-mesh physical information neural network model exceed a preset number. If the coarse-mesh physical information neural network model meets the preset solution conditions or the current training iterations exceed the preset number, it outputs the displacement solution of the solid mechanics problem. Otherwise, it calculates the strain energy density value of each grid cell in the coarse-mesh physical information neural network model and selects several target grid cells with larger strain energy density values ​​for mesh refinement, obtaining a refined grid model. The parameters of the coarse-mesh physical information neural network model are then transferred to the deep neural network of the refined grid model to obtain the target neural network model. This training-judgment-refinement-transfer loop is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem. This solves the problem of how to improve the training efficiency and solution accuracy of physical information deep neural networks in solving complex solid mechanics problems, significantly improving overall training efficiency, reducing computational costs, and ensuring the stability and reliability of the calculation results.

[0037] Specifically, Figure 1 This is a flowchart illustrating an adaptive mesh refinement method for a physical information neural network provided in an embodiment of the present invention.

[0038] like Figure 1 As shown, the adaptive mesh refinement method of this physical information neural network includes the following steps:

[0039] In step S101, it is determined whether there is a solid mechanics calculation requirement for the tensile process of the planar plate.

[0040] In step S102, if there is a computational requirement for a solid mechanics problem, a coarse-grid physical information neural network model is used to discretize the mechanical problem, establish a mapping relationship between node coordinates and node displacements, construct the physical information function and total loss function of the solid mechanics control equations based on the mapping relationship between node coordinates and node displacements, and obtain the total loss function value and its rate of change of the solid mechanics problem by training the coarse-grid physical information neural network model.

[0041] First, if there is a need for solid mechanics calculations of a planar plate under tension, then establish a solid mechanics problem of a planar plate under tension, and discretize the model into element models, such as... Figure 2 As shown, the node coordinates of the element are thus obtained. ).

[0042] Furthermore, such as Figure 3 As shown, a deep neural network is constructed to obtain a coarse-grid physical information neural network model, and the coordinates of the unit nodes are established. ) and nodal displacements ( The mapping relationship between the physical information and the physical information neural network model is established, and the physical information function of the solid mechanics problem is constructed based on the node information of the coarse grid physical information neural network model. The physical information function includes boundary conditions, initial conditions and physical equations (governing equations or energy conservation equations).

[0043] Further, the mean squared error is used to construct residual terms related to the boundary conditions, initial conditions, and physical equations. , and The expression for the total loss function of the solid mechanics problem is obtained as follows:

[0044] + + (1)

[0045] in, , and These are the weighting coefficients.

[0046] It should be noted that, in constructing the coarse-grid physical information neural network model, the method of this invention is characterized by initially using a grid with a large unit size to establish the model. This results in fewer unit nodes, and thus, the node coordinates ( ) and nodal displacements ( The mapping relationship between them is relatively simple and can be solved quickly using a physical information deep neural network.

[0047] The total loss function value and its rate of change of the solid mechanics problem are obtained from the total loss function. Then, based on the total loss function value and its rate of change, it is determined whether the coarse-grid physical information neural network model meets the preset solution conditions, or whether the current training number of the coarse-grid physical information neural network model is greater than or equal to the preset number.

[0048] In step S103, the coarse-grid physical information neural network model is judged to meet the preset solution conditions based on the total loss function value and its rate of change, or it is judged to determine whether the current training number of the coarse-grid physical information neural network model is greater than the preset number.

[0049] The preset number of times is the maximum number of training iterations for the deep neural network that is set in advance.

[0050] Optionally, in some embodiments, determining whether the coarse-grid physical information neural network model meets the preset solution conditions based on the total loss function value and its rate of change includes: determining whether the total loss function value and its rate of change are less than a preset value; if the total loss function value and its rate of change are less than the preset value, then the neural network model based on the coarse-grid model is determined to meet the preset solution conditions; otherwise, the neural network model based on the coarse-grid model is determined not to meet the preset solution conditions.

[0051] The preset value can be a threshold set by the user, a threshold obtained through a finite number of experiments, or a threshold obtained through a finite number of computer simulations. No specific limitation is made here. The preset solution condition can be understood as the coarse-grid physical information neural network model satisfying a predefined solution accuracy. The predefined solution accuracy can be a threshold set by the user, a threshold obtained through a finite number of experiments, or a threshold obtained through a finite number of computer simulations. No specific limitation is made here.

[0052] If the total loss function value and its rate of change of the solid mechanics problem are less than the preset value, then the neural network model based on the coarse mesh model is determined to meet the preset solution conditions. Otherwise, the neural network model based on the coarse mesh model is determined not to meet the preset solution conditions. Alternatively, if the current training number of the coarse mesh physical information neural network model is less than or equal to the preset number, the neural network model based on the coarse mesh model is also determined not to meet the condition of outputting the solution of the solid mechanics problem. In this case, it is necessary to refine the mesh of the coarse mesh physical information neural network model. If the neural network model based on the coarse mesh model can output the solution of the solid mechanics problem, then the displacement solution of the solid mechanics problem is directly output.

[0053] In step S104, if the coarse-grid physical information neural network model meets the preset solution conditions, or the current number of training iterations is greater than the preset number of iterations, the displacement solution of the solid mechanics problem is output. Otherwise, the strain energy density value of each grid cell in the coarse-grid physical information neural network model is calculated, and multiple target grid cells with larger strain energy density values ​​are selected for mesh refinement to obtain a refined grid model. The parameters of the coarse-grid physical information neural network model are then transferred to the deep neural network of the refined grid model to obtain the target neural network model. The above training-judgment-refinement-transfer loop is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem.

