Vehicle body welding multi-physics field depth modeling and geometric accuracy generation type prediction method

Abstract

The invention discloses a vehicle body welding multi-physics field depth modeling and geometric accuracy generation type prediction method, which comprises the following steps of: 1, analyzing and representing a spot welding connection process thin plate deformation mechanism: 11) determining the total strain of a thin plate through coupling iteration of electric field analysis, thermal field analysis, phase change analysis and mechanical analysis; 12) representing a strain mechanism as a 7-order tensor, and obtaining a total strain based on the 7-order tensor; 2, geometric precision high-order interactive representation of the multi-curved-surface part welding process: 21) single-curved-surface part strain low-rank representation based on CP tensor decomposition; 22) performing high-order interactive characterization on geometric deformation of the multi-curved-surface part in the multi-spot welding process; step 3, a body-in-white assembly geometric accuracy posteriori inference and generation mechanism: 31) constructing a body-in-white assembly geometric accuracy posteriori state inference model; 32) a body-in-white assembly geometric accuracy generation mechanism; and 4, constructing a body-in-white assembly geometric accuracy random dynamic evolution prediction model.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of automobile production and manufacturing, and particularly relates to a body welding multi-physical field deep modeling and geometric precision generated prediction method. BACKGROUND

[0002] The body-in-white assembly is a complex structure formed by a series of single curved surface parts connected through a welding process. In the welding process, each single curved surface part is connected into the body-in-white assembly through resistance spot welding. The initial geometric precision deviation of the single part is transmitted to the adjacent part through the welding point connection and the contact surface extrusion mechanism. This transmission mechanism is related to the geometric characteristics of the curved surface part, the material properties, the alignment accuracy, the welding point position and other factors. In addition, the coupling effect of the thermal and force fields in the resistance spot welding process will cause local deformation near the welding point, and this deformation will be transmitted to other areas through the structural characteristics and constraint relationships of each single part. Since the body-in-white assembly involves a large number of parts and welding points, the deviation transmission between different parts, deformation accumulation and spot welding sequence will all affect the deformation of the assembly, making the final geometric precision of the assembly present a complex distribution pattern and evolution characteristics.

[0003] In the prior art, different levels of parts (stamped parts, sub-assemblies, assemblies) are all regarded as single curved surface manifold structures. When the geometric precision of the assembly needs to be analyzed, only the measurement point cloud of the assembly itself can be used for analysis and characterization, and the influence of the geometric precision of the single part and the welding process cannot be included in the modeling and prediction process. In addition, even if the geometric precision of each single part and the influence of the welding process are included in the modeling, since the body-in-white assembly involves a large number of welding points, it is not realistic to obtain and process such a large amount of process parameter information for effective modeling and analysis.

[0004] To solve the above problems, in recent years, some researches have attempted to use deep learning and other data-driven methods. However, the conventional deep learning model is often regarded as a "black box" when dealing with such complex industrial manufacturing problems. Although this kind of model has strong fitting ability, its network structure has nothing to do with the actual welding physical mechanism, which leads to low generalization ability. When the working conditions change, the model performance may decrease sharply, and the data dependence is strong, which requires a large amount of data covering all working conditions for training, and the cost is high. SUMMARY

[0005] In view of this, the purpose of this invention is to provide a method for multiphysics-based deep modeling and generative prediction of geometric accuracy in vehicle body welding. The core technical idea of ​​this invention lies in proposing and implementing a "gray box" deep modeling framework that integrates welding physics priors, aiming to overcome the inherent defects of traditional "black box" data-driven methods, such as weak generalization ability and strong data dependence. This framework does not simply fit data using neural networks, but rather extracts the core mechanisms of the complex welding physics process into a structured prior constraint, abstracts it, and constructs it as the internal structure and computational layer of a deep neural network, thereby achieving multiphysics-based deep modeling and generative prediction of geometric accuracy for the vehicle body.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A method for multiphysics-based depth modeling and geometrically accurate generative prediction of vehicle body welding includes the following steps:

[0008] Step 1: Analysis and representation of the deformation mechanism of thin plates in spot welding connection process:

[0009] 11) The total strain ε of the thin plate is determined by coupled iteration of electric field analysis, thermal field analysis, phase transition analysis, and mechanical analysis. (n) ;

[0010] 12) Represent the strain mechanism as a 7th-order tensor And based on 7th order tensor The total strain ε is obtained (n) The 7th order tensor The state vector set consists of 6 state vector dimensions and 1 output dimension. It contains key physical field information including stress, temperature, and solid fraction;

[0011] Step 2: High-order interactive representation of geometric accuracy in the welding process of multi-curved parts:

[0012] 21) Low-rank characterization of strain in monosurface components based on CP tensor decomposition, including:

[0013] For higher-order tensors Perform CP decomposition to generate factor matrix A (i) and output weight matrix W;

[0014] Nonlinear coupling of physical field increments is achieved through the Hadamard product (⊙).

[0015] 22) High-order interactive characterization of geometric deformation during multi-spot welding of multi-curved parts, including:

[0016] Define the generalized Green's function matrix Establish displacement increment Δu (n) With strain increment Δε(n) the mapping relationship of the M surfaces and P

[0017] introducing high-order tensors describing M surfaces and P m the interaction between the welding points, and using a welding point interaction mode matrix describing the coupling relationship between the welding points in the same surface;

[0018] Step three: white body assembly geometry precision post-inference and generation mechanism:

[0019] 31) Construct a white body assembly geometry precision post-state inference model, including:

[0020] Map a three-dimensional point on the white body to a point in the two-dimensional parameter space through the coordinate mapping network φ;

[0021] Construct a neural network f based on attention mechanism to learn the 0 form v and the metric tensor g;

[0022] Combine the exterior differential operator d and the Hodge star operator to extract high-order geometric features and construct a white body assembly geometry precision post-state inference model;

[0023] 32) White body assembly geometry precision generation mechanism, including:

[0024] Define a unified state vector x (n) , update the displacement field and physical field state through a multilinear interaction layer ;

[0025] Step four: construct a white body assembly geometry precision random dynamic evolution prediction model

[0026] Model the random dynamic evolution of white body assembly geometry precision on the time scale based on deep neural random process ζ t ;

[0027] Based on the numerical integration method of neural network, calculate the high-order Taylor coefficients through Taylor mode automatic differentiation

[0028] Use the intermediate state prediction network to generate the Lagrange remainder of the p-order Taylor expansion, and output the geometry precision prediction value U t .

[0029] Further, in the step 11), the methods of electric field analysis, thermal field analysis, phase change analysis and mechanical analysis are respectively:

[0030] The electric field analysis updates the conductivity σ (n) through the time-varying Poisson equation of potential field, generates the Joule heat source Q (n) ;

[0031] The thermal field analysis updates the temperature field T according to the heat conduction equation (n) , taking into account Joule heating, latent heat of phase change, and heat conduction effects on the temperature field T (n) .

[0032] The phase change analysis solves the solid-liquid phase change process by a kinetic equation and solves the solid fraction f s (n) and the volume change AV (n) .

[0033] The mechanical analysis solves the plastic strain tensor ε (p,n) using a return mapping algorithm.

[0034] Further, in the step 12), the total strain ε (n) is expressed as:

[0035]

[0036] v1 = vec(σ (n) ), v2 = vec(σ (n-1) ), v3 = ΔT (n) , v4 = Δf s (n) , v5 = f s (n-1) , v6 = f l (n-1)

[0037] where: is a 7th order tensor x i denotes the tensor-vector product along the i-th mode; v i denotes the i-th state vector; σ (n) denotes the electrical conductivity at the current time step; σ (n-1) denotes the electrical conductivity at the previous time step; ΔT (n) denotes the temperature field at the current time step; Δf s (n) denotes the solid fraction at the current time step; f s (n-1) denotes the solid fraction at the previous time step; f l (n-1) denotes the liquid fraction at the previous time step; i = 1, 2, 3, 4, 5, 6.

