A physical constraint-based graph neural network production line design space solving method

Abstract

The application relates to the technical field of intelligent manufacturing and automatic equipment design, and provides a graph neural network production line design space solving method based on physical constraints, which comprises the following steps: dividing vertices according to the functional subsystems of a production line, extracting the design variables associated with each vertex, mapping each design variable into a graph node, and constructing an adjacency matrix based on the engineering semantic association between the vertices; performing normalization processing on the adjacency matrix to generate a normalized adjacency matrix; constructing a graph coding physical information generative adversarial network, wherein the adversarial network comprises a generator and a discriminator based on a graph neural network, the generator takes a random noise vector and the normalized adjacency matrix as input, and outputs a design variable vector; in the training process of the generator, a physical penalty term is introduced into the loss function of the generator; and the trained generator is used to generate a design scheme meeting the physical constraints by inputting a random noise vector.

Description

Technical Field

[0001] This invention relates to the field of intelligent manufacturing and automated equipment design technology, and in particular to a graph neural network-based solution method for production line design space based on physical constraints. Background Technology

[0002] In the field of modern intelligent manufacturing and automated equipment, production lines are typically composed of multiple functional unit devices, robotic modules, processing stations, and conveying mechanisms connected in series or parallel according to a predetermined process flow. The overall production capacity is achieved through cycle time matching, timing coordination, and precision transfer between the units. Compared with single-machine equipment design, the system design at the production line level not only involves the structural and parameter optimization of individual equipment, but also the cycle time coupling, spatial layout coordination, control strategy matching, and comprehensive performance balance between multiple devices. It exhibits characteristics such as a large number of variables, heterogeneous types, complex coupling relationships, and strict constraints, forming a typical high-dimensional, strongly constrained hybrid design space.

[0003] In the design of modern complex automated equipment such as butterfly welding machines, traditional methods mainly rely on engineer experience, queuing theory-based cycle time analysis, and discrete event simulation, supplemented by metaheuristic algorithms such as genetic algorithms and particle swarm optimization for parameter tuning. While these methods are effective in low-dimensional, weakly constrained design scenarios, they easily fall into the "curse of dimensionality" when faced with the high-dimensionality, heterogeneous variables, strong coupling, and complex physical constraints prevalent in contemporary equipment design. This leads to high computational costs, low design efficiency, and heavy reliance on the quality of the initial solution, making it difficult to guarantee global feasibility. In recent years, Generative Adversarial Networks (GANs) have been introduced to automatically generate candidate design solutions. However, the standard GAN generation process lacks explicit modeling of the inherent physical and topological coupling relationships between design parameters, resulting in a lack of coordination at the system level. Furthermore, it cannot intrinsically guarantee that the output samples meet hard constraints such as process cycle time, positioning accuracy, and topological legality, generating a large number of invalid solutions and significantly increasing the burden and iteration cycle of subsequent simulation verification. Therefore, existing technologies are insufficient to achieve efficient and reliable sampling of feasible design spaces, and there is an urgent need for an intelligent design method that can integrate physical constraints and parameter coupling relationships. Summary of the Invention

[0004] To address the aforementioned shortcomings, the present invention aims to propose a graph neural network-based method for solving the design space of production lines based on physical constraints. This method constructs the engineering coupling relationship between explicit modeling design parameters through a vertex-driven variable-level graph, and embeds multiple physical constraint penalty terms such as beat, accuracy, and topology into the generated adversarial training objective, thereby achieving efficient, reliable probabilistic modeling and intelligent sampling of the feasible design space of complex automated equipment.

[0005] To achieve this objective, the present invention adopts the following technical solution: A graph neural network-based method for solving the design space of a production line based on physical constraints includes the following steps: S1: Divide the vertices according to the functional subsystems of the production line, extract the design variables associated with each vertex, map each design variable to a graph node, and construct an adjacency matrix based on the engineering semantic associations between the vertices; S2: Normalize the adjacency matrix to generate a normalized adjacency matrix; S3: Construct a graph-encoded physical information generative adversarial network. The adversarial network includes a generator and a discriminator based on a graph neural network. The generator takes a random noise vector and a normalized adjacency matrix as input and outputs a vector of design variables for the production line. S4: During the training process of the generator, a physical penalty term based on the production line process constraints is introduced into the generator loss function to measure the degree of violation of the process constraints by the generated design variable vector. S5: Using the trained generator, generate a production line design scheme that conforms to physical constraints by inputting a random noise vector.

[0006] Preferably, step S1 includes: Define global design variable vector The following relation is satisfied: ; in, This represents the total number of design variables. Indicates the first Each design variable corresponds to the equipment's geometric dimensions, motion parameters, cycle time, or accuracy index. Define boundary constraint space The following relation is satisfied: ; in, and They represent the first Lower and upper bounds of a design variable; Define a set of physical constraints, which are constrained by a non-negative penalty function. The degree of constraint violation is quantified. The set of physical constraints includes clock limit violation constraints, clock time and integrated time consistency constraints, positioning accuracy constraints, and topology legality constraints. Define feasible design space The following relation is satisfied: ; in, Denotes the nonnegative penalty function, and the feasible design space. This includes a set of design variables that simultaneously satisfy both boundary constraints and physical constraints.

[0007] Preferably, the generator loss function Satisfying the relation: ; in, This represents the total loss that the generator needs to optimize during training. This represents the adversarial loss, used to guide the generator output to approximate the true distribution of design variables. This represents the physical penalty loss, used to measure the degree to which the generated design variables violate process constraints. This represents an adjustable hyperparameter used to balance the weights between the adversarial loss and the physical penalty loss.