[0054] This process involves calculating the strain energy density of all grid cells in the coarse-grid physical information neural network model, sorting them by strain energy density from largest to smallest, and selecting a predetermined number of cells with higher strain energy densities for mesh refinement. Generally, the number of selected target grid cells to be refined does not exceed 3%-10% of the total number of cells in the mesh model.

[0055] Understandably, if the coarse-grid physical information neural network model meets the preset solution conditions, or if the current training iterations of the coarse-grid physical information neural network model are greater than or equal to the preset iterations, then the displacement solution of the solid mechanics problem is output. Otherwise, under specific conditions, the strain energy density of each grid cell is calculated based on the grid cell information of the coarse-grid physical information neural network model, and the cells are sorted from largest to smallest according to their strain energy density values. A preset number of cells with larger strain energy densities are selected for adaptive mesh refinement to improve the solution accuracy.

[0056] The characteristic of the total loss function value and its rate of change satisfying specific conditions is that: the total loss function value and its rate of change do not satisfy the preset solution conditions, but the rate of change of the total loss function is very small, and the value of the total loss function is still very large. This indicates that the deep neural network calculation using this grid model has basically converged, but the deep neural network cannot converge to a solution that satisfies the preset solution conditions based on this grid model.

[0057] Therefore, this invention reduces the initial computational load and accelerates the convergence speed by initially using a coarse mesh to train the model. As the computation progresses, the mesh is gradually refined to ultimately obtain a high-precision solution. This avoids the problems of high computational load and slow convergence speed caused by using a fine mesh for training from the beginning in traditional methods.

[0058] This invention calculates the strain energy density of each element in the mesh model and sorts them from largest to smallest. The reason for selecting a predetermined number of elements with high strain energy densities for mesh refinement is primarily because strain energy density reflects the degree of deformation of an element. Generally, strain energy density is the strain energy per unit volume, directly reflecting the degree of deformation of the element. A higher strain energy density indicates more severe deformation and a higher stress level, resulting in a greater impact on the mechanical properties of the overall structure. Furthermore, in solid mechanics problems, high strain energy density regions are typically critical areas such as stress concentration areas, crack tip regions, and contact regions. The calculation accuracy of these regions is crucial for evaluating the mechanical properties of the overall structure.

[0059] Optionally, in some embodiments, multiple target mesh elements with relatively large strain energy density values ​​are selected for mesh refinement. When the target mesh element is a triangular element, the refinement process includes: if the target mesh element shares a boundary with three adjacent elements, a first set of new nodes is added to the midpoint of each edge of the target mesh element, and the target mesh element is subdivided into multiple sub-elements. The first set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent elements, and each adjacent element is subdivided into multiple sub-elements; if the target mesh element shares an edge with two adjacent elements... If a target mesh cell shares a boundary with an adjacent cell, a second set of new nodes is added to the midpoint of each edge of the target mesh cell to be refined, and the target mesh cell to be refined is subdivided into multiple sub-cells. The second set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent cells, and each adjacent cell is subdivided into multiple sub-cells.

[0060] Detailed process such as Figure 4 As shown, the number of elements that need to be refined can be selected according to the actual situation. Generally, the number of selected target mesh elements to be refined should not exceed 3%-10% of the total number of elements in the mesh model.

[0061] Mesh refinement: Depending on the type of the target triangular mesh cells to be refined, the following three cases are handled:

[0062] Internal cell: This cell shares an edge with its three surrounding adjacent cells. For example... Figure 4As shown in Figure (a), the adaptive refinement strategy is carried out in two steps: First, a new node is added to the midpoint of each edge of the internal cell, and the internal cell is subdivided into four small triangles; second, these new nodes are connected to the opposite vertices of the corresponding edges in the adjacent cells, and each adjacent cell is subdivided into two triangles. In this way, the original mesh cell region (including the internal cell and three adjacent cells) is subdivided into a total of 10 triangular cells;

[0063] Single-boundary element: This element shares an edge with its two adjacent elements. For example... Figure 4 As shown in Figure (b), the adaptive refinement strategy is carried out in two steps: First, a new node is added to the midpoint of each edge of the single boundary cell, and the single boundary cell is subdivided into four small triangles; second, these new nodes are connected to the opposite vertices of the corresponding edges in adjacent cells, and each adjacent cell is subdivided into two triangles. In this way, the original mesh cell region (including the single boundary cell and the two adjacent cells) is subdivided into a total of 8 triangular cells;

[0064] Double-boundary element: This element shares an edge with one of its adjacent elements. For example... Figure 4 As shown in Figure (c), the adaptive refinement strategy is carried out in two steps: First, a new node is added to the midpoint of each edge of the double-boundary cell, and the double-boundary cell is subdivided into four small triangles; second, these new nodes are connected to the opposite vertices of the corresponding edges in the adjacent cells, and the adjacent cells are subdivided into two triangles. In this way, the original mesh cell region (including the double-boundary cell and one adjacent cell) is subdivided into a total of 6 triangular cells;

[0065] Furthermore, it should be noted that since adaptive mesh refinement involves refining multiple mesh elements, in addition to refining the target triangular mesh element, its neighboring elements may also be segmented due to refinement. These refined neighboring elements may already contain the target triangular mesh element that originally needed refinement. Therefore, if the target triangular mesh element that needs refinement has already been refined, it does not need to be refined again.

