Graph neural network Poisson equation solving method for isogeometric analysis control grid

Abstract

The invention provides an isogeometric analysis control grid-oriented graph neural network Poisson equation solving method. The method comprises the following steps of S1, matching data set acquisition and input formats; s2, carrying out proportion division on the data set; s3, constructing a graph neural network model; s4, performing model training and parameter initialization; training a model by using a training set, and monitoring a training effect through a verification set by taking Poisson solution error minimization and operation speed maximization as targets; step S5, Poisson solution and performance verification: executing Poisson solution by using the test set, calculating a solution error and a running speed which need to be superior to a file specified standard, and returning to adjust model parameters and retraining if the standard is not reached; and S6, outputting a result. Step S7, Main file integration and interface implementation: writing Main file integration functions from the step S1 to the step S7; according to the method, the problem that the compatibility of a data set and input of an existing method is limited remarkably in the engineering simulation and scientific calculation process can be solved.

Description

Technical Field

[0001] This invention relates to the field of set analysis technology, and in particular to a method for solving the Poisson equation using graph neural networks for isogeometric analysis control grids. Background Technology

[0002] The Poisson equation, as a core partial differential equation describing the distribution of physical fields (such as heat conduction, electrostatic fields, and fluid potential fields), directly determines the reliability of analysis results in fields such as engineering simulation and scientific computing due to the accuracy and efficiency of its numerical solution. It has irreplaceable application value in mechanical design, aerospace, and electronic engineering. With the development of isogeometric analysis techniques, solution methods based on control meshes have gradually become a research hotspot—isogeometric analysis, by maintaining the consistency between the geometric model and the solution mesh, can effectively avoid the errors caused by the separation of geometric discretization and numerical discretization in traditional finite element methods. Existing methods have significant shortcomings in data source selection and format adaptation: On the one hand, most methods lack a clearly defined and unified source of high-quality datasets, resulting in poor data consistency across different research or application scenarios. This leads to a lack of comparable benchmarks for model training and solution results, making it difficult to effectively verify the superiority of the methods. On the other hand, input format support is limited, generally only compatible with traditional geometric model formats such as STEP. This fails to consider potential corruption and data loss issues during data transmission and storage, and there are no alternative formats (such as the commonly used NPZ format for numerical computation) to supplement the data. If the STEP file cannot be preserved, the solution process will be directly interrupted, making it difficult to guarantee the continuity of the data source. Furthermore, existing methods lack sufficient support for auxiliary data formats, failing to adapt to common formats such as .pt, .xyz, and .json, further limiting their applicability to various scenarios. Summary of the Invention

[0003] This invention proposes a graph neural network method for solving the Poisson equation based on isogeometric analysis control grids, which can solve the problem of significant limitations in the compatibility of datasets and inputs in existing methods during engineering simulation and scientific computing.

[0004] The present invention adopts the following technical solution.

[0005] A graph neural network-based method for solving the Poisson equation for isogeometric analysis control grids, used for physical field data set analysis in engineering simulations and scientific computing processes, includes the following steps: Step S1: Dataset acquisition and input format adaptation; Step S2: Divide the dataset proportionally; Step S3: Construction of the graph neural network model; Step S4, Model Training and Parameter Initialization: Train the model using the training set, aiming to minimize the Poisson solution error and maximize the running speed. Monitor the training effect using the validation set. If the validation set error does not decrease for three consecutive cycles, make preliminary adjustments to the basic parameters. Step S5, Poisson Solution and Performance Verification: Perform Poisson solution using the test set, calculate the solution error and running speed. It must be better than the benchmark specified in the file. If it does not meet the benchmark, return to adjust the model parameters and retrain. Step S6: Output the results; Step S7, Main File Integration and Interface Implementation: Write the functions for integrating the Main file from steps S1 to S7, develop the user interface, support NPZ / STEP file upload, mesh generation and vertex display, auxiliary functions, simulation running and real-time status display, and realize "one-click" process execution.

[0006] In step S1, a dataset with a specified file link is used. This dataset employs specialized data related to the isogeometric analysis control mesh and the solution of the Poisson equation; it supports both STEP and NPZ input formats. Step S1 sets a format priority rule, specifically: if data loss, format incompatibility, or inability to parse the STEP file occurs during reading, resulting in the file being unable to be retained, then the NPZ file corresponding to the link code is directly switched as the input data source to ensure the integrity of the dataset and the continuity of the solution process. That is, if the STEP file cannot be retained, the NPZ file corresponding to the link code is used as the main one to ensure the integrity of the data source.

[0007] In step S2, if the dataset is for engine cylinder head heat conduction analysis, the adapted dataset is divided into a training set, a validation set, and a test set in an 8:1:1 ratio. The training set is used for model parameter learning, the validation set is used for model hyperparameter tuning and overfitting suppression, and the test set is used for final Poisson solution performance verification. The dataset is split strictly according to the specific data splitting approach to ensure that the splitting method is comparable to existing technologies. That is, the splitting logic refers to the code approach in the link to ensure comparability with existing technologies. Specifically: If the dataset is for engine cylinder head heat conduction analysis, then in step S2, the adapted dataset is divided into training set, validation set and test set in a ratio of 8:1:1. The training set is used for model parameter learning, the validation set is used for model hyperparameter tuning and overfitting suppression, and the test set is used for final Poisson solution performance verification. The dataset is divided according to the segmentation ideas for data specific to engineering simulation and scientific computing processes. In the data source preparation in step S1, STEP format file: Exports a 3D geometric model of the cylinder head from UGNX, including vertex coordinates and geometric topology information of element connection relationships; NPZ format file: Based on engine bench test data, it organizes boundary conditions and physical parameters, including combustion chamber wall temperature, cooling water temperature, aluminum alloy thermal conductivity, and actual temperature field measurement data; Format compatibility verification: When attempting to read the STEP file, if detailed air passage data is lost during the export process, the system automatically switches to the NPZ file as the core data source. Simultaneously, it uploads a .pt format cylinder head point cloud file for mesh completion and a .csv format operating parameter file to perform multi-format compatibility. When dividing the dataset, ensure that the cylinder head geometric features and physical parameter distribution of the three datasets have a consistency error of less than 2%, align with the dataset division logic of traditional finite element analysis, and ensure the comparability of results.

[0008] In step S3, the control mesh is based on isogeometric analysis. Mesh vertices are set as graph nodes, and cell connection relationships are set as graph edges. The geometric features and physical prior information of the mesh are used as node features to complete the data transformation from control mesh to graph structure. The model construction follows the core idea of ​​the code in the dedicated data, without introducing any other unmentioned network structures. The model is constructed by combining the geometric and physical features of the mesh. The construction idea is referenced in the linked code.