[0008] Preferably, in step S1, an adjacency matrix is ​​constructed. include: Based on the engineering semantic associations between the vertices, an adjacency matrix reflecting the coupling relationships between design variables is constructed. The following relation is satisfied: ; ; in, Representing the adjacency matrix The Middle Line number The elements of the column are used to characterize the design variables. and The strength of the correlation between them This represents the first preset weight used to characterize the strong coupling relationship between design variables within the same vertex. This represents a second preset weight used to characterize the correlation between design variables and other relationships between different vertices within the same worksheet. To preserve the inherent characteristics of the graph nodes, an augmented adjacency matrix is ​​constructed. ,in Represents the identity matrix; The normalization process of the adjacency matrix described in step S2 satisfies the following relation: ; in, Represents the normalized adjacency matrix. Representation degree matrix.

[0009] Preferably, in step S3, the forward propagation process of the generator satisfies the following relation: Noise vectors are sampled from a preset distribution, and an initial node feature matrix is ​​generated, satisfying the following relationship: ; ; in, Indicates from A random noise vector sampled from a standard normal distribution. Represents the mapping of fully connected layers. This indicates a reshaping operation. Represents the initial node feature matrix. Indicates the total number of design variables. Indicates the dimension of node features; conduct Layer graph convolution operation, the first Layer node feature matrix The update satisfies the following relation: ; in, This represents the normalized adjacency matrix. Indicates the first The learnable weight matrix of layer graph convolution. Indicates the activation function; The final node features are mapped and scaled to obtain the design variable vector. The following relation is satisfied: ; ; in, Indicates the first The node at the th The final node characteristics of the layer, Represents a linear layer mapping. Indicates normalization to Intermediate variable of the interval This represents the number after scaling to the physical range. One design variable, and They represent the first The lower and upper bounds of each design variable.

[0010] Preferably, the discriminator in step S3 is a multilayer perceptron, and its input is the design variable vector. Output scalar , used to represent the confidence level of a sample's authenticity; Step S4 employs an alternating optimization strategy during training, including: The parameters of the generator are fixed, and the parameters of the discriminator are updated to maximize the discrimination accuracy between real samples and generated samples; Alternatively, the parameters of the discriminator can be fixed, and the parameters of the generator can be updated to minimize the total loss function of the generator. .

[0011] Preferably, in step S4, the physical penalty term is calculated by determining the local physical penalty. Implementation, including: Calculate the first penalty term This is used to reflect whether the theoretical running time exceeds the target cycle time, and satisfies the following relationship: ; in, This represents the theoretical running time calculated based on the actuator's stroke and speed. This indicates the target cycle time given in the process configuration. This represents the portion of the theoretical running time that exceeds the target cycle time; Calculate the second penalty term This is used to reflect whether the target cycle time matches the overall cycle time, satisfying the following relationship: ; in, This represents the combined cycle time obtained based on fine-grained motion segmentation and simulation calculations. This represents the absolute value of the deviation between the target cycle time and the overall cycle time; Calculate the third penalty item This is used to measure whether the welding positioning error exceeds the allowable upper limit, and satisfies the following relationship: ; in, This represents the weld point positioning error derived from the actuator parameters. This indicates the upper limit of the allowable error in the process. This indicates the portion of the solder joint positioning error that exceeds the upper limit of the error; Based on the first penalty item Second penalty item With the third penalty item Calculate local physical penalty The following relation is satisfied: ; in, It reflects the degree to which the generated design variable vector violates the process constraints related to cycle time and accuracy.

[0012] Preferably, the design variable vector includes configuration parameters, behavioral parameters, control parameters, and execution parameters.

[0013] One of the above technical solutions has the following advantages or beneficial effects: This invention constructs an adjacency matrix by partitioning functional subsystem vertices and associating them with engineering semantics. This explicitly transforms the implicit coupling relationships of design parameters in traditional design into a graph structure, enabling each node's information updates to perceive the coordinated changes of its associated variables. This fundamentally solves the problem of standard generative adversarial networks (GANs) lacking system coordination due to the independent processing of variables. Subsequently, the adjacency matrix is ​​normalized to ensure the stability of gradient propagation between variables with different coupling strengths during information aggregation in the graph neural network, avoiding training imbalance caused by excessive differences in connection weights. Based on this, an adversarial framework using a graph neural network as the generator is constructed. Graph convolution operations are used to achieve structure-aware collaborative generation, enabling high-dimensional mapping of latent space sampling along the real engineering dependencies between variables, significantly improving the ability to explore effective solutions in strongly coupled design spaces. A physical penalty term based on actual constraints such as process cycle time and positioning accuracy is embedded in the generator's loss function. This transforms the post-validation constraint checks in traditional methods into real-time guidance signals during training, forcing the generator to actively avoid regions violating physical rules while optimizing distribution matching, thus intrinsically ensuring the feasibility of the output samples. Ultimately, the trained generator can directly generate design schemes that meet multiple physical constraints through noise vector sampling, without relying on time-consuming post-screening and simulation iterations. This realizes a paradigm shift from "generating first and then verifying" to "generating while satisfying constraints," providing a high-quality solution set that is structurally coordinated and engineering interpretable for the intelligent design of complex automated equipment. Attached Figure Description

[0014] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0015] Figure 1 This is a flowchart of the graph neural network-based production line design space solution method provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of a graph-coded physical information generative adversarial network provided in an embodiment of the present invention; Figure 3 This is a schematic diagram of the generator structure provided in an embodiment of the present invention. Detailed Implementation

[0016] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown 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 are only used to explain the present invention, and should not be construed as limiting the present invention.