[0066] Optionally, in an embodiment, when selecting multiple target mesh elements with larger strain energy density values ​​for mesh refinement, the process includes: obtaining the current area of ​​the target mesh element and the mesh element with the largest area value among its adjacent elements; if the ratio of the current area to the maximum area value is less than a preset ratio, then the mesh element with the largest area value is refined; otherwise, the target mesh element is refined.

[0067] To ensure the quality of the adaptive mesh and avoid excessive differences in the size of adjacent mesh cells, the present invention also takes the following measures when refining the mesh: First, determine the target mesh cell to be refined and find its adjacent cells.

[0068] If the area of ​​the target mesh cell to be refined is less than 2 / 3 of the area of ​​the largest adjacent cell, then the adjacent cell with the largest area is selected as the target mesh cell for refinement. This avoids abrupt changes in mesh cell size, thereby improving computational accuracy and stability.

[0069] Optionally, in some embodiments, when refining the target mesh cell to be refined, the process includes: detecting the minimum area value of the target mesh cell to be refined; if the minimum area value of the target mesh cell to be refined is less than a preset area threshold, then refining the target mesh cell to be refined is stopped.

[0070] It should be understood that, in order to prevent over-refinement, this invention also sets a preset area threshold, i.e., a minimum area threshold, for the target mesh element to be refined, based on the specific solid mechanics problem. When the minimum area value of the target mesh element to be refined is less than or equal to the preset area threshold, that is, when the minimum area value of the target mesh element to be refined is less than or equal to the pre-set minimum area threshold, the target mesh element to be refined is no longer refined, and it is considered that the current mesh element size already meets the requirements of the problem solution accuracy.

[0071] Optionally, in some embodiments, after transferring the parameters of the deep neural network model trained using the coarse mesh model to the deep neural network of the refined mesh model to obtain the target neural network model, the process includes: determining whether the target neural network model meets the preset solution conditions, or determining whether the current training count of the target neural network model is greater than the preset count; if the target neural network model meets the preset solution conditions, or the current training count of the target neural network model is greater than the preset count, then the displacement solution of the solid mechanics problem is output; otherwise, the mesh of the target neural network model is further refined until the target neural network model after mesh refinement meets the preset solution conditions or the current training count of the target neural network model after mesh refinement is greater than the preset count.

[0072] Understandably, the parameters of the deep neural network model in the coarse-grid physical information neural network model are transferred to the deep neural network of the refined grid model to obtain the target neural network model. If the target neural network model meets the preset solution conditions, or the current training count of the target neural network model is greater than the preset count, then the displacement solution of the solid mechanics problem is output. Otherwise, the grid of the target neural network model is refined until the target neural network model after grid refinement meets the preset solution conditions or the current training count of the target neural network model after grid refinement is greater than the preset count. At this time, the high-precision solution of the solid mechanics problem is output.

[0073] Furthermore, this invention employs transfer learning technology, using the neural network model parameters trained in the coarse grid as the initial parameters for the neural network in the refined grid model. In the refined model, parameters can be initialized only for the regions where newly added grid nodes are located, thereby improving the training efficiency of the deep neural network.

[0074] It is important to emphasize that the method of this invention does not perform mesh adaptive training from the outset. Instead, it enters the mesh adaptive training phase only after the initial mesh model training has converged to a certain extent, in order to further improve the solution accuracy. Specifically, this invention adaptively determines when to update the mesh and when to reach the final convergence accuracy by monitoring the value and rate of change of the total loss function. This is based on the fact that the larger the cell size in the model, the larger the computational error usually is. Therefore, the total loss function value is relatively large in the initial training phase. However, when the model has reached its solution capability limit at the current mesh density, although the total loss function value may still be relatively large, its decreasing trend will tend to level off.

[0075] To enable those skilled in the art to further understand the adaptive mesh refinement method of the physical information neural network in the embodiments of this application, the following detailed description is provided in conjunction with specific embodiments.

[0076] Based on the tensile problem of a planar plate in solid mechanics, the overall implementation of this method is described as follows:

[0077] 1) For the tension problem of a planar plate, the problem is discretized using a triangular mesh, such as... Figure 2 As shown, the coordinates of all triangular element nodes in the tension problem of a planar plate are obtained. ).

[0078] 2) such as Figure 3 As shown, a physical information neural network model is established, and the coordinates of the unit nodes are constructed ( ) and nodal displacements ( The mapping relationship between ).

[0079] 3) The grid cell information obtained by discretization based on the physical information neural network model, and the cell node displacements predicted by the physical information neural network model ( The physical information function of the solid mechanics problem under the physical information neural network model is calculated, including boundary conditions, initial conditions, and physical functions (governing equations or energy conservation equations).

[0080] 4) Based on known boundary conditions, initial conditions, and physical constraints, the mean square error is used to construct the residual between the predicted value and the known value of the physical information function, thus obtaining the residual term. , and By weighted summation, the total loss function for this solid mechanics problem is obtained.