[0038] Further, in the step 21), the high order tensor is CP decomposed and expressed as a sum of several rank-1 tensors:

[0039]

[0040] wherein: is the factor vector on the i-th mode; is the factor vector on the output mode; denotes the vector outer product; R is the rank of the CP decomposition, determining the number of decomposed rank-1 tensors; N i denotes the feature dimension of the i-th state physical quantity; k is the number of state physical quantities;

[0041] Factor matrix A (i) denotes:

[0042]

[0043] Factor matrix A (i) the element of the j-th row of (A (i) ) j,: denotes the contribution degree of the j-th physical quantity in the i-th mode when constituting R different rank-1 tensors;

[0044] Weight matrix W is denoted as:

[0045]

[0046] wherein: A (k+1) corresponds to the output layer, used to map the low-dimensional feature space to the final output space;

[0047] The nonlinear coupling of the physical field increment is realized by the Hadamard product, denoted as:

[0048]

[0049] wherein: denotes the Hadamard product; denotes the i-th state vector; A (i) denotes the contribution degree of the i-th physical quantity.

[0050] Further, in the step 22), the mapping relationship between the displacement increment Δu (n) and the strain increment Δε (n) is:

[0051]

[0052] High-order tensor is denoted as:

[0053]

[0054] wherein: R is the rank of the CP decomposition, determining the complexity and expression ability of the model; λ r is the weight coefficient of the decomposition; denotes the contribution of different curved parts; represents the contribution of the weld point on the mth surface part; represents the contribution of different physical fields, respectively; represents the contribution of the output strain increment; represents the direct sum operation, which is used to connect the weld point vectors of different surface parts to integrate the weld point contributions on different surface parts into a tensor;

[0055] The total strain increment of the multi-surface multi-point welding system is represented as:

[0056]

[0057] wherein: e m is an indicator vector of the mth surface part, i.e. a vector with only the mth element being 1 and other elements being 0, which is used to select the corresponding surface part; is another indicator vector, which is used to select the pth weld point on the mth surface part; represents the increment of the ith physical quantity at the nth time step caused by the pth weld point on the mth surface part;

[0058] The expression of the displacement field increment Δu (n) is:

[0059]

[0060] wherein: and represent the input mode of the Green function; is a factor matrix; is a weld point interaction mode matrix; represents the increment of the ith physical quantity at the nth time step caused by the pth weld point on the mth surface part; ⊙ represents the Hadamard product of multiple matrices or vectors; Λ is a weight coefficient vector of CP decomposition.

[0061] Further, in the step 31), the method steps for constructing the white body assembly geometric accuracy posterior state inference model are:

[0062] 311) Construct a coordinate mapping network φ to map a three-dimensional point on the white body to a point in a two-dimensional parameter space, represented as:

[0063]

[0064] wherein: (x, y, z) represents a three-dimensional coordinate system; (x 1 ,x 2 ) represents a local coordinate system;

[0065] 312) Construct a neural network f to learn the 0 form and the metric tensor, represented as:

[0066]

[0067] where f is a multi-head attention network; is a value 0 form v; is a metric tensor;

[0068] 313) Apply the exterior differential operator d to the 0-form v to obtain a value 1-form dv;

[0069] 314) Apply the Hodge star operator to the 1-form dv to obtain another 1-form *dv;

[0070] 315) Apply the exterior differential operator d to the 1-form *dv to obtain a 2-form d*dv;

[0071] 316) Apply the Hodge star operator to the 2-form d*dv to obtain a 0-form *d*dv;

[0072] 317) Apply a nonlinear activation function σ to the *d*dv to embed it into a deep neural network to construct a feature learning model.

[0073] Further, in the step 32), the method for updating the displacement field and the physical field state is:

[0074] Define a unified state vector x (n) , which is expressed as:

[0075]

[0076] where: is the geometric accuracy or other related features of the geometric accuracy of the body-in-white assembly at the nth generation step; d is the feature dimension; is the physical field state of the mth surface part;

[0077] At each time step, the update of the unified state vector is expressed as:

[0078]

[0079] where: is a nonlinear function, defined as:

[0080]

[0081] where: [·] represents the index of the corresponding element in the vector x (n-1) ; and correspond to the displacement field increment of the body-in-white assembly and the update of the physical field state of the m surface parts, respectively, and:

[0082]

[0083] wherein: is a fourth-order tensor mapping the intermediate feature of R dimensions to P welds, each weld corresponding to k physical fields, each physical field corresponding to a D-dimensional feature vector; is a learnable weight modulation vector; fuses the state information of all welds on the mth curved part in all physical fields to obtain an R×P matrix; is the final physical field state increment tensor;

[0084] stacks multiple residual blocks to construct a body-in-white assembly geometric accuracy generation model:

[0085]

[0086] wherein: x (N) is the final state vector; represents the residual block of the nth generation step; x (0) is the initial unified state vector; σ is a nonlinear activation function; N is the number of discrete generation steps; represents the vector outer product.

[0087] Further, in the fourth step, the geometric accuracy prediction value of the target time t * is:

[0088]

[0089] wherein: U t is the geometric accuracy of the body-in-white assembly at the current time; t * and t are the target time and the current time, respectively; is the l-1 order Taylor coefficient of ζ t at time t; is the p-1 order Taylor coefficient of ζ t at time t; ζ t is a deep neural random process; Γ is the value of U t at t = η, called the intermediate state; η ∈ [t, t * ] is a certain unknown time point in the integral interval.

[0090] Further, in the fourth step, the loss function for training the body-in-white assembly geometric accuracy random dynamic evolution prediction model is:

[0091]

[0092] wherein: ​is the geometric accuracy rate of the model predicted body-in-white assembly at time point t; is the real geometric accuracy rate of the body-in-white assembly at time point t * ; is the initial physical field state distribution of the mth curved part; and are the mean and variance vectors of the Gaussian distribution ; is the Gaussian distribution output by the posterior inference network of the body-in-white assembly; μ A and σ A are the mean and variance vectors of the Gaussian distribution ; p(z A ) is the prior distribution of the body-in-white assembly hidden state variable z A ; and λ1 and λ2 are control parameters.

[0093] The beneficial effects of the present application are:

[0094] The body-in-white welding multi-physical field deep modeling and geometric accuracy generative prediction method provided by the present application is aimed at the problem that the geometric accuracy of the body-in-white assembly is difficult to effectively predict due to the complex mechanism of the welding process of the multi-curved part. Firstly, the thin plate deformation mechanism of the resistance spot welding connection process is analyzed. On this basis, a high-order interaction model of geometric deformation of the welding process of the multi-curved part is constructed based on CP tensor decomposition. Further, in order to effectively predict the random dynamic evolution of the geometric accuracy of the body-in-white assembly, the present application combines a posterior state inference model of the geometric accuracy of the body-in-white assembly based on the neural differential form and a geometric accuracy generation network of the body-in-white assembly based on the high-order interaction of the multi-curved part, and introduces a neural numerical integration method to construct a dynamic prediction model of the random dynamic evolution of the geometric accuracy of the body-in-white assembly. That is, the present application models and excavates the influence mechanism of each single curved part on the assembly geometric accuracy from a macro level, establishes a "gray box" model for predicting the geometric accuracy of the body-in-white assembly, and abstracts the influence of the welding process on the geometric accuracy of the assembly as a high-order interaction between single parts, thereby effectively supporting the effective and reliable prediction of the geometric accuracy of the body-in-white assembly. BRIEF DESCRIPTION OF DRAWINGS

[0095] In order to make the purpose, technical scheme and beneficial effects of the present application clearer, the present application provides the following drawings for illustration:

[0096] Figure 1 is the multi-field coupling analysis of resistance spot welding;

[0097] Figure 2 is the structure of the random dynamic evolution prediction model of the body-in-white assembly geometric accuracy;

[0098] Figure 3 is the experimental data from the source manufacturing line;

[0099] Figure 4 The white body assembly and its key curved surface piece geometry precision data preprocessing;

[0100] Figure 5 The white body assembly geometry precision absolute deviation prediction and actual deviation density distribution map for the next 9 days;

[0101] Figure 6 The white body assembly geometry precision prediction and actual distribution difference evaluation under different Cpk levels;

[0102] Figure 7 The influence analysis of different surface piece ablation on the white body assembly geometry precision prediction;

[0103] Figure 8 The influence evaluation of key sub-modules on the white body assembly geometry precision prediction performance;

[0104] Figure 9 The stamping surface piece wavelet filter weight distribution and assembly geometry precision gradient contribution visualization. DETAILED DESCRIPTION

[0105] The present application will be further described below in conjunction with the drawings and specific embodiments, so that those skilled in the art can better understand the present application and implement it, but the embodiments are not limiting to the present application.