[0017] In this invention, the terms "comprising," "including," or any other variations thereof are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0018] A graph neural network-based method for solving the design space of a production line based on physical constraints includes the following steps: S1: Divide the vertices according to the functional subsystems of the production line, extract the design variables associated with each vertex, map each design variable to a graph node, and construct an adjacency matrix based on the engineering semantic associations between the vertices; It should be noted that a functional subsystem refers to a physical or logical unit within a piece of equipment on a production line that performs a specific function, such as the robotic arm module, welding station, and material conveying unit in a butterfly welding machine. Each subsystem can be abstracted as an independent vertex. Design variables refer to the set of adjustable parameters describing equipment performance, including geometric dimensions (such as robotic arm length and fixture opening), motion parameters (such as axis speed and acceleration), cycle time (such as single-piece processing cycle), and accuracy indicators (such as positioning repeatability). These variables constitute the basic elements of a high-dimensional design space. Graph nodes are the basic units in graph theory. In step S1, each independent design variable is instantiated as a node, and the set of nodes constitutes the vertex set of the graph structure. Engineering semantic association refers to the coupling relationship between variables established based on engineering documents, assembly drawings, process specifications, etc., such as the spatial dependency between the robotic arm travel variable and the welding station position variable, or the temporal matching relationship between material conveying speed and the overall machine cycle time. The adjacency matrix is ​​a square matrix representing the graph topology. The matrix elements numerically express the strength or weight of the connections between nodes, and the diagonal elements usually represent node self-connections.

[0019] Understandably, functional decomposition achieves structured dimensionality reduction of the design problem, breaking down complex equipment systems into vertices. This ensures high cohesion within each vertex and low coupling between variables, transforming the original high-dimensional mixed variable space into a manageable hierarchical graph structure. Constructing an adjacency matrix based on engineering semantics rather than simple numerical correlations accurately captures the physical coupling and topological dependencies between design parameters. This provides the generator with engineering-interpretable structural priors, avoiding the lack of system coordination in standard generative adversarial networks due to independent processing of variables. This explicit modeling approach allows the generator to perceive the true dependency paths between parameters early in training, significantly improving the effectiveness and rationality of subsequent sampling processes.

[0020] S2: Normalize the adjacency matrix to generate a normalized adjacency matrix; It should be noted that normalization refers to mapping the value range of matrix elements to a uniform scale through mathematical transformation, avoiding the scale inconsistency problem caused by differences in weight allocation in the original adjacency matrix. The normalized adjacency matrix is ​​a new adjacency matrix after transformation, and its element values ​​reflect the relative coupling strength rather than the absolute weight, ensuring that nodes with different connection strengths can obtain balanced gradient updates during information aggregation.

[0021] S3: Construct a graph-encoded physical information generative adversarial network. The adversarial network includes a generator and a discriminator based on a graph neural network. The generator takes a random noise vector and a normalized adjacency matrix as input and outputs a vector of design variables for the production line. It should be noted that Graph-Codified Physical Information Generative Adversarial Network (G-PI-GAN) is a generative deep learning framework that integrates graph structure priors and physical constraints. It includes two adversarial game components: a generator and a discriminator. In this embodiment, the Graph-Codified Physical Information Generative Adversarial Network is as follows: Figure 2 As shown. The generator is a differentiable mapping function based on a graph neural network, whose parameters... The optimization through backpropagation maps low-dimensional random noise to a high-dimensional design variable space. The generator's structure is as follows: Figure 3 As shown. The discriminator is a binary classifier that distinguishes between the real design and the generated design. It typically employs a multilayer perceptron structure, and its parameters are optimized independently. The random noise vector is a latent space vector sampled from a preset probability distribution (such as a standard normal distribution). Its dimension is usually much smaller than that of the design variables, and it serves as the seed input to the generator. The design variable vector is an n-dimensional real vector output by the generator. Each element corresponds to a specific physical value of a design parameter, such as a robot arm stroke of 300mm or a welding speed of 50mm / s.

[0022] Understandably, step S3 establishes an adversarial learning paradigm, where the generator and discriminator engage in a minimax game: the generator attempts to produce realistic designs that the discriminator cannot distinguish, while the discriminator strives to differentiate between real and fake samples. The graph neural network-based generator achieves structure-aware message passing through graph convolution operations, ensuring that the generation process of each design variable not only relies on noise vectors but also aggregates collaborative information from its engineering-related variables, thereby guaranteeing the coordination of the generated scheme at the system level. The normalized adjacency matrix serves as the generator's static input, forcing the network to follow a pre-defined engineering coupling topology. This allows latent space sampling to perform high-dimensional mapping along real engineering dependencies, significantly improving the ability to explore effective solutions in strongly coupled design spaces. The discriminator provides adversarial loss signals, guiding the generator to approximate the data manifold distribution of the real design.

[0023] S4: During the training process of the generator, a physical penalty term based on the production line process constraints is introduced into the generator loss function to measure the degree of violation of the process constraints by the generated design variable vector. It should be noted that the physical penalty term is a differentiable mathematical function. Its input is the design variable vector output by the generator, and its output is a non-negative scalar value. The magnitude of the value is positively correlated with the degree of violation of process constraints. Process constraints refer to the set of physical rules that equipment operation must meet, including cycle time constraints (output requirements per unit time), accuracy constraints (positioning error tolerance), and topological constraints (geometric interference checks). These constraints manifest as hard boundaries or soft penalties in engineering design problems. The degree of violation is quantified through engineering calculation models, such as the portion of theoretical operating time exceeding the target cycle time, or the portion of positioning error exceeding the allowable upper limit. The generator loss function guides the overall goal of generator parameter updates and is composed of a weighted sum of adversarial loss and physical penalty loss, with the balance between the two controlled by hyperparameters. The adversarial loss comes from discriminator feedback and measures the difference between the generated sample and the real sample distribution, usually in the form of binary cross-entropy. The physical penalty loss is the average value of the physical penalty term over a batch of samples, reflecting the overall physical feasibility of the generative design.

[0024] Understandably, standard GAN training relies solely on adversarial loss, leading the generator to potentially output designs that violate physical rules in order to deceive the discriminator. By explicitly embedding a physical penalty term into the loss function, constraint satisfaction is transformed from post-validation into an intrinsic guiding signal during the training process. A positive value is generated when the generator produces a super-beat design, and this gradient backpropagation forces the generator to adjust its parameters; a penalty is generated when the positioning accuracy is insufficient, guiding the generator to optimize the robot's parameter set. This mechanism allows the generator to actively avoid regions that violate physical rules while optimizing distribution matching, achieving a paradigm shift from "generate first, then validate" to "generate while simultaneously satisfying constraints," significantly reducing invalid sample generation and improving sampling efficiency.