[0081] 5) Determine whether the physical information neural network model has reached the required solution accuracy or whether the current training iterations have reached the preset number of iterations based on the rate of change of the total loss function. If the physical information neural network model has reached the required solution accuracy or the current training iterations have reached the preset number of iterations, output the displacement solution for the solid mechanics problem; otherwise, proceed to the next training iteration.

[0082] 6) Before proceeding to the next training round, this invention also needs to determine whether the coarse-grid physical information neural network model needs to update its grid cell information based on the rate of change and magnitude of the total loss function. Specifically, it uses a strain energy-driven cell adaptation method to update the model's grid cell information. If updating the grid cell information is not required, an optimization algorithm (such as gradient descent) is used to optimize the total loss function and calculate the weights W and biases b used to update the physical information neural network model. If updating the grid cell information of the physical information neural network model is required, the following steps are necessary.

[0083] 7) Based on the unit node displacement information predicted by the physical information neural network model, calculate the strain energy density of each grid unit, and sort them in descending order of strain energy density. Select a preset number of grid units with the highest strain energy density for adaptive grid refinement.

[0084] Depending on the edge sharing between the target mesh element (i.e., high strain energy density element) and its adjacent elements, adaptive refinement is performed in the following three cases:

[0085] Internal cell: This cell shares an edge with its three surrounding adjacent cells. For example... Figure 4 As shown in Figure (a), the adaptive refinement strategy is performed in two steps: First, a new node is added to the midpoint of each edge of the internal cell, and the internal cell is subdivided into four small triangles; second, these new nodes are connected to the opposite vertices of the corresponding edges in adjacent cells, and each adjacent cell is subdivided into two triangles. In this way, the original mesh cell region (including the internal cell and three adjacent cells) is subdivided into a total of 10 triangular cells, as shown in Figure (a). Figure 5 As shown in Figure (b).

[0086] Single-boundary element: This element shares an edge with its two adjacent elements. For example... Figure 4As shown in Figure (b), the adaptive refinement strategy is performed in two steps: First, a new node is added to the midpoint of each edge of the single boundary cell, and the single boundary cell is subdivided into four small triangles; second, these new nodes are connected to the opposite vertices of the corresponding edges in adjacent cells, and each adjacent cell is subdivided into two triangles. In this way, the original mesh cell region (including the single boundary cell and two adjacent cells) is subdivided into a total of 8 triangular cells, as shown in Figure (b). Figure 5 As shown in Figure (c).

[0087] Double-boundary element: This element shares an edge with one of its adjacent elements. For example... Figure 4 As shown in Figure (c), the adaptive refinement strategy is performed in two steps: First, a new node is added to the midpoint of each edge of the double-boundary cell, and the double-boundary cell is subdivided into four small triangles; second, these new nodes are connected to the opposite vertices of the corresponding edges in adjacent cells, and the adjacent cells are subdivided into two triangles. In this way, the original mesh cell region (including the double-boundary cell and one adjacent cell) is subdivided into a total of 6 triangular cells, as shown in Figure (c). Figure 5 As shown in Figure (d).

[0088] 8) Model Update and Initialization: The newly generated mesh cells and node information are precisely replaced in the original model to update the cell information in the corresponding regions, resulting in a new computational model. Then, a deep neural network is used to construct the mapping relationship between node coordinates and node displacements in the new model, and the neural network parameters obtained from training the original model are used to assist the training process of the new model through a transfer learning strategy.

[0089] 9) By repeatedly performing steps 7 and 8 above, the iterative cycle of adaptive unit refinement, model update and transfer learning is realized, gradually improving the grid density and solution accuracy of the physical information neural network model, and finally enabling the physical information neural network model to converge to a solution that meets the solution accuracy requirements.

[0090] Figure 6 This paper demonstrates an optimization strategy for selecting the target mesh cell to be refined during the adaptive mesh refinement process. Figure 6 In the diagram, the area of ​​the target mesh cell to be refined is... The areas of their adjacent units are respectively , ,and To avoid excessive differences in adjacent cell sizes leading to a decrease in mesh quality, this invention employs the following criterion: if... < This indicates that the size of the target mesh cell to be refined is too small. The present invention employs... Figure 6 The refinement scheme I shown applies to the adjacent cells with the largest area. Instead of directly refining the target mesh elements, refinement is performed (Refinement Scheme II). As shown in Scheme II, directly refining the target mesh may result in triangular elements with excessively large aspect ratios, thus reducing computational accuracy. In contrast, Scheme I refines the adjacent elements with the largest areas. This also refines the target unit mesh to a certain extent, while effectively avoiding the generation of mesh units with excessively large aspect ratios, thereby ensuring mesh quality and improving computational accuracy.

[0091] based on Figure 7 This paper describes the specific implementation process of this invention for calculating solid mechanics problems. Traditional physical information deep neural networks rely on dense grid discretization (e.g., ...) when solving solid mechanics problems. Figure 7 As shown in (a) and (b), this leads to limited computational efficiency. The fundamental reason is that traditional methods directly establish a global mapping relationship between node coordinates and displacements by constructing a total loss function with global physical information constraints. As the number of mesh cells and nodes increases, the complexity of the mapping relationship increases exponentially, significantly increasing the computational burden of neural network training.