[0009] In step S3, the graph neural network model employs an attention mechanism and multi-scale feature aggregation technology to construct a lightweight network framework, specifically including: A custom message passing layer can replace torch-geometric's MessagePassing without relying on libraries such as torch-scatter. Set up a NURBS convolutional layer with attention mechanism, define x[num_nodes,in_channels] as node features, edge_index[2,num_edges] as edge indices, and edge_attr[num_edges,edge_dim] as edge features, and assign different weights to different nodes and edges through attention mechanism; A multi-scale feature aggregation network, Multi_Scale_GNN, is constructed. The input layer of this network converts the in_channels of the input nodes into hidden_channels features. The hidden layer performs multiple nonlinear transformations on the input features to learn complex geometric features. The output layer converts the hidden_channels features into the final out_channels features, while checking and repairing the edge feature dimensions. A multi-scale feature extraction layer and a fully connected layer constitute an end-to-end grid quality evaluation system. This evaluation system takes node features x, edge index edge_index, and edge features edge_attr as input in dictionary form, and finally outputs quality scores [num_nodes, num_quality_metrics] and transmits them to the fully connected layer.

[0010] Step S4 specifically involves: inputting the training set data into the graph neural network, with the dual optimization objectives of minimizing the error in solving the Poisson equation and maximizing the running speed, using the mean squared error as the loss function, updating the model parameters through the gradient descent algorithm, and using the validation set to monitor the model training results in real time. If the solution error on the validation set does not show a decreasing trend for three consecutive iterations, training is paused and basic parameters, including the learning rate and batch size, are initially adjusted.

[0011] In step S4, the model training process combines a decoupled optimization process with an incremental optimization strategy, specifically including: The control grid quality assessment module and the grid optimization operation are separated into parallel and independent modules, and data interaction is achieved through an asynchronous communication mechanism; The grid quality evaluation module uses the PyTorchGeometric library to load the trained node data, edge data, and grid quality labels, and creates the evaluate_model function to calculate the average loss, mean squared error, coefficient of determination, average inference time, and generate a model evaluation report. The incremental optimization module takes as input PyTorch-saved graph data (.pt file) and NURBS data in .json format. It iteratively optimizes low-quality grid regions based on grid quality assessment results, setting a grid quality pass / fail value of 0.5. Nodes scoring below this pass / fail value are considered low-quality nodes. The Adam optimizer is used on these low-quality nodes, with a maximum optimization step count of 100, a learning rate of 0.01, and a loss function calculated as the sum of squared distances between low-quality nodes and their neighboring grid control points. Alternating between global and local optimization is employed, with a maximum global iteration count. Optimization stops when no low-quality grid node is found in a global iteration or when the number of iterations exceeds the maximum global iteration count.

[0012] Step S5 specifically involves: inputting test machine data into a trained and stable graph neural network model, performing numerical solutions to the Poisson equation, recording the total running time of the solution process, calculating two core indicators—Poisson solution error and running speed, comparing the calculation results with the benchmark data, and if the results are not better than the benchmark, returning to the model training and parameter initialization steps to adjust the model parameters, retraining the model, and comparing the calculation results with the benchmark data again, until the solution results are better than the specified benchmark in both error and speed.

[0013] In step S6, a full model mesh file in STEP / STL format is generated, and Poisson solution error data, running speed data, and a comparative heatmap of "true solution - graph neural network predicted solution - convolutional neural network predicted solution" drawn with reference to the Poisson section of the linked code are output simultaneously. Specifically, the control mesh of the same geometric analysis is divided into all dimensions through a mesh generation algorithm to generate mesh data covering the overall structure of the model. This data is then converted into STEP or STL format to visualize the mesh of the entire model. Three types of results are output simultaneously: error data, running speed data, and heatmap. The error data records the absolute error and relative error in tabular form, the running speed data records the total time taken from the input of the test set to the output of the result, and the heatmap generates a comparative heatmap of "true solution - graph neural network predicted solution - convolutional neural network predicted solution".

[0014] In step S7, a Main file is created, integrating functions including dataset reading, proportional partitioning, model training, Poisson solution, and result output into a unified execution module. A visual operation interface is developed based on the Main file, supporting the uploading of NPZ or STEP files, and optionally point cloud files and data files. Point cloud file formats are .pt, .xyz, and .ply, while data file formats are .json and .csv. Ensure that the interface operation corresponds one-to-one with the backend function to achieve "one-click" Poisson equation solution and mesh visualization.

[0015] The beneficial effects of this invention are as follows: 1. By explicitly using a dedicated dataset for the control grid of the specified isogeometric analysis, the problem of poor data consistency in different scenarios can be avoided, providing a comparable benchmark for model training and solution results, and effectively verifying the superiority of the method; Dual format adaptation + priority mechanism: Supports dual core inputs of STEP (traditional geometric format) and NPZ (numerical calculation format). When the STEP file cannot be retained due to corruption or data loss, it will automatically switch to the NPZ file. It is also compatible with auxiliary formats such as .pt, .xyz (point cloud) and .json, .csv (data file), which completely solves the problem of process interruption caused by the single input of the existing method and adapts to the data source requirements of multiple scenarios.

[0016] 2. Scientific partitioning ensures generalization ability: The training set (parameter learning), validation set (hyperparameter tuning + overfitting suppression), and test set (performance verification) are precisely partitioned in an 8:1:1 ratio, strictly following the partitioning logic in the linked code to avoid performance evaluation distortion caused by data partitioning bias, thus ensuring the model's generalization ability; Dual-objective training + dynamic hyperparameter tuning: With the dual objectives of "minimizing Poisson solution error and maximizing running speed", the mean squared error loss function and gradient descent algorithm are used, combined with real-time monitoring of the validation set (if the error does not decrease for 3 consecutive cycles, the learning rate and batch size are adjusted), effectively avoiding model overfitting / underfitting and laying the foundation for high-precision solution.

[0017] 3. Enables full model mesh visualization: Generates complete mesh files in STEP / STL format through a full-dimensional mesh generation algorithm, overcoming the shortcomings of existing technologies that only visualize partial surfaces. It can clearly display the mesh topology and element distribution of key areas such as holes and grooves inside the model, facilitating the evaluation of the correlation between mesh quality and solution error; Multi-dimensional result presentation: Simultaneously outputs a quantized error table (absolute / relative error), running speed data, and a heatmap comparing the "actual solution - GNN predicted solution - CNN predicted solution", supporting both quantitative analysis and intuitive display of solution accuracy differences, meeting the result evaluation needs in different scenarios.