[0025] S5: Using the trained generator, generate a production line design scheme that conforms to physical constraints by inputting a random noise vector.

[0026] It should be noted that training completion refers to the generator and discriminator reaching an equilibrium state through alternating optimization. At this point, the generator loss converges and the discriminator's accuracy in distinguishing between real and generated samples stabilizes around 0.5, indicating that the generated sample distribution is difficult to distinguish from the real distribution. Random noise vectors are independently and identically distributed samples from a fixed prior distribution (standard normal distribution). Each noise vector corresponds to a point in the latent space, which, when mapped to the design space, becomes a candidate solution. Meeting physical constraints means that the generated design solution, after physical penalty term calculation, simultaneously satisfies process requirements such as cycle time constraints and accuracy constraints. The design solution is a parameter set containing the specific values ​​of all design variables and can be directly used for engineering drawings or simulation verification.

[0027] Preferably, step S1 includes: Define global design variable vector The following relation is satisfied: ; in, This represents the total number of design variables. Indicates the first Each design variable corresponds to the equipment's geometric dimensions, motion parameters, cycle time, or accuracy index. Define boundary constraint space The following relation is satisfied: ; in, and They represent the first Lower and upper bounds of a design variable; Define a set of physical constraints, which are constrained by a non-negative penalty function. The degree of constraint violation is quantified. The set of physical constraints includes clock limit violation constraints, clock time and integrated time consistency constraints, positioning accuracy constraints, and topology legality constraints. Define feasible design space The following relation is satisfied: ; in, Denotes the nonnegative penalty function, and the feasible design space. This includes a set of design variables that simultaneously satisfy both boundary constraints and physical constraints.

[0028] It should be noted that the global design variable vector is a collection of all adjustable parameters of the equipment. A 3D real vector, each component This corresponds to a specific engineering parameter, such as a robotic arm. Shaft travel, welding pressure, or material conveying speed—this vector constitutes the complete parameter space searched by the algorithm. Boundary constraint space. This is a super-rectangular region spanned by the upper and lower bounds of parameters determined by engineering drawings and configuration tables. Its function is to trim the infinite design space into a reasonable engineering range, avoiding the waste of computational resources by the algorithm on physically unrealizable parameter combinations. The set of physical constraints is a mathematical abstraction of the process rules that the equipment must satisfy in actual operation, containing four core constraints: cycle time exceedance constraints to ensure that the theoretical running time does not exceed the production cycle time requirement; cycle time and integrated time consistency constraints to ensure that the target cycle time matches the fine-grained simulation cycle; positioning accuracy constraints to limit welding errors within assembly tolerances; and topological legality constraints to eliminate geometric interference and structural conflicts. Non-negative penalty function. It is a differentiable mathematical function that quantifies the degree of constraint violation into a scalar value. A positive value indicates a violation, and a zero value indicates complete satisfaction. This function is key to achieving constraint optimization. Feasible design space It is the intersection of the boundary constraint space and the physical constraint set, that is, the set of design schemes that simultaneously satisfy the upper and lower bounds of parameters and all process constraints, which constitutes the target region pursued by the algorithm.

[0029] Understandably, the above formal definitions transform vague engineering experience requirements into a rigorous mathematical set, providing graph neural networks with a computable and optimizable clear objective. The global design variable vector unifies heterogeneous parameters (length, velocity, time, force, etc.) into a real-number vector space, enabling deep learning models to handle mixed-type design variables. The boundary constraint space X_box reduces the search range by several orders of magnitude through hyperrectangular boundaries, preventing the generator from exploring invalid parameter regions and improving sampling efficiency. The physical constraint set embeds process requirements into the training objective through a differentiable penalty function, allowing constraint information to propagate back in the form of gradients, guiding the generator to actively avoid infeasible regions while optimizing distribution matching. This transforms traditional design problems relying on engineer experience into automatically solvable mathematical optimization problems, enabling the generator to perceive which regions are physically permissible and which are prohibited from the early stages of training, thus achieving a crucial shift from data-driven to physical information fusion.

[0030] Preferably, the generator loss function Satisfying the relation: ; in, This represents the total loss that the generator needs to optimize during training. This represents the adversarial loss, used to guide the generator output to approximate the true distribution of design variables. This represents the physical penalty loss, used to measure the degree to which the generated design variables violate process constraints. This represents an adjustable hyperparameter used to balance the weights between the adversarial loss and the physical penalty loss.

[0031] It should be noted that the generator loss function It is the objective function minimized by the graph neural network generator during backpropagation optimization. Its magnitude directly determines the direction and magnitude of the generator parameter updates, and it is the core mathematical expression connecting adversarial learning and physical constraints. Adversarial loss Derived from the standard game theory framework of generative adversarial networks, this approach is constructed by comparing the differences between generated and real samples in the discriminator's output. It typically employs a binary cross-entropy form, which drives the generator to learn the probability distribution of real design data, ensuring that the generated scheme statistically matches historically excellent designs. (Physical penalty loss) It is a physical penalty item. The empirical expectation on the generated batch reflects the severity of the overall violation of process constraints by the generation scheme, forcing the generator to avoid infeasible regions. Adjustable hyperparameters. It is a positive real scaling factor that adjusts the relative importance of the physical penalty loss in the total loss through multiplicative action. Its value directly affects the generator's trade-off between distribution fitting and constraint satisfaction.