[0092] In contrast, the adaptive mesh refinement method proposed in this invention (such as...) Figure 7 of (a) Figure 7 (c) Figure 7 of (d), Figure 7 (e) shows that efficiency breakthroughs are achieved through the following innovative processes:

[0093] 1. Coarse mesh initialization ( Figure 7 (c): In the first stage, a coarse grid is used to discretize the computational domain. At this time, the number of nodes is small, the coordinate-displacement mapping relationship is simple, and the neural network can quickly converge to the approximate solution under the current grid model and obtain the neural network parameters (weights W and biases b).

[0094] 2. Strain energy density-guided mesh refinement ( Figure 7 (d): Based on the displacement field of the coarse-grid solution, the strain energy density of each grid element is calculated. The grid elements are sorted in descending order of strain energy density values, and the top 50 high-strain-energy elements are selected. Figure 5 and Figure 6 The mesh refinement strategy shown performs local mesh densification, generating... Figure 7 The refined mesh model in (e).

[0095] 3. Parameter inheritance iterative optimization: The parameters (weights W and bias b) of the deep neural network in the coarse mesh model are transferred to the mesh model after mesh refinement as initial values. At this time, only the network parameters of the newly added node regions need to be randomly initialized. The network parameters are quickly adapted through the transfer learning mechanism, so that the physical information deep neural network based on the refined mesh model can converge quickly.

[0096] 4. Accuracy convergence determination: Repeat steps 2-3 until the refined mesh model after migration parameters meets the preset solution conditions or reaches the preset number of times.

[0097] Compared with the prior art, the present invention has the following significant advantages and advantages:

[0098] 1. High training efficiency: By adopting an adaptive refinement strategy based on strain energy density, the mesh density can be automatically increased in stress concentration areas to improve local solution accuracy. Combined with the transfer learning strategy, the training convergence of the new physical information neural network model is accelerated, thereby significantly improving the overall training efficiency and reducing the computational cost.

[0099] 2. Good solution accuracy: By dynamically adjusting the distribution of grid cells, it is possible to capture complex physical phenomena in solid mechanics problems more accurately, effectively improve the solution accuracy, and obtain more reliable numerical results.

[0100] 3. Controllable Mesh Quality: By limiting the size difference between adjacent mesh cells, abrupt changes in mesh cell size are avoided, ensuring a smooth mesh transition. Simultaneously, by setting a minimum area threshold for mesh cells, over-refinement is prevented, avoiding wasted computational resources. This effectively controls mesh quality, avoiding cells with excessively high aspect ratios, thereby ensuring the stability and reliability of the computational results.

[0101] According to the adaptive mesh refinement method of the physical information neural network proposed in this embodiment of the invention, if there is a computational requirement for a solid mechanics problem, a coarse-mesh physical information neural network model is used to discretize the mechanics problem, establish a mapping relationship between node coordinates and node displacements, construct the physical information function and total loss function of the solid mechanics governing equations based on the mapping relationship between node coordinates and node displacements, obtain the total loss function value and its rate of change of the solid mechanics problem by training the coarse-mesh physical information neural network model, and determine whether the coarse-mesh physical information neural network model meets the preset solution conditions based on the total loss function value and its rate of change, or determine the current state of the coarse-mesh physical information neural network model. The system checks whether the number of training iterations exceeds a preset limit. If the coarse-grid physical information neural network model meets the preset solution conditions, or if the current number of training iterations exceeds the preset limit, then it outputs the displacement solution of the solid mechanics problem. Otherwise, it calculates the strain energy density value of each grid cell in the coarse-grid physical information neural network model and selects multiple target grid cells with larger strain energy density values ​​for mesh refinement, obtaining a refined grid model. The parameters of the coarse-grid physical information neural network model are then transferred to the deep neural network of the refined grid model to obtain the target neural network model. This training-judgment-refinement-transfer loop is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem. This solves the problem of how to improve the training efficiency and solution accuracy of physical information deep neural networks in solving complex solid mechanics problems, significantly improving overall training efficiency, reducing computational costs, and ensuring the stability and reliability of the calculation results.

[0102] Next, with reference to the accompanying drawings, an adaptive mesh refinement system for a physical information neural network according to an embodiment of the present invention is described.

[0103] Figure 8 This is a block diagram of the adaptive mesh refinement system of the physical information neural network according to an embodiment of the present invention.

[0104] like Figure 8 As shown, the adaptive mesh refinement system 10 of the physical information neural network includes: a first judgment module 100, a calculation module 200, a second judgment module 300, and a mesh refinement module 400.

[0105] The system comprises the following modules: a first judgment module 100, used to determine whether there is a computational requirement for a solid mechanics problem involving a planar plate under tension; a calculation module 200, used to discretize the mechanics problem using a coarse-grid physical information neural network model if such a requirement exists, establishing a mapping relationship between node coordinates and node displacements, constructing the physical information function and total loss function of the solid mechanics governing equations based on this mapping relationship, and obtaining the total loss function value and its rate of change of the solid mechanics problem through training the coarse-grid physical information neural network model; and a second judgment module 300, used to determine whether the coarse-grid physical information neural network model meets preset solution conditions based on the total loss function value and its rate of change, or to determine whether the coarse-grid physical information neural network model meets preset solution conditions. The system checks whether the current number of training iterations of the coarse-grid physical information neural network model is greater than the preset number of iterations. The mesh refinement module 400 is used to output the displacement solution of the solid mechanics problem if the coarse-grid physical information neural network model meets the preset solution conditions or the current number of training iterations is greater than the preset number of iterations. Otherwise, it calculates the strain energy density value of each mesh element of the coarse-grid physical information neural network model and selects multiple target mesh elements with larger strain energy density values ​​for mesh refinement to obtain a refined mesh model. The parameters of the coarse-grid physical information neural network model are then transferred to the deep neural network of the refined mesh model to obtain the target neural network model. The above training-judgment-refinement-transfer loop is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem.