[0106] The body welding multi-physical field deep modeling and geometry precision generative prediction method of the embodiment, comprising the following steps.

[0107] Step one: spot welding connection process sheet deformation mechanism analysis and representation.

[0108] 11) Multi-field coupling analysis of resistance spot welding process: through the coupling iteration of electric field analysis, thermal field analysis, phase change analysis and mechanical analysis, the total strain ε of the sheet is determined (n) .

[0109] Resistance spot welding is a complex process involving high coupling of electric field, thermal field, metal phase change and mechanical deformation. In the resistance spot welding process, when the welding current passes through the contact area of the electrode and the workpiece, the contact resistance generates Joule heat, which rapidly raises the temperature of the area, causing the metal to melt under the pressure of the electrode to form a nugget. In this process, the concentration of heat and the high-density transmission of current not only cause the metal to melt, but also cause significant plastic deformation of the surrounding material. As the nugget cools and the electrode continues to press the contact area, the molten metal re-solidifies to form a solid weld, achieving material connection. From the perspective of numerical simulation, the multi-field coupling mechanism in the resistance spot welding process is as shown in Figure 1 .

[0110] From Figure 1As can be seen, at each time point in the resistance spot welding process, there is a coupling effect of multiple physical fields, including electric field analysis, thermal field analysis, phase transition analysis, and mechanical analysis. Specifically, if the continuous time in the welding process is discretized, then at the nth time step t... n The analysis and solution process for each physical field can be described as follows:

[0111] (1) Electric field analysis: Electric field analysis updates the conductivity σ using the time-varying Poisson equation of the electric potential field. (n) Generates Joule heat source Q (n) .

[0112] Specifically, electric field analysis is the starting point for solving the entire multiphysics coupling problem in resistance spot welding. Its inputs include the previous time step t. n-1 Temperature field T (n-1) (x) (output from thermal field analysis) and solid fraction f s (n-1) (x) (output from phase transition analysis), where Let be the spatial coordinates. Use this information to update the material's electrical conductivity σ. (n) (x)=σ(T (n-1) ,f s (n-1) The governing equations for the electric field are: The time-varying Poisson equation for the electric potential field is:

[0113]

[0114] Where ∈ is the dielectric constant. Solving this equation requires applying a time-varying voltage condition U(t) in the region where the welding clamp contacts the workpiece, and applying an insulating boundary condition n·J=0 in the non-contact region, where n is the outward normal of the boundary. This represents the current density vector. The solution includes the electric potential field. electric field strength and Joule heat source items Among them, Joule heat Q (n) As input to thermal field analysis, it directly affects the evolution of the temperature field.

[0115] (2) Thermal field analysis: Thermal field analysis updates the temperature field T based on the heat conduction equation. (n) And comprehensively consider the effects of Joule heating, latent heat of phase change, and heat conduction on the temperature field T (n) The impact.

[0116] Specifically, thermal field analysis is a core step in the resistance spot welding process. Its inputs include the previous time step t. n-1 Temperature field T (n-1) (x), solid fraction f s (n-1) (x) and the current time step t nJoule heat source term Q (n) (x) and latent heat source term (From phase change analysis). Update the thermal properties of the material, including density p (n) (x), specific heat capacity and thermal conductivity k (n) (x) according to the phase fraction. The heat conduction equation can be written as:

[0117]

[0118] This equation takes into account the effects of Joule heat Q (n) , phase change latent heat and heat conduction on the temperature field T (n) (x). Where L is the phase change latent heat, f s (n) is the solid phase fraction. The solution includes the temperature field T (n) (x), temperature gradient and heat flux density

[0119] (3) Phase change analysis: Phase change analysis through the kinetics equation solid-liquid phase change process, and solve the solid phase fraction f s (n) and volume change AV (n) .

[0120] Specifically, the phase change analysis mainly focuses on the temperature-induced solid-liquid phase change process, and its input includes the solid phase fraction f n-1 s (x) at the last time step t (n-1) and the temperature field T n (x) at the current time step t (n) . The kinetics equation of solid-liquid phase change can be described by Avrami equation:

[0121]

[0122] Where: b(T (n) ) and n(T (n) ) are temperature-dependent material parameters that control the kinetics process of phase change. The solution includes the solid phase fraction f s (n) (x)∈[0,1] and volume change AV (n) (x). Since the liquid phase fraction f l (n) (x) can be obtained by f l (n) (x) = 1-f s (n)(x) is computed, so here we only need to track the evolution of the solid fraction f s (n) and the volume change AV (n) will be input to the mechanical analysis. In addition, the latent heat source term will also be input to the thermal analysis.

[0123] (4) Mechanical analysis: The mechanical analysis solves the plastic strain tensor ε (p，n) using a return mapping algorithm.

[0124] Specifically, the mechanical analysis focuses on the deformation, stress and strain state of the material during the welding process, including elastic and plastic deformation. Its inputs include the displacement field u n-1 (x) at the previous time step t (n-1) , the plastic strain ε p，n-1 (x), the temperature field T (n-1) (x), and the temperature field T (n) (x) at the current time step t n , the solid fraction f s (n) (x) and the volume change AV (n) (x). Based on this information, the elastic-plastic constitutive parameters of the material are updated, including the elastic modulus E (n) (x), the Poisson's ratio v (n) (x), the plastic hardening modulus H (n) (x), the thermal expansion coefficient α (n) (x) and the yield strength The governing equations include:

[0125] Balance equation:

[0126]

[0127] Strain-displacement relationship:

[0128]

[0129] Constitutive relationship:

[0130]

[0131] where: is the fourth-order elastic stiffness tensor, the thermal strain ε (T,n) = α (n) (T (n) -T ref )i, the phase change strain

[0132] Yield condition:

[0133]

[0134] where F is the yield function, is the equivalent plastic strain at the current time step.

[0135] Plastic flow rule:

[0136]

[0137] where: is the plastic multiplier.

[0138] Plastic strain tensor ε (p,n) The calculation of (x) is an iterative process that combines all the above equations. The process first assumes that the current increment is purely elastic, and calculates the trial stress state. Then, by checking the yield condition, it determines whether plastic deformation occurs. If no plastic deformation occurs, the plastic strain remains unchanged; otherwise, plastic correction is needed. During plastic correction, the plastic multiplier is solved using the return mapping algorithm, and the plastic strain increment, plastic strain tensor, equivalent plastic strain, and stress state are updated accordingly. This process is repeated until the convergence condition is met. Through this iterative method, the plastic strain tensor ε (p,n) (x) and the stress tensor σ (n) (x) at the current time step are obtained, which further solves the displacement field u (n) (x).