[0018] 4. Fully integrated functionality: The entire process of "data reading-partitioning-training-solving-output" is integrated through the Main file, avoiding the tedious operation of running each code module one by one, reducing errors in step connection (such as parameter passing errors), and shortening the operation time by more than 50%; Visual "one-click" interface: Supports file upload, mesh display, auxiliary functions (saving ANSYS templates, creating sample STL), simulation running and real-time status display, requiring no professional coding skills, and non-technical personnel can quickly get started, significantly improving the applicability of the method. Attached Figure Description

[0019] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments: Appendix Figure 1 This is a schematic diagram of the method flow in an embodiment of the present invention; Appendix Figure 2 This is a schematic diagram of the operation interface in an embodiment of the present invention. Detailed Implementation

[0020] As shown in the figure, a graph neural network method for solving the Poisson equation for control grids with isogeometric analysis is presented. This method is used for the analysis of physical field data sets in engineering simulations and scientific computing processes, and includes the following steps: Step S1: Dataset acquisition and input format adaptation; Step S2: Divide the dataset proportionally; Step S3: Construction of the graph neural network model; Step S4, Model Training and Parameter Initialization: Train the model using the training set, aiming to minimize the Poisson solution error and maximize the running speed. Monitor the training effect using the validation set. If the validation set error does not decrease for three consecutive cycles, make preliminary adjustments to the basic parameters. Step S5, Poisson Solution and Performance Verification: Perform Poisson solution using the test set, calculate the solution error and running speed. It must be better than the benchmark specified in the file. If it does not meet the benchmark, return to adjust the model parameters and retrain. Step S6: Output the results; Step S7, Main File Integration and Interface Implementation: Write the functions for integrating the Main file from steps S1 to S7, develop the user interface, support NPZ / STEP file upload, mesh generation and vertex display, auxiliary functions, simulation running and real-time status display, and realize "one-click" process execution.

[0021] In step S1, a dataset with a specified file link is used. This dataset employs specialized data related to the isogeometric analysis control mesh and the solution of the Poisson equation; it supports both STEP and NPZ input formats. Step S1 sets a format priority rule, specifically: if data loss, format incompatibility, or inability to parse the STEP file occurs during reading, resulting in the file being unable to be retained, then the NPZ file corresponding to the link code is directly switched as the input data source to ensure the integrity of the dataset and the continuity of the solution process. That is, if the STEP file cannot be retained, the NPZ file corresponding to the link code is used as the main one to ensure the integrity of the data source.

[0022] In step S2, if the dataset is for engine cylinder head heat conduction analysis, the adapted dataset is divided into a training set, a validation set, and a test set in an 8:1:1 ratio. The training set is used for model parameter learning, the validation set is used for model hyperparameter tuning and overfitting suppression, and the test set is used for final Poisson solution performance verification. The dataset is split strictly according to the specific data splitting approach to ensure that the splitting method is comparable to existing technologies. That is, the splitting logic refers to the code approach in the link to ensure comparability with existing technologies.

[0023] Specifically: If the dataset is for engine cylinder head heat conduction analysis, then in step S2, the adapted dataset is divided into training set, validation set and test set in a ratio of 8:1:1. The training set is used for model parameter learning, the validation set is used for model hyperparameter tuning and overfitting suppression, and the test set is used for final Poisson solution performance verification. The dataset is divided according to the segmentation ideas for data specific to engineering simulation and scientific computing processes. In the data source preparation in step S1, STEP format file: Exports a 3D geometric model of the cylinder head from UGNX, including vertex coordinates and geometric topology information of element connection relationships; NPZ format file: Based on engine bench test data, it organizes boundary conditions and physical parameters, including combustion chamber wall temperature, cooling water temperature, aluminum alloy thermal conductivity, and actual temperature field measurement data; Format compatibility verification: When attempting to read the STEP file, if detailed air passage data is lost during the export process, the system automatically switches to the NPZ file as the core data source. Simultaneously, it uploads a .pt format cylinder head point cloud file for mesh completion and a .csv format operating parameter file to perform multi-format compatibility. When dividing the dataset, ensure that the cylinder head geometric features and physical parameter distribution of the three datasets have a consistency error of less than 2%, align with the dataset division logic of traditional finite element analysis, and ensure the comparability of results.

[0024] In step S3, the control mesh is based on isogeometric analysis. Mesh vertices are set as graph nodes, and cell connection relationships are set as graph edges. The geometric features and physical prior information of the mesh are used as node features to complete the data transformation from control mesh to graph structure. The model construction follows the core idea of ​​the code in the dedicated data, without introducing any other unmentioned network structures. The model is constructed by combining the geometric and physical features of the mesh. The construction idea is referenced in the linked code.

[0025] In step S3, the graph neural network model employs an attention mechanism and multi-scale feature aggregation technology to construct a lightweight network framework, specifically including: A custom message passing layer can replace torch-geometric's MessagePassing without relying on libraries such as torch-scatter. Set up a NURBS convolutional layer with attention mechanism, define x[num_nodes,in_channels] as node features, edge_index[2,num_edges] as edge indices, and edge_attr[num_edges,edge_dim] as edge features, and assign different weights to different nodes and edges through attention mechanism; A multi-scale feature aggregation network, Multi_Scale_GNN, is constructed. The input layer of this network converts the in_channels of the input nodes into hidden_channels features. The hidden layer performs multiple nonlinear transformations on the input features to learn complex geometric features. The output layer converts the hidden_channels features into the final out_channels features, while checking and repairing the edge feature dimensions. A multi-scale feature extraction layer and a fully connected layer constitute an end-to-end grid quality evaluation system. This evaluation system takes node features x, edge index edge_index, and edge features edge_attr as input in dictionary form, and finally outputs quality scores [num_nodes, num_quality_metrics] and transmits them to the fully connected layer.

[0026] Step S4 specifically involves: inputting the training set data into the graph neural network, with the dual optimization objectives of minimizing the error in solving the Poisson equation and maximizing the running speed, using the mean squared error as the loss function, updating the model parameters through the gradient descent algorithm, and using the validation set to monitor the model training results in real time. If the solution error on the validation set does not show a decreasing trend for three consecutive iterations, training is paused and basic parameters, including the learning rate and batch size, are initially adjusted.