[0106] Optionally, in some embodiments, when the target mesh cell to be refined is a triangular cell, the mesh refinement module 400 is further configured to: if the target mesh cell to be refined shares a boundary with three adjacent cells, add a first set of new nodes at the midpoint of each edge of the target mesh cell to be refined, subdivide the target mesh cell to be refined into multiple sub-cells, connect the first set of new nodes to the opposite vertices of the corresponding edges in the adjacent cells, and subdivide each adjacent cell into multiple sub-cells; if the target mesh cell to be refined shares a boundary with two adjacent cells, add a second set of new nodes at the midpoint of each edge of the target mesh cell to be refined, subdivide the target mesh cell to be refined into multiple sub-cells, connect the second set of new nodes to the opposite vertices of the corresponding edges in the adjacent cells, and subdivide each adjacent cell into multiple sub-cells; if the target mesh cell to be refined shares a boundary with one adjacent cell, add a third set of new nodes at the midpoint of each edge of the target mesh cell to be refined, subdivide the target mesh cell to be refined into multiple sub-cells, connect the third set of new nodes to the opposite vertices of the corresponding edges in the adjacent cells, and subdivide the adjacent cells into multiple sub-cells.

[0107] Optionally, in some embodiments, the second judgment module 300 is used to: determine whether the total loss function value and its rate of change are less than a preset value; if the total loss function value and its rate of change are less than the preset value, then determine that the neural network model based on the coarse grid model meets the preset solution conditions; otherwise, determine that the neural network model based on the coarse grid model does not meet the preset solution conditions.

[0108] Optionally, in some embodiments, when selecting multiple target mesh elements with larger strain energy density values ​​for mesh refinement, the mesh refinement module 400 is further configured to: obtain the current area of ​​the target mesh element to be refined and the mesh element with the largest area value among the adjacent elements; if the ratio of the current area to the maximum area value is less than a preset ratio, then the mesh element with the largest area value is refined; otherwise, the target mesh element to be refined is refined.

[0109] Optionally, in some embodiments, when refining the target mesh cell to be refined, the mesh refinement module 400 is further configured to: detect the minimum area value of the target mesh cell to be refined; if the minimum area value of the target mesh cell to be refined is less than a preset area threshold, then the refining of the target mesh cell to be refined is stopped.

[0110] Optionally, in some embodiments, after transferring the parameters of the deep neural network model trained using the coarse mesh model to the deep neural network of the refined mesh model to obtain the target neural network model, the mesh refinement module 400 is further used to: determine whether the target neural network model meets the preset solution conditions, or determine whether the current training number of the target neural network model is greater than the preset number; if the target neural network model meets the preset solution conditions, or the current training number of the target neural network model is greater than the preset number, then output the displacement solution of the solid mechanics problem; otherwise, continue to refine the mesh of the target neural network model until the target neural network model after mesh refinement meets the preset solution conditions or the current training number of the target neural network model after mesh refinement is greater than the preset number.

[0111] It should be noted that the explanation of the aforementioned embodiment of the adaptive mesh refinement method for physical information neural networks also applies to the adaptive mesh refinement system for physical information neural networks in this embodiment, and will not be repeated here.

[0112] According to the adaptive mesh refinement system of the physical information neural network proposed in this embodiment of the invention, if there is a computational requirement for a solid mechanics problem, a coarse-mesh physical information neural network model is used to discretize the mechanical problem, establish a mapping relationship between node coordinates and node displacements, and construct the physical information function and total loss function of the solid mechanics governing equations based on the mapping relationship between node coordinates and node displacements. By training the coarse-mesh physical information neural network model, the total loss function value and its rate of change of the solid mechanics problem are obtained. Based on the total loss function value and its rate of change, it is determined whether the coarse-mesh physical information neural network model meets the preset solution conditions, or to determine the current state of the coarse-mesh physical information neural network model. The system checks whether the number of training iterations exceeds a preset limit. If the coarse-grid physical information neural network model meets the preset solution conditions, or if the current number of training iterations exceeds the preset limit, then it outputs the displacement solution of the solid mechanics problem. Otherwise, it calculates the strain energy density value of each grid cell in the coarse-grid physical information neural network model and selects multiple target grid cells with larger strain energy density values ​​for mesh refinement, obtaining a refined grid model. The parameters of the coarse-grid physical information neural network model are then transferred to the deep neural network of the refined grid model to obtain the target neural network model. This training-judgment-refinement-transfer loop is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem. This solves the problem of how to improve the training efficiency and solution accuracy of physical information deep neural networks in solving complex solid mechanics problems, significantly improving overall training efficiency, reducing computational costs, and ensuring the stability and reliability of the calculation results.

[0113] Figure 9 This is a schematic diagram of an electronic device provided in an embodiment of the present invention. The electronic device may include:

[0114] The memory 901, the processor 902, and the computer program stored on the memory 901 and capable of running on the processor 902.

[0115] When the processor 902 executes the program, it implements the adaptive mesh refinement method for the physical information neural network provided in the above embodiments.