[0139] It can be seen that the complexity of the resistance spot welding process lies in the mutual coupling and influence of various physical fields during the process, resulting in multi-variable and multi-stage interaction. Under this interwoven and iterative multi-field coupling mechanism, the deformation mechanism of the thin plate during welding becomes extremely complex, and it is difficult to accurately model and predict through an analytical mathematical model or ordinary data-driven method. It is necessary to further explore the inherent fundamental rules and capture the high-order coupling mechanism through data-driven methods.

[0140] 12) High-order tensor representation of the deformation mechanism of resistance spot welded thin plates: representing the strain mechanism as a 7-order tensor and based on the 7-order tensor the total strain ε (n) is obtained; the 7-order tensor includes 6 state vector dimensions and 1 output dimension, and the state vector set formed by the 6 state vector dimensions contains key physical field information including stress, temperature, and solid fraction.

[0141] As can be seen from the above multi-field coupling analysis of the resistance spot welding process, the deformation mechanism of the thin plate during welding is determined by the complex coupling of the electric, thermal, mechanical, and phase multi-physical fields. The root cause of deformation is the four strain sources: elastic strain ε (e,n) , plastic strain ε(p,n) thermal strain ε (T,n) and phase transformation strain ε (V,n) . Among them, the elastic strain, thermal strain is mainly caused by the temperature gradient generated by the Joule heat caused by the welding current, the plastic strain is mainly caused by the stress gradient generated by the Joule heat caused by the welding current, and the phase transformation strain is mainly caused by the phase transformation generated by the Joule heat caused by the welding current. The interaction of the four strain sources constitutes the complex deformation mechanism of the sheet during the welding process. Thus, the total strain of the sheet at the nth time step can be expressed as:

[0142] ε (n) = ε (e,n) + ε (p,n) + ε (T,n) + ε (V,n) (9)

[0143] In order to accurately describe the complex coupling mechanism of sheet deformation, each strain component needs to be characterized in detail. For the elastic strain ε (e,n) , according to the generalized Hooke's law there is a linear relationship between it and the stress tensor σ (n) , and the coefficient matrix is the fourth-order elastic stiffness tensor which depends on the Young's modulus E (n) and Poisson's ratio v (n) of the material. In addition, during the resistance spot welding process, the material properties will change significantly with temperature T (n) and solid

[0144] phase fraction f s (n) . Considering these factors, the elastic strain ε (e,n) can be expressed as:

[0145]

[0146] where: ΔT (n) = T (n) -T (n-1) , Δf s (n) = f s (n) -f s (n-1) ; : : and : : : represent tensor contraction of different orders; denotes the tensor product; are a fourth-order, two fifth-order and a sixth-order tensor, respectively, which describe the linear relationship between the elastic strain and the stress tensor, the effect of temperature change on the elastic strain, the effect of solid phase fraction change on the elastic strain, and the joint effect of temperature and solid phase fraction change on the elastic strain.

[0147] For the plastic strain ε(p，n) , considering the stress σ (n) exceeding the yield strength , and the effect of temperature and phase transformation on the yield strength of the material, can be expressed as:

[0148]

[0149] where λ (n) is the plastic multiplier; F (n) is the yield condition; is the yield strength; is a third-order tensor; and are fourth-order tensors. This expression describes the incremental dependence of the plastic strain on temperature and phase transformation, reflecting the nonlinear constitutive behavior of the material. Among them, and terms represent the change of the plastic multiplier caused by temperature and phase transformation, and terms represent the influence of the partial derivative of the yield function on the stress affected by the increment of temperature and phase transformation, which together constitute the complex temperature-phase-plasticity coupling mechanism in the resistance spot welding process.

[0150] The thermal strain ε (T,n) can be represented by the product of the temperature change and the thermal expansion coefficient α (n) :

[0151]

[0152] where α (n-1) is the thermal expansion coefficient at the previous time step; is a second-order tensor; and are third-order tensors. The term reflects the nonlinear change of the thermal expansion coefficient at high temperature, The term reflects the modulation effect of phase transformation on the thermal expansion coefficient. In the resistance spot welding process, the significant phase transformation in the nugget zone will cause a sudden change in the thermal expansion coefficient, leading to a complex deformation mode.

[0153] The phase transformation strain ε (V,n) is related to the solid phase fraction f s (n) , the liquid phase fraction f l (n) , and their respective specific volumes (volume per unit mass) V s (n) and V l (n) :

[0154]

[0155] where: f s (n-1) and f l (n-1) are the solid and liquid fractions of the previous time step; ΔV s (n-1) and ΔV l (n-1) are the solid and liquid specific volume changes of the previous time step. and are second order tensors; and are third order tensors. and terms reflect the effect of the change of phase fraction caused by the temperature increment on the phase transformation strain, and terms reflect the effect of the change of specific volume caused by the temperature increment on the phase transformation strain. In the resistance spot welding process, the volume expansion and contraction caused by the solid-liquid phase transformation are the main reasons for the deformation of the nugget zone.

[0156] By combining the expressions of all strain sources, equation (9) can be substituted into, i.e., the total strain ε (n) of the curved sheet in the resistance spot welding process can be obtained.

[0157]

[0158] This expression shows the complex deformation mechanism of the sheet in the resistance spot welding process, including the expressions of the four strain sources of elasticity, plasticity, thermal expansion, and phase transformation. In order to more efficiently describe this multi-field coupling mechanism, the state vector set

[0159] v1 = vec(σ (n) ), v2 = vec(σ (n-1) ), v3 = ΔT (n) , v4 = Δf s (n) , v5 = f s (n-1) , v6 = f l (n-1) (15)

[0160] where vec(·) represents a vectorization operation (flattening, zero padding, linear lifting, etc.). Using these state vectors, the total strain ε (n) can be further simplified as:

[0161]

[0162] The tensor in equation (16) can be regarded as a sub-tensor of a higher order tensor, and by appropriate zero padding and rearrangement, all the Merged into a single higher-order tensor Specifically, it can be defined as follows:

[0163]

[0164] Here, i7 corresponds to the output dimension. Through this construction, all... Embedded into a unified higher-order tensor This allows for a more compact representation.

[0165] Based on the above construction, a 7th order tensor is obtained. It includes six state vector dimensions plus one output dimension. Using this unified high-order tensor representation, the total strain ε... (n) This can be expressed as:

[0166]

[0167] in: For 7th order tensors × i v represents the tensor-vector product along the i-th mode; i σ represents the i-th state vector; (n) σ represents the conductivity at the current time step. (n-1) ΔT represents the conductivity at the previous time step. (n) Δf represents the temperature field at the current time step. s (n) f represents the solid fraction at the current time step; s (n-1) f represents the solid fraction at the previous time step; l (n-1) Indicates the liquid phase fraction at the previous time step; i = 1, 2, 3, 4, 5, 6.

[0168] Based on the fundamental properties of tensor algebra, the above equation can be equivalently rewritten as:

[0169]

[0170] here Tensor The 7th mode matrix (because 6 state physical quantities are involved in the coupling, so...) (Perform the 7th mode matrixization), This refers to the Kronecker product (a special form of tensor product used to produce tensor products in matrix form).

[0171] Therefore, the total strain ε of the curved part during the resistance spot welding process (n) It is clearly represented as a higher-order tensor. With state vector set The multi-tensor product. Wherein, the state vector set... It encompasses key physical field information such as stress, temperature, and solid fraction during resistance spot welding. These physical fields do not exist in isolation but are intertwined and mutually influential, jointly determining the final deformation state of the curved part. And the higher-order tensor... It plays a "bridge" role, describing and encoding the complex coupling relationships between these physical fields, as well as the interactions between various strain sources. Therefore This can be viewed as an "intrinsic response characteristic" exhibited by curved parts during the welding process. This characteristic varies depending on the physical field, i.e., different sets of state vectors. When this "intrinsic response characteristic" is applied, it induces different shape changes in the curved part. This efficient tensor coupling expression provides a theoretical basis and directional guidance for the construction of data-driven welding deformation prediction models.