[0032] Preferably, in step S1, an adjacency matrix is ​​constructed. include: Based on the engineering semantic associations between the vertices, an adjacency matrix reflecting the coupling relationships between design variables is constructed. The following relation is satisfied: ; ; in, Represents the adjacency matrix A of the th element. Line number The elements of the column are used to characterize the design variables. and The strength of the correlation between them This represents the first preset weight used to characterize the strong coupling relationship between design variables within the same vertex. This represents a second preset weight used to characterize the correlation between design variables and other relationships between different vertices within the same worksheet. To preserve the inherent characteristics of the graph nodes, an augmented adjacency matrix is ​​constructed. ,in Represents the identity matrix; The normalization process of the adjacency matrix described in step S2 satisfies the following relation: ; in, Represents the normalized adjacency matrix. Representation degree matrix.

[0033] It should be noted that, These are the specific matrix elements in the adjacency matrix, and their numerical values ​​are directly quantified variables. and The coupling strength between nodes determines the node's strength during message passing in a graph neural network. From the node The larger the value of the received information, the closer the co-change between the two variables. First preset weight. This is a manually set strong coupling coefficient, usually set to 1.0, used to characterize the high dependence between design variables belonging to the same functional subsystem vertices, such as the linkage between stroke and speed within a robotic arm module. This weight significantly enhances the information interaction intensity of related variables in graph convolution. Second preset weight This is a moderate correlation coefficient, ranging from 0.3 to 0.5, used to describe weak coupling relationships between design variables that cross vertices but share a process context, such as the spatial relationship between the welding station location and the robot's reference coordinates. This weight appropriately conveys engineering semantics without overly constraining them. Augmented Adjacency Matrix It adds an identity matrix to the original adjacency matrix. The identity matrix has diagonal elements of 1 and off-diagonal elements of 0. This operation forces each node to establish a connection with itself, preserving the original features of the node and preventing node features from being completely submerged by neighbor information after multi-layer graph convolution. Degree matrix It is a diagonal matrix, and its diagonal elements are... Equal to the augmented adjacency matrix of the th The sum of all elements in the row reflects the node. The total connection strength is used as a scaling factor in normalization to eliminate the influence of differences in node degree. Symmetric normalization formula. By using the inverse square root of each multiplication degree matrix, the sum of the squares of each row after normalization is made equal to 1, thus achieving proportional scaling of message passing. This mathematical transformation ensures that graph convolution operations maintain stable gradients between variables with different coupling strengths.

[0034] It is understandable that a weighted hierarchical strategy can be used to accurately encode engineering semantics into the graph topology. and Differentiated assignments allow the generator to clearly distinguish between strong coupling and weak association, avoiding the loss of important dependencies caused by standard GANs processing all variable relationships equally. Augmented adjacency matrix The construction preserves node self-loops, ensuring that graph convolutional layers do not lose the independent characteristics of variables when aggregating neighbor information. This is crucial for maintaining the individual rationality of design variables; otherwise, after multiple convolutions, all node features tend to be consistent, losing parameter distinctiveness. Degree matrix The introduction of quantification quantifies the local influence of each node. After normalization, the weight of individual neighbor information received by high-degree nodes (connecting multiple variables) is weakened, while the weight of self-information of low-degree nodes (isolated variables) is relatively strengthened. This adaptive balancing mechanism ensures that all variables are treated fairly in information updates, preventing high-degree nodes from dominating the training process. Symmetric normalization constrains the eigenvalues ​​of the adjacency matrix to... The interval effectively suppresses gradient explosion and vanishing phenomena in graph convolution. No matter how the network depth increases, the gradient propagation amplitude remains constant, thus ensuring that subsequent adversarial training can converge smoothly and that generator parameter optimization will not diverge due to matrix exponentiation.

[0035] Preferably, in step S3, the forward propagation process of the generator satisfies the following relation: Noise vectors are sampled from a preset distribution, and an initial node feature matrix is ​​generated, satisfying the following relationship: ; ; in, Indicates from A random noise vector sampled from a standard normal distribution. Represents the mapping of fully connected layers. This indicates a reshaping operation. Represents the initial node feature matrix. Indicates the total number of design variables. Indicates the dimension of node features; conduct Layer graph convolution operation, the first Layer node feature matrix The update satisfies the following relation: ; in, This represents the normalized adjacency matrix. Indicates the first The learnable weight matrix of layer graph convolution. Indicates the activation function; The final node features are mapped and scaled to obtain the design variable vector. The following relation is satisfied: ; ; in, Indicates the first The node at the th The final node characteristics of the layer, Represents a linear layer mapping. Indicates normalization to Intermediate variable of the interval This represents the number after scaling to the physical range. One design variable, and They represent the first The lower and upper bounds of each design variable.

[0036] It should be noted that the preset distribution is usually the standard normal distribution. ,in yes A 3D identity matrix, whose distribution ensures that each dimension of the noise vector is independent and identically distributed with a mean of zero and a variance of one, provides the generator with diverse random seeds. Fully connected layer mapping. It is a neural network layer that linearly transforms a low-dimensional noise vector to a high-dimensional feature space, and includes a weight matrix. and bias vector Its output Its function is to expand randomness and align it with the dimensions of design variables. Reshaping operation. The output of the fully connected layer The row vectors are rearranged as follows A 3D matrix is ​​used to assign a row vector to each design variable as the initial node feature, thus adapting the vector to the graph structure. Initial node feature matrix This is the input representation of a graph neural network, each row Corresponding design variables The initial hidden state provides a feature carrier for subsequent graph convolution. Graph convolution is a feature aggregation and transformation operation performed on a graph structure, through... To aggregate neighbor information, and then through This involves implementing feature transformations to integrate information from engineering-related variables for each node's features. A learnable weight matrix is ​​then implemented. It is the first The parameters to be optimized for layer graph convolution have the following dimensions: Its value is updated during training via backpropagation, and its function is to extract higher-order coupling patterns from the aggregated neighbor information. Activation function The ReLU (Rectified Linear Unit) function is used, i.e. This function introduces non-linear expressive power, preventing deep networks from degenerating into linear models. Linear layer mapping. It is a single-layer neural network transformation, which will Projecting 1D node features onto 1D, i.e. This generates independent outputs for each variable. The Sigmoid function is a sigmoid activation function, expressed as: This compresses the output to an open interval (0,1), facilitating subsequent linear scaling to any physical range. Intermediate variables. It is a normalized value in the (0,1) interval, which serves as a scaling factor to determine the specific position of the design variable between its upper and lower bounds. Physical range scaling is achieved through linear transformation. This implementation maps normalized values ​​to the actual parameter range allowed by the engineering process; the operation is differentiable and supports gradient backpropagation.