[0116] Furthermore, electronic devices also include:

[0117] Communication interface 903 is used for communication between memory 901 and processor 902.

[0118] The memory 901 is used to store computer programs that can run on the processor 902.

[0119] The memory 901 may include high-speed RAM memory, and may also include non-volatile memory, such as at least one disk storage device.

[0120] If the memory 901, processor 902, and communication interface 903 are implemented independently, then the communication interface 903, memory 901, and processor 902 can be interconnected via a bus to complete communication between them. The bus can be an Industry Standard Architecture (ISA) bus, a Peripheral Component Interconnect (PCI) bus, or an Extended Industry Standard Architecture (EISA) bus, etc. Buses can be categorized as address buses, data buses, control buses, etc. For ease of representation, Figure 9 The bus is represented by a single thick line, but this does not mean that there is only one bus or one type of bus.

[0121] Optionally, in a specific implementation, if the memory 901, processor 902, and communication interface 903 are integrated on a single chip, then the memory 901, processor 902, and communication interface 903 can communicate with each other through an internal interface.

[0122] The processor 902 may be a central processing unit (CPU), an application specific integrated circuit (ASIC), or one or more integrated circuits configured to implement embodiments of the present invention.

[0123] This invention also provides a computer program product on which a computer program is stored, which, when executed by a processor, implements the above-described adaptive mesh refinement method for a physical information neural network.

[0124] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0125] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "N" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0126] Any process or method description in the flowchart or otherwise herein can be understood as representing a module, segment, or portion of code comprising one or more N executable instructions for implementing custom logic functions or processes, and the scope of preferred embodiments of the invention includes additional implementations in which functions may be performed not in the order shown or discussed, including substantially simultaneously or in reverse order depending on the functions involved, as should be understood by those skilled in the art to which embodiments of the invention pertain.

[0127] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequential list of executable instructions for implementing logical functions, and can be specifically implemented in any computer program product for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a processor-included system, or other system that can fetch and execute instructions from, an instruction execution system, apparatus, or device). For the purposes of this specification, "computer program product" can be any means that can contain, store, communicate, propagate, or transmit a program for use by, or in conjunction with, an instruction execution system, apparatus, or device. More specific examples of computer program products (a non-exhaustive list) include the following: an electrical connection having one or N wires (electronic device), a portable computer disk drive (magnetic device), random access memory (RAM), read-only memory (ROM), erasable and editable read-only memory (EPROM or flash memory), fiber optic device, and portable optical disc read-only memory (CDROM). Furthermore, the computer program product can even be paper or other suitable medium on which the program can be printed, because the program can be obtained electronically, for example, by optically scanning the paper or other medium, followed by editing, interpreting, or otherwise processing as necessary, and then stored in a computer memory.

[0128] It should be understood that various parts of the present invention can be implemented in hardware, software, firmware, or a combination thereof. In the above embodiments, the N steps or methods can be implemented in software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware as in another embodiment, it can be implemented using any of the following techniques known in the art, or a combination thereof: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0129] Those skilled in the art will understand that all or part of the steps of the methods described in the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer program product, and when executed, the computer program includes one or a combination of the steps of the method embodiments.

[0130] Furthermore, the functional units in the various embodiments of the present invention can be integrated into a processing module, or each unit can exist physically separately, or two or more units can be integrated into a module. The integrated module can be implemented in hardware or as a software functional module. If the integrated module is implemented as a software functional module and sold or used as an independent product, it can also be stored in a computer program product.

[0131] The computer program product mentioned above can be a read-only memory, a disk, or an optical disk, etc. Although embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

Claims

1. An adaptive mesh refinement method for a physical information neural network, characterized in that, Includes the following steps: There is a computational need to determine whether a solid mechanics problem exists that involves a planar plate under tension. If there is a computational requirement for the solid mechanics problem, a coarse-grid physical information neural network model is used to discretize the mechanical problem, establish a mapping relationship between node coordinates and node displacements, construct the physical information function and total loss function of the solid mechanics control equation based on the mapping relationship between node coordinates and node displacements, and obtain the total loss function value and its rate of change of the solid mechanics problem by training the coarse-grid physical information neural network model. Based on the total loss function value and its rate of change, determine whether the coarse-grid physical information neural network model meets the preset solution conditions, or determine whether the current number of training iterations of the coarse-grid physical information neural network model is greater than the preset number of iterations; If the coarse-grid physical information neural network model satisfies the preset solution conditions, or if the current number of training iterations is greater than the preset number of iterations, then the displacement solution of the solid mechanics problem is output. Otherwise, the strain energy density value of each grid cell in the coarse-grid physical information neural network model is calculated, and multiple target grid cells with larger strain energy density values ​​are selected for mesh refinement to obtain a refined grid model. The parameters of the coarse-grid physical information neural network model are then transferred to the deep neural network of the refined grid model to obtain the target neural network model. The above training-judgment-refinement-transfer loop is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem. The step of selecting multiple target mesh elements with relatively large strain energy density values ​​for mesh refinement, when the target mesh element is a triangular element, includes: If the target mesh cell to be refined shares a boundary with three adjacent cells, then a first set of new nodes is added to the midpoint of each edge of the target mesh cell to be refined, and the target mesh cell to be refined is subdivided into multiple sub-cells. The first set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent cells, and each adjacent cell is subdivided into multiple sub-cells. If the target mesh cell to be refined shares a boundary with two adjacent cells, then a second set of new nodes is added to the midpoint of each edge of the target mesh cell to be refined, and the target mesh cell to be refined is subdivided into multiple sub-cells. The second set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent cells, and each adjacent cell is subdivided into multiple sub-cells. If the target mesh cell to be refined shares a boundary with an adjacent cell, a third set of new nodes is added to the midpoint of each edge of the target mesh cell to be refined, and the target mesh cell to be refined is subdivided into multiple sub-cells. The third set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent cell, and the adjacent cell is subdivided into multiple sub-cells.