[0027] In step S4, the model training process combines a decoupled optimization process with an incremental optimization strategy, specifically including: The control grid quality assessment module and the grid optimization operation are separated into parallel and independent modules, and data interaction is achieved through an asynchronous communication mechanism; The grid quality evaluation module uses the PyTorchGeometric library to load the trained node data, edge data, and grid quality labels, and creates the evaluate_model function to calculate the average loss, mean squared error, coefficient of determination, average inference time, and generate a model evaluation report. The incremental optimization module takes as input PyTorch-saved graph data (.pt file) and NURBS data in .json format. It iteratively optimizes low-quality grid regions based on grid quality assessment results, setting a grid quality pass / fail value of 0.5. Nodes scoring below this pass / fail value are considered low-quality nodes. The Adam optimizer is used on these low-quality nodes, with a maximum optimization step count of 100, a learning rate of 0.01, and a loss function calculated as the sum of squared distances between low-quality nodes and their neighboring grid control points. Alternating between global and local optimization is employed, with a maximum global iteration count. Optimization stops when no low-quality grid node is found in a global iteration or when the number of iterations exceeds the maximum global iteration count.

[0028] Step S5 specifically involves: inputting test machine data into a trained and stable graph neural network model, performing numerical solutions to the Poisson equation, recording the total running time of the solution process, calculating two core indicators—Poisson solution error and running speed, comparing the calculation results with the benchmark data, and if the results are not better than the benchmark, returning to the model training and parameter initialization steps to adjust the model parameters, retraining the model, and comparing the calculation results with the benchmark data again, until the solution results are better than the specified benchmark in both error and speed.

[0029] In step S6, a full model mesh file in STEP / STL format is generated, and Poisson solution error data, running speed data, and a comparative heatmap of "true solution - graph neural network predicted solution - convolutional neural network predicted solution" drawn with reference to the Poisson section of the linked code are output simultaneously. Specifically, the control mesh of the same geometric analysis is divided into all dimensions through a mesh generation algorithm to generate mesh data covering the overall structure of the model. This data is then converted into STEP or STL format to visualize the mesh of the entire model. Three types of results are output simultaneously: error data, running speed data, and heatmap. The error data records the absolute error and relative error in tabular form, the running speed data records the total time taken from the input of the test set to the output of the result, and the heatmap generates a comparative heatmap of "true solution - graph neural network predicted solution - convolutional neural network predicted solution".

[0030] In step S7, a Main file is created, integrating functions including dataset reading, proportional partitioning, model training, Poisson solution, and result output into a unified execution module. A visual operation interface is developed based on the Main file, supporting the uploading of NPZ or STEP files, and optionally point cloud files and data files. Point cloud file formats are .pt, .xyz, and .ply, while data file formats are .json and .csv. Ensure that the interface operation corresponds one-to-one with the backend function to achieve "one-click" Poisson equation solution and mesh visualization.

[0031] Example 1: Step 1: Dataset Acquisition and Input Format Adaptation Dataset Acquisition Process Data source download: Access the dataset link specified in the patent (https: / / codeocean.com / capsule / 5929976 / tree / v1), download the approximately 2.8GB compressed file, and extract it to obtain the "geometric_data" folder, which contains two types of core files: 1000 STEP files (named from mesh_001.step to mesh_1000.step): Each file stores geometric topology information such as vertex coordinates and cell connection relationships of the control mesh for geometric analysis.

[0032] 500 NPZ files (named from data_001.npz to data_500.npz): Each file contains preprocessed numerical data, including the boundary conditions of the Poisson equation (Dirichlet / Neumann type), the true solution field values, and the physical characteristics of the mesh (such as edge length).

[0033] Data deduplication: By verifying the file hash (calculating a unique hash value for each file), duplicate sample files are removed, and finally 1400 valid samples are retained (1000 STEP files + 500 NPZ files with no duplicates).

[0034] Format adaptation implementation logic Dual-format reading and exception handling STEP file reading: First, attempt to read the STEP file to verify its integrity—it must contain both the "vertex data" and "cell data" core fields. If the reading is successful, directly extract the vertex coordinates (for building graph nodes) and cell connections (for building graph edges); if the reading fails (e.g., file corruption, missing key fields), it will automatically switch to reading the corresponding NPZ file (replacing the STEP file name suffix .step with .npz).

[0035] NPZ file reading: When reading an NPZ file, first check whether it contains the four core fields "vertices", "edges", "poisson_boundary", and "true_solution". If any field is missing, it is determined to be an invalid data source, the file processing is terminated and an error message is displayed.

[0036] Format compatibility verification scheme Test design: 100 files were randomly selected from 1000 STEP files as test samples. Ten of these files were artificially corrupted (by modifying the file header information and deleting some vertex data) to simulate the scenario of "file corruption / unable to parse" in real-world applications.

[0037] Verification results: All 90 intact STEP files were successfully read, with an average of 512 vertices and 1024 cells per file, achieving a 100% adaptation success rate; 10 corrupted STEP files triggered an exception handling mechanism, automatically loading the corresponding NPZ files without data loss, and subsequent solution processes could be seamlessly connected.

[0038] Step 2: Divide the dataset proportionally Data filtering criteria To select 1000 "complete samples" from 1400 valid samples, the following three conditions must be met simultaneously: The number of vertices in the control mesh should be ≥256 (to ensure that the mesh complexity matches the actual application scenario). The boundary conditions of the Poisson equation include both Dirichlet (first type of boundary) and Neumann (second type of boundary) types (covering common boundary scenarios). The true solution values ​​are fully labeled (without missing or outliers, ensuring the effectiveness of the labels during model training).

[0039] Dataset partitioning process The first partitioning: The 1000 complete samples were split into 800 training / validation samples and 200 test samples according to the ratio of "training / validation set: test set = 8:2". A "format-based stratification" strategy was used during the partitioning—ensuring that the ratio of STEP files to NPZ files in the test set is consistent with the format distribution of the overall samples (avoiding the concentration of samples of a single format in the test set, which would affect the fairness of validation). At the same time, a fixed random seed (value 42) was used to ensure the reproducibility of the partitioning results.

[0040] The second partitioning: The 800 training and validation sets were split again in a ratio of "training set: validation set = 9:1", resulting in 720 training set samples (72% of the total samples), 80 validation set samples (8% of the total samples), and 200 test set samples (20% of the total samples).

[0041] Verification of the rationality of the division The consistency of the distribution of each dataset was verified using three core metrics to ensure fairness in the training, validation, and testing processes. The results are as follows: Conclusion: The consistency error of the key metrics for each dataset is less than 5%, which meets the requirement of "comparable to existing technologies" and can be used for subsequent model training and validation.

[0042] Step 3: Graph Neural Network Model Construction Core Network Architecture (Lightweight GNN) Custom message passing layer Functional positioning: To replace the MessagePassing layer of traditional graph neural networks and realize feature interaction between graph nodes and neighboring nodes.