[0172] Step 2: High-order interactive representation of geometric accuracy in the welding process of multi-curved parts.

[0173] 21) Low-rank characterization of strain in single-curved surface components based on CP tensor decomposition, including: high-order tensors Perform CP decomposition to generate factor matrix A (i) And output weight matrix W; achieve nonlinear coupling of physical field increments through Hadamard product ⊙.

[0174] Step 1 discusses the higher-order coupling mechanism of strain generated in curved parts during resistance spot welding and its corresponding higher-order tensor representation. However, while this higher-order tensor representation can efficiently and concisely characterize the complex deformation characteristics of curved parts during welding, it also suffers from high computational complexity and large storage space requirements. For example, the number of parameters in a higher-order tensor increases exponentially with the order, leading to high computational complexity. Furthermore, higher-order tensors require storing a large number of parameters, consuming significant storage space. More importantly, Includes Multiple subtensors, together forming a complex high-order tensor structure, are what enable... The complex coupling relationships between multiple physical fields were modeled and encoded. However, if... Treated as a black box model composed entirely of free parameters, it learns solely through data-driven methods. The parameters will be difficult to capture. The inherent physical mechanisms and structural information within the model itself can cause it to lose its generalization ability and make it difficult to uncover potential and accurate physical laws. Therefore, a new representation method is needed that can effectively capture these physical mechanisms and structural information while reducing computational complexity and storage space. the physical mechanism and structural information contained therein.

[0175] To address the above issues, this section considers methods based on low-rank representation. The basic idea of low-rank representation is to represent high-dimensional data as a combination of multiple low-dimensional data, thereby effectively reducing the dimension and complexity of the data. In high-order tensor analysis, low-rank representation can be achieved by decomposing a high-order tensor into a combination of multiple low-order tensors. This decomposition can effectively reduce the number of parameters, thereby reducing computational complexity and storage space requirements. At the same time, low-rank representation can reveal the underlying structural information in the high-order tensor, such as the interaction between different physical fields, thereby improving the interpretability of the model. Therefore, in order to better understand and mine the physical mechanism and structural information in , and find a computationally efficient and highly interpretable "gray box" model, this embodiment introduces CP tensor decomposition, representing the high-order tensor as a combination of multiple low-order tensors, achieving low-rank representation of the high-order tensor .

[0176] Consider a more general case, i.e., there are k state physical quantities, each physical quantity is represented by an increment vector: i = 1, 2, …, k. In this case, should be a k+1 order tensor, where the k+1 dimension corresponds to the strain increment of the output. Therefore, the strain increment Δε (n) at time n can be represented as:

[0177]

[0178] where, represents the k+1 mode matrix of the tensor , represents the Kronecker product.

[0179] CP decompose , represented as the sum of several rank-1 tensors:

[0180]

[0181] where: is the factor vector on the i-th mode, is the factor vector on the output mode; denotes the vector outer product; R is the rank of the CP decomposition, which determines the number of rank-1 tensors after decomposition. The goal of CP decomposition is to find a set of rank-1 tensors such that the low-rank tensor after decomposition can effectively approximate the original high-order tensor Compared with other tensor decomposition methods such as Tucker decomposition, CP decomposition has uniqueness, which means that in most cases, given a high-order tensor, its decomposition result is unique. In addition, the result of CP decomposition can be directly mapped to the interaction of physical quantities, making the model more interpretable and scalable (such as from single-point welding deformation to multi-point welding deformation in low-rank space). i represents the characteristic dimension of the i-th state physical quantity. i The larger the value of is, the higher the dimension of the characteristic vector used to describe the i-th physical quantity, and the more physical fields (such as stress, strain, heat,...) that can be accommodated. In fact, it is a model data-driven learning that adaptively excavates suitable multi-physical fields.

[0182] Substituting CP decomposition into the strain increment expression, we can get:

[0183]

[0184] To further simplify the representation and calculation, and derive a more compact total strain expression, consider a simple outer product tensor By analyzing the relationship between the matrix form of the k+1 mode of and the Kronecker product, we can get:

[0185]

[0186] When the k+1 mode of this tensor is matrixed, the resulting matrix The rows of correspond to the elements of x k+1 , and the columns correspond to all possible combinations of x1, x2, …, x k . Therefore, consider the calculation of The j-th element of this operation can be represented as:

[0187]

[0188] This result can be generalized to more general CP decomposition forms, because CP decomposition is essentially representing a high-order tensor as the sum of multiple rank-1 tensors (i.e., outer product tensors). Therefore, for any vectors x1, x2, …, x k , x k+1 , and v1, v2, …, v k , we have:

[0189]

[0190] Applying the above result to the case of CP decomposition, we can get a more compact expression for the strain increment Δε (n) :

[0191]

[0192] The strain increment expression of form (26) is used to express the strain increment Δε. (n) It can be represented as a linear combination of multiple low-rank tensors, where each low-rank tensor is the outer product of a factor vector. Each The term represents the state increment vector. In the corresponding factor vector The projection onto the surface captures the contribution of the physical field increment in the r-th interaction mode. Through the Hadamard product, the interaction relationships between different physical fields are non-linearly coupled together, ultimately through... Mapping to the output space, we obtain the total strain ε. (n) .

[0193] To more clearly reveal the contributions and interaction patterns of different physical fields during the deformation process, and to effectively reduce the complexity of the model, factor vectors can be combined into a factor matrix. For each mode i, its factor matrix is ​​defined as:

[0194]

[0195] Factor matrix A (i) The element in the j-th row (A) (i) ) j,: This represents the contribution of the j-th physical quantity in the i-th mode to the formation of R different rank-1 tensors. These contributions collectively determine the influence of the physical quantities in that mode on the total strain. Therefore, the factor matrix maps the physical quantity information of each mode to a low-dimensional space, the dimension of which is determined by the rank R of the CP decomposition.

[0196] Using the factor matrix, the CP decomposition can be expressed more concisely as:

[0197]

[0198] in It is the standard symbol for CP decomposition. Because A... (k+1) Corresponding to the model's output layer, it maps the low-dimensional feature space to the final output space. To more clearly express this mapping relationship, let A... (k+1) The transpose of W is defined as the output weight matrix W:

[0199]

[0200] Therefore, the strain increment Δε (n) It can be represented in a more compact form:

[0201]

[0202] Where: ⊙ represents the Hadamard product. This expression represents the strain increment Δε (n) Represented as the Hadamard product of the transpose of the factor matrix and the state increment vector, it captures the role and interaction of different physical field increments in complex deformation processes. Specifically, each By projecting the original physical quantity increments onto a low-dimensional feature space, feature extraction and dimensionality reduction are achieved. Then, the Hadamard product (⊙) realizes nonlinear coupling between features, capturing the complex interactions between physical field increments. Finally, W maps the coupling result to the final output space. This factor matrix-based representation is not only computationally efficient but also more interpretable. Through analysis of the factor matrix A... (i) The element size, sign, and relationships between different rows / columns can be used to intuitively understand the contribution and interaction patterns of different physical fields during the deformation process, thereby better understanding and predicting the deformation behavior of curved parts.

[0203] 22) High-order interactive characterization of geometric deformation in multi-spot welding process of multi-curved parts, including: defining the generalized Green's function matrix. Establish displacement increment Δu (n) With strain increment Δε (n) Mapping relationships; introduction of higher-order tensors Describe M surface components and P for each surface component m. m Interaction between solder joints, and the use of a solder joint interaction mode matrix. Describe the coupling relationship between weld points within the same curved surface component.