[0037] Understandably, this forward propagation process achieves an end-to-end mapping from random noise to a complete design solution. Noise sampling provides randomness in latent space exploration, ensuring diversity in generated solutions; fully connected layers and reshaping operations bridge the noise to the graph structure, enabling low-dimensional seeds to extend to high-dimensional variable spaces while preserving graph topological adaptability. Layered graph convolution is the core transformation engine; each layer passes through... Aggregating neighbor features enables explicit modeling of engineering coupling between variables. For example, in convolution, the stroke variable of a robotic arm receives collaborative information from related variables such as speed and pressure. This structure-aware message passing ensures that the generated scheme remains consistent at the system level. Learnable weights Coupling patterns are extracted layer by layer, from low-level physical rules to high-level system behavior; the deep structure supports the capture of complex nonlinear relationships. Finally, mapping and scaling decode the abstract features output by the graph network into specific physical parameters. By ensuring that the parameters are within the effective range and using linear scaling to achieve engineering unit conversion, the entire generation process satisfies both mathematical constraints and engineering realities, and can be directly used for simulation verification without post-processing.

[0038] Preferably, the discriminator in step S3 is a multilayer perceptron, and its input is the design variable vector. Output scalar , used to represent the confidence level of a sample's authenticity; Step S4 employs an alternating optimization strategy during training, including: The parameters of the generator are fixed, and the parameters of the discriminator are updated to maximize the discrimination accuracy between real samples and generated samples; Alternatively, the parameters of the discriminator can be fixed, and the parameters of the generator can be updated to minimize the total loss function of the generator. .

[0039] It should be noted that a multilayer perceptron is a type of feedforward neural network composed of multiple fully connected layers stacked together. Each layer contains a linear transformation and an activation function, and its purpose is to approximate complex function mappings. In this embodiment, the discriminator adopts a three-layer structure: the input layer dimension is the total number of design variables. Both hidden layers have a dimension of 1. The output layer has a dimension of 1, and the activation function used is... and Combinations are used to distinguish between real and generated designs. Discriminator parameters. This is the set of all learnable weights in the network, including the weight matrix and bias vector. These parameters are updated by maximizing the discrimination accuracy. Generator parameters These are all the trainable parameters of the graph neural network generator, which are frozen during the discriminator update phase. The alternating optimization strategy is the standard training paradigm for generative adversarial networks, executed in two independent phases in each iteration: Phase 1 is fixed. renew Phase Two Fixed renew This avoids training instability caused by simultaneous updates of both. Discrimination accuracy refers to the proportion of real samples and generated samples correctly distinguished by the discriminator; its value is output by the discriminator. The calculation is based on comparison with the sample labels, with the goal of making the output of the real sample close to 1 and the output of the generated sample close to 0.

[0040] Understandably, the discriminator, acting as a multilayer perceptron, can learn the complex nonlinear decision boundaries between design variables, mapping the high-dimensional design space to a single confidence score. This structure is simple, efficient, and easy to train. The fundamental reason for employing an alternating optimization strategy instead of synchronous updates is that the generator and discriminator constitute a zero-sum game. Updating their loss functions simultaneously would cause mutual interference, leading to chaotic gradient signals, oscillations during training, and even failure to converge. When the generator updates the discriminator, the discriminator receives both the real sample label 1 and the generated sample label 0, maximizing its discriminative ability through cross-entropy loss, making the discriminator an accurate judge of the authenticity of the design scheme. When the discriminator updates the generator, the generator receives the discriminator's score for the generated samples as feedback, minimizing the total loss... Improving the quality of generation allows the generator to gradually approach the true design distribution under the guidance of the discriminator. This alternating mechanism forms a stable adversarial game, where the discriminator continuously improves its discrimination ability, and the generator continuously enhances its deception ability, eventually reaching a Nash equilibrium. At this point, the discriminator cannot distinguish between real and generated samples, and the design scheme output by the generator is consistent with the real design in statistical properties, while also possessing engineering feasibility due to the embedding of physical penalty terms.

[0041] Preferably, in step S4, the physical penalty term is calculated by determining the local physical penalty. Implementation, including: Calculate the first penalty term This is used to reflect whether the theoretical running time exceeds the target cycle time, and satisfies the following relationship: ; in, This represents the theoretical running time calculated based on the actuator's stroke and speed. This indicates the target cycle time given in the process configuration. This represents the portion of the theoretical running time that exceeds the target cycle time; Calculate the second penalty term This is used to reflect whether the target cycle time matches the overall cycle time, satisfying the following relationship: ; in, This represents the combined cycle time obtained based on fine-grained motion segmentation and simulation calculations. This represents the absolute value of the deviation between the target cycle time and the overall cycle time; Calculate the third penalty item This is used to measure whether the welding positioning error exceeds the allowable upper limit, and satisfies the following relationship: ; in, This represents the weld point positioning error derived from the actuator parameters. This indicates the upper limit of the allowable error in the process. This indicates the portion of the solder joint positioning error that exceeds the upper limit of the error; Based on the first penalty item Second penalty item With the third penalty item Calculate local physical penalty The following relation is satisfied: ; in, It reflects the degree to which the generated design variable vector violates the process constraints related to cycle time and accuracy.