[0043] Core logic: Input the "node features" and "edge index" of the graph data (the edge index records the connection relationship between the source node and the target node); Features are extracted from the source node (neighbor nodes) and the target node (current node) respectively, and the two types of features are concatenated into a "node-neighbor feature pair"; By performing a linear transformation on the "node-neighbor feature pairs", the "message features" of each edge are obtained. Sum all the "message features" of the same target node to aggregate the neighbor features, and output the aggregated node features.

[0044] NURBS convolutional layers with attention mechanism Functional localization: Combining the geometric properties of non-uniform rational B-splines (NURBS), attention weights are used to distinguish the importance of different neighbors, thereby improving the accuracy of feature extraction.

[0045] Core logic: In addition to node features and edge indices, the input should include "edge attributes" (such as edge length, which reflects the geometric characteristics of the NURBS mesh). Calculate attention weights: Concatenate the source node features, target node features, and edge attributes, and obtain the attention weight of each edge through linear transformation and Softmax normalization (the larger the weight, the stronger the influence of the corresponding neighbor on the target node). Weighted feature aggregation: The source node features and edge attributes are multiplied by the attention weights respectively to obtain "weighted neighbor features" and "weighted edge attributes". After concatenation, a linear transformation is performed to output node features with uniform dimensions. Aggregate features by indexing the target node, ensuring that each node only receives weighted features from its neighbors.

[0046] Multi-Scale Feature Aggregation Network (Multi-Scale GNN) Network structure: It adopts an architecture of "input layer → 3 multi-scale convolutional layers → output layer", as follows: Input layer: The 6-dimensional node features (x / y / z coordinates, cell area, boundary condition labels) are transformed into 128-dimensional features through linear transformation (to improve feature expressiveness), and then nonlinearity is introduced through the ReLU activation function.

[0047] Multi-scale convolutional layers: First layer: An attention-based NURBS convolutional layer is used to keep the 128-dimensional features unchanged and enhance the differential aggregation of neighbor features; Second layer: Continue to use attention NURBS convolutional layers to reduce the feature dimension to 64 dimensions (reduce computation and avoid overfitting). The third layer further reduces the feature dimension to 32 dimensions through the attention NURBS convolutional layer, extracting more abstract multi-scale features.

[0048] Output layer: The 32-dimensional features are reduced to 1 dimension through linear transformation, and the field value prediction results of the Poisson equation (i.e., the solution value of each node) are output.

[0049] Grid quality assessment system Evaluation logic: Input graph structure data (node ​​features, edge indices, edge attributes) and true solution labels. By comparing the model predictions with the true solutions, three core metrics are output: Mean Squared Error (MSE): Calculated as the average of the squared differences between the predicted and actual values, reflecting the overall error magnitude; Coefficient of determination (R²): Calculated by “1 - (sum of squared prediction errors / sum of squared deviations from true values)”, with a value range of [0,1]. The closer R² is to 1, the better the model fit. Quality score: Calculated by "1-MSE", with a value range of [0,1]. The higher the score, the better the mesh quality (the smaller the MSE, the higher the quality score). 0.5 is set as the "mesh quality pass threshold". Meshes with a value lower than this are judged as low-quality meshes and need further optimization.

[0050] Step 4: Model Training and Parameter Initialization Training parameter settings Training process implementation Data loading preprocessing Convert the split training and validation sets into graph data format: Node features (x): integrate vertex coordinates (3D), cell area (1D), and boundary condition labels (2D, with Dirichlet / Neumann each occupying 1D), for a total of 6 dimensions; Edge index (edge_index): Records the connection relationships between nodes, in the format "[source node index list, target node index list]"; Edge attribute (edge_attr): The edge length is the core attribute, which reflects the geometric characteristics of the NURBS mesh; Label (y): The true solution field value of the Poisson equation is used, and each node corresponds to one label value.

[0051] Training phase In each training epoch, the training set is divided into multiple batches with a batch size of 32, and each batch is input into the model. For each batch, the gradient of the model parameters is first cleared to zero, and then the predicted value is calculated through forward propagation and compared with the true label to obtain the MSE loss; The gradient of the loss with respect to each parameter is calculated by backpropagation, and the model parameters are updated using the Adam optimizer to minimize the loss. The losses of each batch are summed up, and the average loss of the training round (weighted by the number of samples) is calculated at the end.

[0052] Verification phase After each training round, switch the model to "evaluation mode" (without updating parameters) and input validation set samples; Calculate the average MSE loss and R² value on the validation set to evaluate the model's generalization ability; If the current validation set MSE is less than the historical best value, save the current model parameters (as the "optimal model") and reset the early stopping count; If the validation set MSE does not decrease for three consecutive rounds, the early stopping mechanism is triggered: the learning rate is reduced to 1 / 10 of its original value (e.g., from 0.001 to 0.0001), the optimizer is reinitialized, and training continues.

[0053] Incremental optimization (local optimization of low-quality mesh) After every 10 rounds of training, the quality of the validation set grid is evaluated, and low-quality nodes with a quality score of <0.5 are collected. For low-quality nodes, a "local optimization" strategy is adopted: using the Adam optimizer with a learning rate of 0.01, only the prediction error of low-quality nodes is calculated, and the model parameters are adjusted by gradient descent within 100 steps. Optimize stopping conditions: Stop when the error of low-quality nodes drops below 1e-6, or reaches 100 steps, to avoid over-optimization; Optimization results: Tests showed that the proportion of low-quality nodes decreased from the initial 15% to 3%, and the overall quality of the mesh was significantly improved.

[0054] Key metrics of training results Training termination round: Round 48 (validation set MSE shows no decrease for 3 consecutive rounds and reaches the minimum value of 0.008). Final training loss: 0.0092 (mean MSE of training set); Validation set R²: 0.987 (excellent fit, indicating strong model generalization ability); Optimal model size: approximately 1.2MB (for easy subsequent deployment and invocation).

[0055] Step 5: Poisson Solution and Performance Verification Test set solution process Data loading: Load 200 test set samples and convert them into graph data format consistent with the training set; Model inference: Load the trained "optimal model", switch to evaluation mode, input test set samples, and calculate the predicted solution of the Poisson equation for each sample; Time statistics: Record the solution time for each sample (the time taken from input model to output predicted solution), and calculate the average solution time for all samples; Error calculation: Mean Absolute Error: Calculates the average absolute difference between the predicted and actual values ​​for all samples; Average relative error: Calculated by "(average absolute error / average true value) × 100%", reflecting the magnitude of the relative error.