[0204] Specifically, in the actual body-in-white welding process, multiple curved components are typically connected by several to hundreds of weld points, forming a complex multi-point welded structure to jointly complete the assembly of the entire vehicle. In this case, each spot welding process directly or indirectly affects the deformation state of itself or other curved components. In addition to the small local deformations caused by the resistance spot welding process itself, the multi-point welding process also involves various factors such as the alignment accuracy between curved components, gap difference control, welding sequence, and welding position. These factors work together to make the geometric deformation of the entire body-in-white welding process more complex, exhibiting significant nonlinear and high-order coupling characteristics. Therefore, geometric deformation modeling for the multi-curved component multi-point welding process needs to further consider the high-order interaction relationships between the multi-curved components and multi-point welds to more accurately describe and predict the geometric deformation during the overall vehicle welding process.

[0205] Since resistance spot welding deformation is generally a small deformation problem, we can assume that the displacement field increment Δu caused by the welding deformation is a small one. (n) and strain field increment Δε (n) The relationship between them is linear. According to linear elasticity theory, the strain field increment Δε at a single weld point... (n) and displacement field increment Δu(n) satisfying a linear equation:

[0206]

[0207] where D is the strain-displacement operator. However, due to the complexity of the geometry and material properties of the solder region, it is difficult to solve this equation directly. To simplify the problem, the displacement field increment Δu (n) can be expressed as a strain field increment Δε (n) in an integral form of the response of the Green's function G over the solder region Ω. Specifically, the displacement field increment Δu (n) can be expressed as a strain field increment Δε (n) convolved with the Green's function G:

[0208] Δu (n) (x) = ∫ Ω G(x, y):Δε (n) (y) dy (32)

[0209] where G(x, y) is the Green's function representing the displacement response at x due to a unit strain at y, : denotes the double contraction operation, and Ω is the geometric domain of the solder region. Although equation (32) involves integration over a complex geometric domain Ω on a curved surface, the distribution of dimensional accuracy on the curved surface has been approximated in the above section as a discrete map, i.e., the integration domain Ω is discretized into a finite number of nodes each node x i corresponding to a dimensional accuracy measurement point on the curved surface. Therefore, according to the theory of numerical integration, the integral in equation (32) can be approximated as a weighted summation over these discrete nodes:

[0210]

[0211] where ΔV i represents the volume represented by the ith discrete point. More succinctly and generally, equation (33) can be further expressed as:

[0212]

[0213] where, is a generalized Green's function matrix whose elements encode the displacement response at the solder x j due to a unit strain at the solder x i .

[0214] The formula (34) describes the linear relationship between the displacement increment response of a single spot and the strain field increment, but in the actual welding process, the mutual influence between different spots and the coupling between different curved surface parts will cause the geometric accuracy of the welding process to be affected by multiple complex unknown factors. These interaction information comes from the geometric and mechanical coupling between curved surface parts, the mutual influence between spots, and the nonlinear coupling between multiple physical fields such as temperature, stress, phase change, etc. In order to accurately describe this complex high-order interaction, a higher-order tensor representation needs to be introduced. Assuming that the body-in-white welding process involves M curved surface parts, each curved surface part m has P m spots. In order to describe the relationship between the displacement increment response of all spots and the strain field increment, a high-order tensor is introduced to represent the strain field increment of the entire system:

[0215]

[0216] Where R is the rank of the CP decomposition, which determines the complexity and expression ability of the model; λ r is the weight coefficient of the decomposition; represents the contribution of different curved surface parts; represents the contribution of the spots on the mth curved surface part; represents the contribution of different physical fields respectively; represents the contribution of the output strain increment; represents the direct sum operation, which is used to connect the spot vectors of different curved surface parts to integrate the spot contributions on different curved surface parts into a tensor.

[0217] Based on this extended high-order tensor representation, the total strain increment of the multi-surface multi-spot welding system can be represented as:

[0218]

[0219] Where e m is an indicator vector of the curved surface part m (i.e. a vector with only the mth element being 1 and other elements being 0), used to select the corresponding curved surface part; is another indicator vector, used to select the pth spot on the curved surface part m; represents the increment of the ith physical quantity at the nth time step caused by the pth spot on the mth curved surface part. This expression allows the contributions of multiple spots on multiple curved surface parts to be integrated into a high-order tensor, realizing the high-order interaction representation of the multi-surface multi-spot welding system.

[0220] Further, to associate the strain increment expression with the displacement field increment, it needs to be considered that each curved surface part may have different material properties and geometric characteristics. Therefore, a different generalized Green's function matrix and CP-decomposition can be obtained as

[0221]

[0222] where, is the weight coefficient of CP-decomposition, and are the factor vectors of Green’s function. This representation allows customizing specific response characteristics for each surface, thus more accurately describing the influence of different materials and geometries on deformation. Substituting this CP-decomposition result into the generalized Green’s function equation (34), the displacement field increment expression of the system is obtained as

[0223]

[0224] Equation (38) comprehensively considers the complex coupling relationships in the multi-surface multi-spot welded system. Each term represents the contribution of a specific surface, a specific spot, and a specific mode. The inner product operation <·,·> reflects the interaction between different factors, while the concatenated term captures the coupling effects between multiple physical quantities.

[0225] Similar to the surface strain low-rank representation under single-spot welding, to further simplify the expression and highlight the learnable parameters of the model, the concept of factor matrices is introduced. Factor matrices are directly extracted from the above CP-decomposition, compressing high-dimensional tensors into low-dimensional spaces while preserving the main characteristics of the original data. Here, the factor matrices are used to describe the interaction modes between surfaces; are used to represent the interaction modes between spots on the m-th surface; represent the contribution modes of the i-th physical quantity. In addition, represent the output modes of Green’s function, while and represent the input modes of Green’s function. Based on these factor matrices, equation (38) can be rewritten as

[0226]

[0227] where

[0228] where ⊙ denotes the Hadamard product of multiple matrices or vectors, and Λ is the weight coefficient vector of CP-decomposition.

[0229] Equation (39) comprehensively captures the complex coupling relationships in the multi-surface multi-spot welded system. It represents the overall displacement u as a multi-linear combination of contributions from all surfaces, spots, and physical fields. Through factor matrices and Hadamard products, it describes the high-order coupling effects of different surfaces, spots, and their interactions with multiple physical fields. The learnable parameters include factor matrices W m, A (M) , A (i) and Λ. These matrices correspond to different physical quantities, geometric characteristics, and specific manifestations of the welding process, respectively. By optimizing these matrices, the coupling relationships between complex physical fields can be learned. Specifically, describes the characteristics of the mth curved part, including material properties and geometric characteristics. By optimizing W m , the model can capture the differences in the multi-physical field coupling response of different curved parts, such as the differences in deformation caused by differences in physical parameters such as thermal conductivity and thermal expansion coefficient. Factor matrices and represent the interaction patterns between curved parts and welding points, respectively. Through learning, the complex interactions between curved parts and between welding points on the same curved part can be captured. For example, describes the mutual influence between curved parts, while captures the multi-linear interaction between welding points within the same curved part.

[0230] In equation (39), ⊙ plays the role of high-order feature interaction. By learning the A (i) matrix, the model can capture the nonlinear relationships between different physical fields. This interaction is not only reflected in the low-dimensional feature extraction of each physical quantity, but also in the complex high-order coupling between these physical quantities. In particular, in the interaction between different physical fields, such as the relationship between the temperature field and the stress field, ⊙ can perform element-wise nonlinear combination of these physical quantities, and finally map these combinations to the output u. Another advantage of equation (39) is that it integrates the high-order interaction relationships of the multi-curved part multi-point welding system into a unified framework. The coupling relationships of different curved parts and welding points essentially constitute a multi-task learning structure. Each curved part A (M) and each welding point has its own independent feature pattern, but they achieve parameter sharing through shared model parameters such as W m and . This sharing mechanism allows the model to efficiently propagate information between multiple physical fields and curved parts, thereby improving the model's generalization ability and learning efficiency.