2. The adaptive mesh refinement method for physical information neural networks according to claim 1, characterized in that, The step of determining whether the coarse-grid physical information neural network model meets the preset solution conditions based on the total loss function value and its rate of change includes: Determine whether the total loss function value and its rate of change are less than a preset value; If the total loss function value and its rate of change are less than the preset value, then the coarse-grid physical information neural network model is determined to meet the preset solution conditions; otherwise, the coarse-grid physical information neural network model is determined not to meet the preset solution conditions.

3. The adaptive mesh refinement method for physical information neural networks according to claim 1, characterized in that, When refining the mesh by selecting multiple target mesh elements with relatively large strain energy density values, the following steps are included: Obtain the current area of ​​the target mesh cell to be refined and the mesh cell with the largest area value among its neighboring cells; If the ratio of the current area to the maximum area is less than a preset ratio, then the grid cells of the maximum area are refined; otherwise, the target grid cells to be refined are refined.

4. The adaptive mesh refinement method for physical information neural networks according to claim 1, characterized in that, When refining the target mesh element to be refined, the following steps are included: Detect the minimum area value of the target mesh cell to be refined; If the minimum area value of the target mesh cell to be refined is less than a preset area threshold, then the refinement of the target mesh cell to be refined is stopped.

5. The adaptive mesh refinement method for physical information neural networks according to claim 1, characterized in that, After initializing the deep neural network model with parameters obtained from training the coarse-grid model to refine the deep neural network model, and obtaining the target neural network model, the process includes: Determine whether the target neural network model meets the preset solution conditions, or determine whether the current number of training iterations of the target neural network model is greater than the preset number of iterations; If the target neural network model satisfies the preset solution conditions, or the current number of training iterations of the target neural network model is greater than the preset number of iterations, then the displacement solution of the solid mechanics problem is output; otherwise, the mesh of the target neural network model is refined until the refined target neural network model satisfies the preset solution conditions or the current number of training iterations of the refined target neural network model is greater than the preset number of iterations.

6. An adaptive mesh refinement system for a physical information neural network, characterized in that, include: The first judgment module is used to determine whether there is a calculation requirement for a solid mechanics problem involving a planar plate under tension. The calculation module is used to discretize the mechanical problem using a coarse-grid physical information neural network model if there is a calculation requirement for the solid mechanics problem, establish a mapping relationship between node coordinates and node displacements, construct the physical information function and total loss function of the solid mechanics control equation based on the mapping relationship between node coordinates and node displacements, and obtain the total loss function value and its rate of change of the solid mechanics problem by training the coarse-grid physical information neural network model. The second judgment module is used to determine whether the coarse grid physical information neural network model meets the preset solution conditions based on the total loss function value and its rate of change, or to determine whether the current training number of the coarse grid physical information neural network model is greater than the preset number. The mesh refinement module is used to output the displacement solution of the solid mechanics problem if the coarse mesh physical information neural network model satisfies the preset solution conditions or the current training count is greater than the preset count; otherwise, it calculates the strain energy density value of each mesh element of the coarse mesh physical information neural network model, selects multiple target mesh elements with larger strain energy density values ​​for mesh refinement, obtains a refined mesh model, and uses the parameters of the coarse mesh physical information neural network model to transfer to the deep neural network of the refined mesh model to obtain the target neural network model. The above training-judgment-refinement-transfer loop iteration is repeated until the target neural network model outputs a high-precision solution to the solid mechanics problem. When the target mesh element to be refined is a triangular element, the mesh refinement module: If the target mesh cell to be refined shares a boundary with three adjacent cells, then a first set of new nodes is added to the midpoint of each edge of the target mesh cell to be refined, and the target mesh cell to be refined is subdivided into multiple sub-cells. The first set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent cells, and each adjacent cell is subdivided into multiple sub-cells. If the target mesh cell to be refined shares a boundary with two adjacent cells, then a second set of new nodes is added to the midpoint of each edge of the target mesh cell to be refined, and the target mesh cell to be refined is subdivided into multiple sub-cells. The second set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent cells, and each adjacent cell is subdivided into multiple sub-cells. If the target mesh cell to be refined shares a boundary with an adjacent cell, a third set of new nodes is added to the midpoint of each edge of the target mesh cell to be refined, and the target mesh cell to be refined is subdivided into multiple sub-cells. The third set of new nodes is connected to the opposite vertices of the corresponding edges in the adjacent cell, and the adjacent cell is subdivided into multiple sub-cells.

7. An electronic device, characterized in that, It includes a memory, a processor, and a computer program stored in the memory and executable on the processor, the processor executing the program to implement the adaptive mesh refinement method for a physical information neural network as described in any one of claims 1-5.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, The computer program is executed by a processor to implement the adaptive mesh refinement method for a physical information neural network as described in any one of claims 1-5.

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