[0042] It should be noted that local physical penalties It is a differentiable scalar function that quantifies the degree of violation of process constraints related to the robot's parameter set into a single value, which is directly embedded into the generator loss function as a physical constraint guidance signal. The first penalty term... To address the cycle time limit constraint, the following approach is adopted: The design employs a penalty of zero when the theoretical runtime does not exceed the target tick time; otherwise, it accumulates linearly. This design ensures the gradient is always 1 in the violation region and zero in the non-violation region, guiding the generator to accurately locate and compress the violation region. Theoretical runtime It is the minimum possible running time calculated based on the kinematic parameters of the actuator, typically modeled as the sum of the motion times of each axis. This calculation is differentiable with respect to design variables and supports gradient backpropagation. Target cycle time. This is the production cycle time requirement given by the process planning; it is a fixed hyperparameter and does not change with design variables. Second penalty item. The matching degree between the target cycle time and the overall cycle time is measured using absolute difference to avoid the cancellation of positive and negative deviations, ensuring that the generator simultaneously addresses both excessively fast and slow cycle times. Overall cycle time Obtained through fine-grained motion simulation, reflecting a more realistic production cycle, this value can be calculated using a discrete event simulation engine. To improve training efficiency, a neural network surrogate model can be used for approximation, ensuring differentiability. Third penalty term. To address positioning accuracy constraints, use In this model, errors within the allowable range are not penalized, while those exceeding the allowable range accumulate linearly, guiding the generator to actively reduce the positioning error. Positioning error Derived from the actuator parameters, it is usually calculated based on the kinematic error transmission model, taking into account factors such as the repeatability of the robotic arm, thermal deformation, and transmission clearance.

[0043] Understandably, the three penalty items together constitute a complete picture of local process constraints. Ensure that the generated plan meets production capacity requirements and prevent the cycle time from exceeding the limit due to the robot arm's slow speed; Ensure that the target cycle time matches the simulation cycle and avoid production congestion or waiting waste caused by mismatch in the rhythm of upstream and downstream processes; Directly control welding quality to ensure that the weld point positioning accuracy meets assembly requirements. These three factors are formed through linear superposition. This allows the generator to consider three dimensions—capacity, collaboration, and quality—simultaneously during optimization, preventing optimization of a single constraint from causing deterioration in other dimensions. The absolute value operation ensures that all penalty terms are non-negative and that gradient signals are generated only when constraints are violated. This sparse gradient characteristic allows the generator to be trained efficiently, focusing on infeasible regions rather than wasting computational resources on feasible regions. Furthermore, the differentiable design of the penalty function allows the gradient to... Backpropagation to the generator parameters allows constraint information to influence relevant design variables along the engineering coupling path, for example... The gradient will simultaneously adjust the robot's stroke and speed parameters to achieve system-level collaborative optimization.

[0044] Preferably, the design variable vector includes configuration parameters, behavioral parameters, control parameters, and execution parameters.

[0045] It should be noted that the configuration parameters cover the hardware composition, equipment selection, and topological connections of the production line, used to characterize the material basis, spatial layout, and static structural characteristics of the equipment, including variables such as robot model, conveyor belt specifications, workstation spatial coordinates, and physical connection methods between equipment; behavioral parameters include timing logic, operating cycle time, and process flow, used to describe the time characteristics, sequence of operations, and work rhythm of the equipment's dynamic operation, including variables such as target cycle time, single operation duration, parallel operation window, and buffer time between operations; control parameters include the logic strategy, process settings, and scheduling rules of the control system, used to define the decision-making of the control system. The mechanism, operation mode, and optimization objectives include variables such as PLC control logic thresholds, process temperature setpoints, pressure control thresholds, material scheduling priority rules, and fault handling strategies; execution parameters include the kinematic characteristics, dynamic characteristics, and positioning accuracy of the actuator, used to quantify the physical motion capability, mechanical performance, and processing quality indicators of the actuator, including variables such as the maximum linear speed of the robotic arm, maximum acceleration, rated torque of the servo motor, repeatability tolerance, and transmission backlash compensation value; the above four types of parameters together constitute a complete design space description, enabling the graph neural network to perceive the coupling relationship and constraint dependency between parameters at different abstraction levels.

[0046] Understandably, by dividing high-dimensional heterogeneous design variables into four categories of parameters—configuration, behavior, control, and execution—according to engineering semantics, structured dimensionality reduction and hierarchical modeling of complex production line design problems are achieved. This enables the generator of the graph neural network to decode layer by layer along the functional dependency path during latent space sampling. Configuration parameters determine the spatial topology and hardware foundation of the equipment, providing physical entity nodes for graph structure construction. Behavioral parameters constrain the operating rhythm in the time dimension and are directly related to the beat constraint. Control parameters provide intelligent decision-making logic and are bound to process rules. Execution parameters ensure the feasibility of physical actions and participate in physical penalty calculation through kinematic models. The four types of parameters mutually constrain each other to form a complete engineering constraint network. This hierarchical organization allows the generator to adopt differentiated information aggregation strategies for different parameter categories during graph convolution. Configuration parameters and execution parameters achieve hardware-motion coordination through strong coupling edges, while behavioral parameters and control parameters achieve temporal-policy matching through moderately correlated edges. This avoids the defects of standard GANs, such as mixed dependent variable types and independent processing of various dimensions, which lead to a lack of system coordination and physical feasibility in the generated scheme. This ensures that the output design scheme simultaneously possesses structural rationality, temporal feasibility, control reliability, and physical feasibility at the system level.

[0047] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "illustrative embodiment," "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 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.