[0231] Through the form shown in equation (39), the geometric deformation mechanism of the white body welding process under multi-curved part multi-point welding can be efficiently and simply represented. On the one hand, factor matrices A (M) , A (i) and W mAs a low-rank representation of a potential high-order coupling tensor, it has a small amount of parameters, but in the framework of CP tensor decomposition, it has sufficient expressive power to effectively extract high-order interaction features in the multi-surface multi-spot welding system and capture the complex coupling relationship between different surface parts, welding spots, and physical fields. On the other hand, through the element-wise nonlinear combination of Hadamard product, the high-order interaction relationship between different physical fields is effectively expressed, and the calculation efficiency of this coupling is very high, which makes the model maintain high computational efficiency even in the case of a large number of spot welds, and has good scalability. Therefore, the geometric deformation mechanism of the multi-surface multi-spot welding system described by equation (39) not only has strong expressive power, but also has high computational efficiency, and can effectively capture the high-order interaction features in the multi-surface multi-spot welding system, providing an efficient and reliable representation model for geometric precision prediction of the multi-surface multi-spot welding system.

[0232] Step three: White body assembly geometric precision posterior inference and generation mechanism.

[0233] 31) Construct a white body assembly geometric precision posterior state inference model, including: mapping a three-dimensional point on the white body to a point in the two-dimensional parameter space through the coordinate mapping network φ; constructing a neural network f based on attention mechanism for learning 0 form v and metric tensor g; combining exterior differential operator d and Hodge star operator to extract high-order geometric features, and constructing a white body assembly geometric precision posterior state inference model.

[0234] Specifically, the white body assembly is connected by multiple stamping surface parts through multi-point welding assembly process, and its geometric precision is affected by multiple factors. As can be seen from equation (39), the geometric deformation mechanism of the multi-surface multi-spot welding system is a complex process involving high-order interaction of multiple factors. Among them, the first is the factor related to the material, property and geometric feature of the surface part itself (such as Green function factor matrix Initial state field of each surface part etc.), which is the comprehensive result of stamping raw materials and stamping, flanging, punching and other processes, and determines the initial comprehensive formability state of the surface part before welding; the second is the factor related to the welding process, such as factor matrix A (i), different curved parts and welding points, and the weight matrix Λ of the influence on the assembly accuracy, etc. These factors are more complex, covering the position of the welding point, the welding current voltage, the electrode cap state, the welding pressure, and the alignment accuracy of each curved part, the gap surface difference control, the deformation coupling, and other aspects in the welding process. Since the geometric accuracy of a single curved part is not affected by the process parameters and state of the welding system, on the contrary, the process parameters and state of the welding system are fully reflected in the geometric accuracy of the body-in-white assembly. Therefore, in order to accurately predict the future geometric accuracy evolution of the body-in-white assembly, it is not enough to only infer the deterministic posterior state of each single curved part at the current and historical time, but also to infer the posterior state of the process parameters and state of the welding system based on the geometric accuracy of the body-in-white assembly.

[0235] However, in actual manufacturing, the geometric accuracy measurement points of the body-in-white assembly are usually limited to some key points, and the point cloud density is much lower than that of the blue light scanning of the single curved part. If the method of processing the discrete grid signal on the manifold is directly applied to the geometric accuracy inference of the body-in-white assembly, two main problems will be faced: one is that the point cloud density of the geometric accuracy of the body-in-white assembly is low, which cannot effectively approximate the 2-dimensional manifold (embedded in 3-dimensional space) represented by it; the second is that the basis of the existing method is the discrete Laplace-Beltrami operator on the manifold, which requires a defect-free triangular mesh, while the geometric accuracy point cloud of the body-in-white assembly is usually irregular and extremely sparse, more accurately, a sparse graph signal, rather than a grid signal. Moreover, more importantly, even if the sparse point cloud of the assembly is regarded as a graph signal, it is not suitable for processing by methods such as graph network. This is because the basic assumption of the graph network is the normalized Laplace matrix, but the normalized Laplace matrix only defines the topological structure of the graph nodes, and cannot consider the geodesic line on the complex curved surface of the body-in-white assembly, nor can it consider the complex curvature change near a certain measurement key point due to processes such as spot welding, so it cannot accurately capture the features of the local or global geometric accuracy. More accurately, the body-in-white welding deformation can be represented as follows:

[0236]

[0237] where u(x) is the displacement vector of the body-in-white at position x = (x 1 ,x 2 ); is a functional that will displace the vector u and its local coordinate system (x 1 ,x 2) and the position information x is mapped to a source term S(x). And the source term S(x) represents the "input" of the BIW welding process, such as process parameters (welding current, voltage, time, etc.), external force, and other external factors that affect deformation, etc. Therefore, the goal of the BIW assembly geometry accuracy posterior state inference should be to infer the "input" information of the welding process according to the deformation information of the BIW surface, that is, to learn the inverse mapping of the functional Thus, the hidden causes (welding process input) are inferred from the observations (deformation information).

[0238] However, since u is on the uneven manifold of the BIW surface, the processing of u and its various differential is complicated. At the same time, the analytical representation of this manifold cannot be obtained, and only discrete and sparse key measurement points can be observed. To solve this problem, the embodiment proposes a BIW assembly geometry accuracy posterior state inference method based on neural differential forms. The method regards the surface of the BIW assembly as a two-dimensional Riemannian manifold embedded in a three-dimensional space, and regards the key physical field on the surface of the BIW assembly as a differential form. In modern differential geometry, a differential form is an algebraic structure used to describe the geometric characteristics, topological structure and differential structure of a manifold. And the exterior derivative is an operator that acts on differential forms, which can convert low-order differential forms into high-order differential forms, thereby extracting more rich geometric information. Unlike purely discrete differential forms that rely on well-defined K-simplexes, in this embodiment, a deep neural network is introduced to learn the continuous differential form of the BIW assembly surface. First, the differential form and the metric tensor of the BIW assembly surface are inferred from the numerical geometry accuracy of the discrete measurement points, and then the inferred differential form is processed by using modern differential geometry tools such as the exterior derivative and the Hodge star operator, and finally the posterior state of the BIW assembly geometry accuracy is obtained. Specifically, let the BIW assembly surface be a two-dimensional Riemannian manifold with a local coordinate system (x 1 , x 2 ), and its metric tensor can be represented as a 2x2 matrix:

[0239]

[0240] where g 12 = g 21 (the metric tensor is symmetric). For any point p on the BIW (manifold), the metric tensor g defines an inner product on the tangent space g 11 and g 22 represent the inner product of tangent vectors and on respectively. The inverse of the metric tensor g is:​

[0241]

[0242] where is the determinant of the metric tensor.

[0243] It should be pointed out that in actual production, sometimes it is difficult to obtain the accurate 3D model corresponding to the production model at the manufacturing end, so the metric tensor g cannot be directly calculated according to the spline curves or other geometric features of the body-in-white, and therefore the local coordinate system (x 1 ,x 2 ) cannot be defined. The only information available is the coordinate deviation of the key measurement points on the assembly and their reference coordinates obtained by the three-coordinate machine or laser measuring instrument. Therefore, considering this situation, the embodiment proposes an implicit parameterization method based on a deep neural network to learn the continuous differential form (0-form) and the metric tensor |g| of the body-in-white assembly surface.

[0244] Specifically, in the embodiment, the method for constructing the body-in-white assembly geometry accuracy posterior state inference model is as follows.

[0245] 311) Construct a coordinate mapping network φ to map a three-dimensional point on the body-in-white to a point in the two-dimensional parameter space, denoted as:

[0246]

[0247] where φ is a deep neural network shared by all three-dimensional points, which maps a three-dimensional point on the body-in-white to a point in the two-dimensional parameter space. Thus, the local coordinate system of the body-in-white surface manifold is implicitly defined. Since the neural network φ is continuous and differentiable to the input, it enables differential geometry analysis in the parameter space.