[0056] Performance comparison with benchmark methods We selected two existing mainstream Poisson solution methods (IGA-Graph-Net and PINNs(5K)) as benchmarks and compared them from three dimensions: "solution speed", "accuracy" and "mesh quality". The results are as follows: Verification results: This invention significantly outperforms existing benchmark methods in three core dimensions: solution speed, accuracy, and mesh quality. It does not require returning adjustment parameters and can be directly applied to real-world scenarios.

[0057] It can be used directly in real-world scenarios.

[0058] Full model mesh file generation Taking the first sample in the test set as an example, generate a mesh file containing the Poisson solution results: Data extraction: Extract vertex coordinates (3D), edge connectivity, and Poisson prediction field values ​​from graph data; Format conversion: STEP file: Using PyVista, vertex coordinates and edge connectivity are organized according to the STEP format specification, and the predicted solution field values ​​are embedded as additional attributes, and saved as full_model_mesh.step; STL file: Converts vertex and edge data into triangular patches (adapting to STL format requirements), retains the predicted solution field values, and saves them as full_model_mesh.stl (for easy visualization with tools such as MeshLab). Verification: Opening the STL file with MeshLab clearly shows the geometry of the mesh and the distribution of the predicted solution values, with no data loss or formatting errors.

[0059] Error and speed data output Error data table (CSV format): Field design: Includes three fields: "Sample Number", "Mean Absolute Error", and "Relative Error (%)". Data source: Error calculation results for each test set sample; Save path: output / error_result.csv, supports opening and analysis directly with Excel or Python.

[0060] Running speed log (TXT format): The content includes: total number of test samples, total solution time, average solution time per sample, and test date; Save the file to output / speed_log.txt for easy recording and tracking of solution performance.

[0061] Comparison of heat map drawing To visually demonstrate the accuracy advantages of the method of this invention, heatmaps of three solutions are compared: Data selection: Select the 10th sample in the test set and obtain the "true solution", "predicted solution of the GNN of this invention", and "predicted solution of the traditional CNN" (as an additional benchmark). Drawing logic: Horizontal axis / vertical axis: X and Y coordinates (mm) of the grid vertex, respectively; Color mapping: Jet color mapping is used, and the color depth corresponds to the magnitude of the solution field value (value range 0-50). Subgraph layout: 1 row and 3 columns, namely "True Solution Heatmap", "GNN Predicted Solution Heatmap", and "CNN Predicted Solution Heatmap"; Results analysis: The heatmap of the GNN predicted solution is highly consistent with the actual solution, with minimal difference in color distribution; while the CNN predicted solution shows obvious color deviation in the grid edge region, which intuitively verifies the accuracy advantage of the present invention. Save path: output / heatmap_comparison.png (300dpi high-definition format, easy for report display).

[0062] Step 7: Main file integration and interface implementation End-to-end integration logic The entire process of "data loading → model training → Poisson solution → result output" is integrated into an interactive visual interface, which is mainly divided into two parts: "background task thread" and "front-end interface". Background task thread Functionality: Perform time-consuming operations (such as data loading and model inference) in the background without blocking the front-end interface interaction; Core steps: Data loading (progress 10%): Reads the user-selected NPZ / STEP file and converts it into graph data format; Model loading (progress 20%): Load the pre-trained optimal model and initialize the inference environment; Poisson solution (progress 30%-70%): Process data in batches, calculate predicted solutions and record time; Results output (80% complete): Generation of mesh file, error table, and heatmap; Completion notification (progress 100%): Send a completion signal and update the interface log.

[0063] Front-end interface design The interface is divided into four core areas, allowing non-technical personnel to operate it quickly: File upload area: Features: Supports multiple selections of NPZ / STEP files and displays the number of selected files; Operation: Click the "Browse (NPZ / STEP)" button to select a file. The interface will automatically display the mesh information (number of vertices, number of edges, file type) of the first file.

[0064] Grid information area: Function: Displays the grid key parameters of the selected file; Contents: number of vertices, number of edges, boundary condition type, file format.

[0065] Function button area: “RunSimulation”: Starts the Poisson solution with one click and triggers a background thread; "SaveANSYSTemplate": Generates an ANSYS software import template (including mesh node / element information) for subsequent finite element analysis; "CreateSampleSTL": Quickly generate sample STL files (based on test set samples) for grid preview.

[0066] Status display area: Progress bar: Displays the real-time progress of background tasks (0-100%). Log text box: Prints operation logs in real time (such as "Data loading completed" and "Solution time 0.049s"), making it easy for users to understand the process status.

[0067] Interface Functionality Testing File Upload: Upload 10 test set NPZ files. The interface successfully displays "10 files selected" and shows the mesh information of the first file (512 vertices, 1024 edges, NPZ format). One-click simulation: Click “RunSimulation”, the progress bar smoothly rises from 0 to 100%, the log updates the process nodes in real time, and the total time for 10 samples is 9.8s (average 0.98s per sample, including data loading time). Accessibility features: "SaveANSYSTemplate": Generates an .ans format template file containing mesh node coordinates and element connection relationships, which can be directly imported into ANSYS; “CreateSampleSTL”: Generates an STL file within 3 seconds, which can be opened with MeshLab to clearly view the mesh structure; User-friendliness: Non-technical personnel can complete the entire process of "uploading files → running simulation → viewing results" in 5 minutes without writing code or configuring parameters.

[0068] Example 2 Example of heat conduction analysis for engine cylinder head (mechanical design scenario) Step 1: Dataset Acquisition and Input Format Adaptation Data source preparation: STEP format file: Export the 3D geometric model of the cylinder head (including details such as air passages, bolt holes, and cooling water jacket) from UGNX (mechanical design CAD software), named cylinder_head.step, which contains geometric topology information such as vertex coordinates and unit connection relationships; NPZ format file: Based on engine bench test data, the boundary conditions and physical parameters are organized, including combustion chamber wall temperature (500℃), cooling water temperature (80℃), aluminum alloy thermal conductivity (202W / (m・K)), and real temperature field measurement data (obtained through infrared thermometry), named cylinder_head_thermal.npz; Format compatibility verification: When attempting to read the STEP file, due to the loss of detailed air passage data during the export process, the system automatically switched to the NPZ file as the core data source. At the same time, it uploaded the .pt format cylinder head point cloud file (for mesh completion) and the .csv format operating condition parameter file (speed 3000rpm, load 75%) to complete the multi-format compatibility.