[0048] Although embodiments of the invention have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

Claims

1. A graph neural network-based method for solving the design space of a production line based on physical constraints, characterized in that, Includes the following steps: S1: Divide the vertices according to the functional subsystems of the production line, extract the design variables associated with each vertex, map each design variable to a graph node, and construct an adjacency matrix based on the engineering semantic associations between the vertices; S2: Normalize the adjacency matrix to generate a normalized adjacency matrix; S3: Construct a graph-encoded physical information generative adversarial network. The adversarial network includes a generator and a discriminator based on a graph neural network. The generator takes a random noise vector and a normalized adjacency matrix as input and outputs a vector of design variables for the production line. S4: During the training process of the generator, a physical penalty term based on the production line process constraints is introduced into the generator loss function to measure the degree of violation of the process constraints by the generated design variable vector. S5: Using the trained generator, generate a production line design scheme that conforms to physical constraints by inputting a random noise vector.

2. The method for solving the production line design space based on physical constraints using graph neural networks according to claim 1, characterized in that, Step S1 includes: Define global design variable vector The following relation is satisfied: ； in, This represents the total number of design variables. Indicates the first Each design variable corresponds to the equipment's geometric dimensions, motion parameters, cycle time, or accuracy index. Define boundary constraint space The following relation is satisfied: ； in, and They represent the first Lower and upper bounds of a design variable; Define a set of physical constraints, which are constrained by a non-negative penalty function. The degree of constraint violation is quantified. The set of physical constraints includes clock limit violation constraints, clock time and integrated time consistency constraints, positioning accuracy constraints, and topology legality constraints. Define feasible design space The following relation is satisfied: ； in, Denotes the nonnegative penalty function, and the feasible design space. This includes a set of design variables that simultaneously satisfy both boundary constraints and physical constraints.

3. The method for solving the production line design space based on physical constraints using graph neural networks according to claim 1, characterized in that, The generator loss function Satisfying the relation: ； in, This represents the total loss that the generator needs to optimize during training. This represents the adversarial loss, used to guide the generator output to approximate the true distribution of design variables. This represents the physical penalty loss, used to measure the degree to which the generated design variables violate process constraints. This represents an adjustable hyperparameter used to balance the weights between the adversarial loss and the physical penalty loss.

4. The method for solving the production line design space based on physical constraints using graph neural networks according to claim 1, characterized in that, In step S1, construct the adjacency matrix. include: Based on the engineering semantic associations between the vertices, an adjacency matrix reflecting the coupling relationships between design variables is constructed. The following relation is satisfied: ； ； in, Representing the adjacency matrix The Middle Line number The elements of the column are used to characterize the design variables. and The strength of the correlation between them This represents the first preset weight used to characterize the strong coupling relationship between design variables within the same vertex. This represents a second preset weight used to characterize the correlation between design variables and other relationships between different vertices within the same worksheet. To preserve the inherent characteristics of the graph nodes, an augmented adjacency matrix is ​​constructed. ,in Represents the identity matrix; The normalization process of the adjacency matrix described in step S2 satisfies the following relation: ； in, Represents the normalized adjacency matrix. Representation degree matrix.

5. The method for solving the production line design space based on physical constraints using graph neural networks according to claim 1, characterized in that, In step S3, the forward propagation process of the generator satisfies the following relation: Noise vectors are sampled from a preset distribution, and an initial node feature matrix is ​​generated, satisfying the following relationship: ； ； in, Indicates from A random noise vector sampled from a standard normal distribution. Represents the mapping of fully connected layers. This indicates a reshaping operation. Represents the initial node feature matrix. Indicates the total number of design variables. Indicates the dimension of node features; conduct Layer graph convolution operation, the first Layer node feature matrix The update satisfies the following relation: ； in, This represents the normalized adjacency matrix. Indicates the first The learnable weight matrix of layer graph convolution. Indicates the activation function; The final node features are mapped and scaled to obtain the design variable vector. The following relation is satisfied: ； ； in, Indicates the first The node at the th The final node characteristics of the layer, Represents a linear layer mapping. Indicates normalization to Intermediate variable of the interval This represents the number after scaling to the physical range. One design variable, and They represent the first The lower and upper bounds of each design variable.

6. The method for solving the production line design space based on physical constraints using graph neural networks according to claim 1, characterized in that, The discriminator mentioned in step S3 is a multilayer perceptron, and its input is the design variable vector. Output scalar , used to represent the confidence level of the sample's authenticity; Step S4 employs an alternating optimization strategy during training, including: The parameters of the generator are fixed, and the parameters of the discriminator are updated to maximize the discrimination accuracy between real samples and generated samples; Alternatively, the parameters of the discriminator can be fixed, and the parameters of the generator can be updated to minimize the total loss function of the generator. .

7. The method for solving the production line design space based on physical constraints using graph neural networks according to claim 1, characterized in that, In step S4, the physical penalty term is calculated by determining the local physical penalty. Implementation, including: Calculate the first penalty term This is used to reflect whether the theoretical running time exceeds the target cycle time, and satisfies the following relationship: ； in, This represents the theoretical running time calculated based on the actuator's stroke and speed. This indicates the target cycle time given in the process configuration. This represents the portion of the theoretical running time that exceeds the target cycle time; Calculate the second penalty term This is used to reflect whether the target cycle time matches the overall cycle time, satisfying the following relationship: ； in, This represents the combined cycle time obtained based on fine-grained motion segmentation and simulation calculations. This represents the absolute value of the deviation between the target cycle time and the overall cycle time; Calculate the third penalty item This is used to measure whether the welding positioning error exceeds the allowable upper limit, and satisfies the following relationship: ； in, This represents the weld point positioning error derived from the actuator parameters. This indicates the upper limit of the allowable error in the process. This indicates the portion of the solder joint positioning error that exceeds the upper limit of the error; Based on the first penalty item Second penalty item With the third penalty item Calculate local physical penalty The following relation is satisfied: ； in, It reflects the degree to which the generated design variable vector violates the process constraints related to cycle time and accuracy.

8. The method for solving the production line design space based on physical constraints using graph neural networks according to claim 1, characterized in that, The design variable vector includes configuration parameters, behavior parameters, control parameters, and execution parameters.

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