[0069] Step 2: Divide the dataset proportionally Sample expansion: 100 sets of cylinder head heat transfer data under different operating conditions (speed 2000-4000 rpm, load 50%-100%) were collected and divided according to the patent 8:1:1 ratio: Training set (80 sets): covering the entire working condition range, used for model parameter learning; Validation set (10 sets): used for hyperparameter tuning and overfitting suppression, with a focus on extreme operating conditions (4000 rpm, 100% load). Test set (10 sets): Selected data from actual vehicle installation conditions for final performance verification; Reasonableness of division: Ensure that the cylinder head geometric features (wall thickness, cross-sectional area of ​​cooling water jacket) and physical parameter distribution of the three sets of datasets have a consistency error of <2%, align with the dataset division logic of traditional finite element analysis, and ensure the comparability of results.

[0070] Step 3: Graph Neural Network Model Construction Graph structure transformation: Taking the geometric analysis control mesh such as the cylinder head as the core, the mesh vertices (a total of 8192) are mapped to graph nodes, and the mesh units (connection relationships such as air passage walls and coolant jacket walls) are mapped to graph edges. The node features include three-dimensional coordinates (x / y / z), wall thickness (10-25mm), thermal conductivity, and boundary condition labels (high temperature surface / cooling surface). Network architecture adaptation: It adopts a patented lightweight GNN framework, and through NURBS convolutional layers with attention mechanism, it assigns higher weights to complex topological regions of cylinder head (such as the corners of the air passage and around bolt holes) to enhance the feature extraction of key regions; through a multi-scale feature aggregation network, it learns the heat conduction law under different working conditions without the need to introduce additional complex network structures.

[0071] Step 4: Model Training and Parameter Initialization Training objective: To achieve the dual goals of minimizing the temperature field solution error and maximizing the solution speed, thus matching the "rapid iterative optimization" requirement in mechanical design; Parameter settings: The optimizer is Adam, the initial learning rate is 0.001, the batch size is 32, the loss function is mean squared error (MSE), and the grid quality pass value is set to 0.5; Dynamic parameter tuning: When training reached the 35th round, the validation set error did not decrease for three consecutive rounds (MSE=0.011), so the learning rate was adjusted to 0.0001, and training continued until the 42nd round when convergence occurred. The final validation set R²=0.989, and the model size was 1.1MB (for easy embedding into mechanical design simulation platforms).

[0072] Step 5: Poisson Solution and Performance Verification Test set solution: Input 10 sets of actual vehicle loading condition data into the trained model, execute the heat conduction Poisson equation solution, and record the average solution time and temperature error per sample; Performance Comparison: Key metrics were compared with the traditional finite element method (ANSYS Workbench), and the results are as follows: Verification conclusion: The patented method is significantly superior to the traditional method in both speed and accuracy, requires no secondary parameter adjustment, and can be directly used for cylinder head thermal design optimization.

[0073] Step 6: Output Results Mesh visualization files: Generate STEP format cylinder head temperature field mesh files (which can be directly imported into UGNX and aligned with the design model) and STL format files (use MeshLab to view the temperature distribution of the inner wall of the cooling water jacket), clearly showing the temperature topology of key areas such as air passage corners and bolt holes; Quantization result output: Error table (CSV format): Records the absolute temperature error (1.2-2.8℃) and relative error (0.008%-0.012%) at each key location (combustion chamber wall, cooling water jacket wall, bolt seat). Speed ​​log (TXT format): Total solution time for 10 samples: 5.2s, average time per sample: 0.052s; Comparison of heatmaps: A three-color heatmap of "experimental measured actual temperature - GNN predicted temperature - ANSYS predicted temperature" is plotted to visually show the high consistency between the GNN prediction results and the actual temperature (airway area deviation <2℃).

[0074] Step 7: Main file integration and interface implementation One-click operation: Mechanical design engineers can upload cylinder head STEP / NPZ files, optional point cloud and operating condition data files through the patented visualization interface, select the "heat conduction analysis" function, and click "run simulation". The interface displays the progress in real time (data loading → model inference → result generation). Auxiliary functions: The "Save ANSYS Template" function generates mesh and temperature data files that can be directly imported into ANSYS, avoiding tedious format conversion operations; the "Create Sample STL" function allows for a quick preview of the optimized cylinder head cooling fin layout, and even non-technical personnel can complete the entire process within 10 minutes.

[0075] Improved design efficiency: Reduced cylinder head heat conduction simulation time from 30 minutes to 5.2 seconds, supports rapid comparison of multiple schemes (such as different heat sink thicknesses and coolant jacket shapes) during the design phase, and shortened the R&D cycle by 40%; Optimization and accuracy improvement: Based on high-precision temperature field data, the cross-sectional area of ​​the cooling water jacket was adjusted (from 80mm² to 100mm²) and heat dissipation fins around the air passage were added (5mm thick), which reduced the maximum temperature of the cylinder head from 485℃ to 452℃, reduced thermal stress by 18%, and avoided the risk of cracking. Enhanced format compatibility: Resolves issues related to corrupted CAD files (STEP) and incompatibility between numerical data (NPZ) and auxiliary files (.pt / .csv) formats in mechanical design, enabling seamless integration of multi-tool collaborative design (UGNX → patented solver → ANSYS).

[0076] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of this patent should be determined by the appended claims.

Claims

1. A method for solving the Poisson equation using a graph neural network for control grids with isogeometric analysis, characterized by: The solution method is used for the analysis of physical field data sets in engineering simulation and scientific computing processes, and includes the following steps: Step S1: Dataset acquisition and input format adaptation; Step S2: Divide the dataset proportionally; Step S3: Graph Neural Network Model Construction; Step S4, Model Training and Parameter Initialization: Train the model using the training set, aiming to minimize the Poisson solution error and maximize the running speed, and monitor the training effect using the validation set; Step S5, Poisson Solution and Performance Verification: Perform Poisson solution using the test set, calculate the solution error and running speed. It must be better than the benchmark specified in the file. If it does not meet the benchmark, return to adjust the model parameters and retrain. Step S6: Output the results; Step S7, Main File Integration and Interface Implementation: Write the functions for integrating the Main file from step S1 to step S7. The functions include NPZ / STEP file upload, mesh generation and vertex display, auxiliary functions, simulation running and real-time status display.

2. The method for solving the Poisson equation using a graph neural network oriented towards isogeometric analysis control grids according to claim 1, characterized in that: In step S1, a dataset with a specified file link is used. This dataset employs specialized data related to the isogeometric analysis control mesh and the solution of the Poisson equation; it supports both STEP and NPZ input formats. Step S1 sets a format priority rule, specifically: if data loss, format incompatibility, or unparsable problems occur when reading the STEP file, resulting in the file being unable to be retained, then the NPZ file corresponding to the link code is directly switched as the input data source to ensure the integrity of the dataset and the continuity of the solution process.

3. The method for solving the Poisson equation using a graph neural network oriented towards isogeometric analysis control grids as described in claim 2, characterized in that: If the dataset is for engine cylinder head heat conduction analysis, then in step S2, the adapted dataset is divided into training set, validation set, and test set in a ratio of 8:1:

1. The training set is used for model parameter learning, the validation set is used for model hyperparameter tuning and overfitting suppression, and the test set is used for final Poisson solution performance verification. The dataset is divided according to the segmentation approach for data specific to engineering simulation and scientific computing processes. In the data source preparation in step S1, STEP format file: Exports a 3D geometric model of the cylinder head from UGNX, including vertex coordinates and geometric topology information of element connection relationships; NPZ format file: Based on engine bench test data, it organizes boundary conditions and physical parameters, including combustion chamber wall temperature, cooling water temperature, aluminum alloy thermal conductivity, and actual temperature field measurement data; Format compatibility verification: When attempting to read the STEP file, if detailed air passage data is lost during the export process, the system automatically switches to the NPZ file as the core data source. Simultaneously, it uploads a .pt format cylinder head point cloud file for mesh completion and a .csv format operating parameter file to perform multi-format compatibility. When dividing the dataset, ensure that the consistency error of the cylinder head geometric features and physical parameter distribution of the three datasets is less than 2%.

4. The method for solving the Poisson equation using a graph neural network oriented towards isogeometric analysis control grids according to claim 1, characterized in that: In step S3, the control mesh is based on isogeometric analysis. Mesh vertices are set as graph nodes, and cell connection relationships are set as graph edges. The geometric features and physical prior information of the mesh are used as node features to complete the data transformation from control mesh to graph structure. The model is constructed by combining the geometric and physical features of the mesh.

5. The method for solving the Poisson equation using a graph neural network oriented towards isogeometric analysis control grids according to claim 4, characterized in that: In step S3, the graph neural network model employs an attention mechanism and multi-scale feature aggregation technology to construct a lightweight network framework, specifically including: A custom messaging layer is used to replace torch-geometric MessagePassing. Set up a NURBS convolutional layer with attention mechanism, define x[num_nodes,in_channels] as node features, edge_index[2,num_edges] as edge indices, and edge_attr[num_edges,edge_dim] as edge features, and assign different weights to different nodes and edges through attention mechanism; A multi-scale feature aggregation network, Multi_Scale_GNN, is constructed. The input layer of this network converts the in_channels of the input nodes into hidden_channels features. The hidden layer performs multiple nonlinear transformations on the input features to learn complex geometric features. The output layer converts the hidden_channels features into the final out_channels features, while checking and repairing the edge feature dimensions. A multi-scale feature extraction layer and a fully connected layer constitute an end-to-end grid quality evaluation system. This evaluation system takes node features x, edge index edge_index, and edge features edge_attr as input in dictionary form, and finally outputs quality scores [num_nodes, num_quality_metrics] and transmits them to the fully connected layer.

6. The method for solving the Poisson equation using a graph neural network oriented towards isogeometric analysis control grids according to claim 1, characterized in that: Step S4 specifically involves: inputting the training set data into the graph neural network, with the dual optimization objectives of minimizing the error in solving the Poisson equation and maximizing the running speed, using the mean squared error as the loss function, updating the model parameters through the gradient descent algorithm, and using the validation set to monitor the model training results in real time. If the solution error on the validation set does not show a decreasing trend for three consecutive iterations, training is paused and basic parameters, including the learning rate and batch size, are initially adjusted.

7. The method for solving the Poisson equation using a graph neural network oriented towards isogeometric analysis control grids according to claim 6, characterized in that: In step S4, the model training process combines a decoupled optimization process with an incremental optimization strategy, specifically including: The control grid quality assessment module and the grid optimization operation are separated into parallel and independent modules, and data interaction is achieved through an asynchronous communication mechanism; The grid quality evaluation module uses the PyTorchGeometric library to load the trained node data, edge data, and grid quality labels, and creates the evaluate_model function to calculate the average loss, mean squared error, coefficient of determination, average inference time, and generate a model evaluation report. The incremental optimization module takes as input PyTorch-saved graph data (.pt file) and NURBS data in .json format. It iteratively optimizes low-quality grid regions based on grid quality assessment results, setting a grid quality pass / fail value of 0.

5. Nodes scoring below this pass / fail value are considered low-quality nodes. The Adam optimizer is used on these low-quality nodes, with a maximum optimization step count of 100, a learning rate of 0.01, and a loss function calculated as the sum of squared distances between low-quality nodes and their neighboring grid control points. Alternating between global and local optimization is employed, with a maximum global iteration count. Optimization stops when no low-quality grid node is found in a global iteration or when the number of iterations exceeds the maximum global iteration count.

8. The method for solving the Poisson equation using a graph neural network oriented towards isogeometric analysis control grids according to claim 1, characterized in that: Step S5 specifically involves: inputting test machine data into a trained and stable graph neural network model, performing numerical solutions to the Poisson equation, recording the total running time of the solution process, calculating two core indicators—Poisson solution error and running speed, comparing the calculation results with the benchmark data, and if the results are not better than the benchmark, returning to the model training and parameter initialization steps to adjust the model parameters, retraining the model, and comparing the calculation results with the benchmark data again, until the solution results are better than the specified benchmark in both error and speed.

9. The method for solving the Poisson equation using a graph neural network oriented towards isogeometric analysis control grids according to claim 1, characterized in that: In step S6, a full model mesh file in STEP / STL format is generated, and Poisson solution error data, running speed data, and a comparative heatmap of "true solution - graph neural network predicted solution - convolutional neural network predicted solution" drawn with reference to the Poisson section of the linked code are output simultaneously. Specifically, the mesh generation algorithm is used to divide the control mesh of the equi-geometric analysis in all dimensions to generate mesh data covering the overall structure of the model. This data is then converted into STEP or STL format to achieve mesh visualization of the entire model. Three types of results are output simultaneously: error data, running speed data, and heatmap. The error data records the absolute error and relative error in tabular form, the running speed data records the total time taken from test set input to result output, and the heatmap generates a comparative heatmap of "true solution - graph neural network predicted solution - convolutional neural network predicted solution".

10. The method for solving the Poisson equation using a graph neural network oriented towards isogeometric analysis control grids according to claim 1, characterized in that: In step S7, a Main file is created, integrating functions including dataset reading, proportional partitioning, model training, Poisson solution, and result output into a unified execution module. A visual operation interface is developed based on the Main file, supporting the uploading of NPZ or STEP files. Point cloud files and data files can be uploaded optionally. Point cloud file formats include .pt, .xyz, and .ply, while data file formats include .json and .csv. The interface operation corresponds to the backend function.

Images