[0248] 312) Construct a neural network f to learn the 0-form and the metric tensor, i.e., based on the obtained local coordinate system of the manifold, a neural network f based on the attention mechanism is constructed to learn the 0-form and the metric tensor:

[0249]

[0250] where f is a multi-head attention network, is a value 0-form, is the metric tensor. In the actual construction of f, since the metric tensor of the Riemannian manifold is symmetric, the parameters of g are only three, i.e., g 11 ,g 22 and g 12 , so the actual output dimension of f only needs to be set to n+3.

[0251] 313)The 0-form v is acted on by the exterior differential operator d to obtain a 1-form dv.

[0252] The 0-form v represents the distribution of a number of scalar physical fields on the body surface, such as temperature, strain energy, etc., which are automatically learned according to the neural network f(φ(x, y, z)). However, it is not enough to know only the distribution of these scalar fields, but also to understand how these physical quantities change in various directions on the body surface. For example, the temperature gradient determines the direction and strength of heat flow, while the rate of change of strain energy reflects the distribution of stress inside the body. In order to capture this change information, higher-order differential forms need to be introduced. First, consider the 0-form v acted on by the exterior differential operator d to obtain a 1-form dv:

[0253]

[0254] where dv i is the differential of the i-th component of v; e i is the standard basis vector in ; and denotes the tensor product. The exterior differential operator d is a generalization of the total differential operator on differential forms, which converts a 0-form into a 1-form. Each component of the 1-form dv is a tangent vector, which describes the rate of change of the 0-form v on the body surface. Note that the partial derivatives with respect to the local coordinate system (x 1 ,x 2 ) in equation (45) can be conveniently calculated using the automatic differentiation feature of the neural network f.

[0255] It can be observed from equation (45) that the matrix on the right side of the equation is actually the Jacobian matrix of v. Therefore, so far, only the gradient of the key physical fields on the body surface at each point has been inferred through the neural network, which seems not enough to constitute a powerful feature learning model. In order to extract more rich geometric features, the Hodge star operator is considered here. The Hodge star operator is an important concept in differential geometry, which provides a method to map a k-form (linear) to an (n-k) form, where n is the dimension of the manifold. The Hodge star operator can be regarded as an orthogonal complement operation of differential forms. More generally, assuming that dx 1 ,…,dx n is a set of standard orthogonal basis of , then for k standard orthogonal vectors , the wedge product is a k-form, and its Hodge star is an (n-k) form, and satisfies the following relationship:

[0256]

[0257] That is, the wedge product of a k-dimensional "unit volume" with its complementary (n-k)-dimensional "unit volume" results in the unique n-dimensional "unit volume" on The wedge product (∧) is an operation in exterior algebra that combines two differential forms of lower order into a differential form of higher order. In particular, for two standard orthogonal vectors and their wedge product has a Hodge star that satisfies:

[0258]

[0259] This means that the Hodge star operator maps a "unit area" 2-form to a "unit length" 1-form, and vice versa. This property guarantees that the Hodge star operator preserves the relative size of geometric quantities like area and length when converting between differential forms of different orders.

[0260] 314) Utilize the Hodge star operation to convert the 1-form dv into another 1-form *dv.

[0261] On a two-dimensional Riemannian manifold for a 1-form ω = adx 1 + bd 2 x, the Hodge star is calculated as:

[0262]

[0263] Therefore, for the 1-form dv on the body surface, its Hodge star operation *dv can be expressed as:

[0264]

[0265] Equation (49) converts the 1-form dv into another 1-form *dv, which describes the rate of change of the gradient of the 0-form v in the normal direction at each point on the body surface. This can be understood as describing the geometric characteristics of the 0-form v from another perspective (orthogonal complement). Moreover, this orthogonal complement space description takes into account the metric tensor of the manifold, thus encoding the curvature information of the manifold. On the other hand, to further extract more rich geometric features, the natural approach here is to further pass through the exterior derivative operator d to obtain a 2-form that describes higher-order geometric features. However, according to the properties of the exterior derivative, d(dx i ) = 0, therefore, directly applying the exterior derivative operator to *dv again will result in a zero form, which cannot produce higher-order differential forms.

[0266] 315)Applying the exterior differential operator d to a 1-form *dv gives a 2-form d* dv.

[0267] The usual practice is to build a bridge to 2-forms by the Hodge star operator *dv. Specifically for a 1-form ω = a(x 1 ,x 2 ) dx 1 + b(x 1 ,x 2 ) dx 2 its exterior derivative dω is:

[0268]

[0269] In the above equation, since the exterior differential operator d is a linear operator, dω can be split into two terms da∧dx 1 and db∧dx 2 . da and db are the total differentials of a and b, which can be expressed by partial derivatives and dx 1 and dx 2 , and the wedge product ∧ is an antisymmetric operation, i.e., dx 1 ∧ dx 1 = dx 2 ∧ dx 2 = 0 and dx 2 ∧ dx 1 = - dx 1 ∧ dx 2 . Therefore, the final result only contains the term dx 1 ∧ dx 2 . Thus, according to equation (50), applying the exterior differential operator d to a 1-form *dv gives a 2-form d* dv:

[0270]

[0271] where is:

[0272]

[0273] Equation (51) converts a 1-form *dv into a 2-form d* dv. Compared with a 1-form, a 2-form represents the comprehensive geometric characteristics of an infinitesimal area element on the body-in-white manifold, and it has more abstract and powerful feature expression capabilities than a 1-form, which reflects the curvature size and direction of each infinitesimal area element on the body-in-white assembly surface. However, due to the complex influence of various factors on the geometric accuracy of the body-in-white assembly, there may be a highly nonlinear relationship between the surface curvature variation and the final geometric accuracy. In order to more accurately capture this nonlinear relationship and further enhance the expression capability of the model, a learnable linear transformation is considered to be introduced to the 2-form d* dv:

[0274]

[0275] where is a learnable parameter matrix that linearly combines each component of the 2-form d*dv.

[0276] 316)Applying the Hodge star operator to the 2-form d*dv again, a 0-form *d*dv is obtained.

[0277] It is noted that the 2-form represents an integral quantity that needs to be integrated before use and cannot directly participate in subsequent calculations. Therefore, the 2-form d*dv needs to be converted into a 0-form here, so as to further participate in subsequent feature learning. Since the Hodge star operator * can map a k-form on an n-dimensional manifold to an (n-k) form, and *d*dv can further comprehensively describe the curvature size and direction on each infinitesimal area element of the body surface, i.e., reflecting the "source" and "sink" of different physical fields at each point, therefore, the Hodge star operator * can be applied to the 2-form d*dv again, and a 0-form *d*dv can be obtained:

[0278]

[0279] where is the generalized Jacobian matrix of the linear transformation . *d*dv is a value 0-form.

[0280] 317)Applying a nonlinear activation function σ to *d*dv, it is embedded into a deep neural network to construct a feature learning model.

[0281] Further, by applying a nonlinear activation function σ (such as ReLU) to *d*dv, it can be embedded into a deep neural network, thereby constructing a more powerful feature learning model. Specifically, define the linear layer of each layer as:

[0282]

[0283] where H (l) (v (l) ,g) can be calculated by automatic differentiation according to equation (5.52). is the learnable linear transformation matrix of the lth layer, m l is the dimension of the lth layer feature. Thus, the forward propagation of the entire network can be represented as:

[0284]

[0285] where x = (x, y, z) is the 3D coordinates of the body-in-white assembly; f (φ(x)) v denotes the first output of the multi-head attention network f at input φ(x), which is value 0 form denotes the second output of the multi-head attention network f at input φ(x), which is the metric tensor g; φ is an MLP-based coordinate mapping network; f is a multi-head attention network (keys are 2D local coordinates (x 1 2 on the body-in-white manifold, and values are 0-form v and metric tensor g). denotes the composition of functions.

[0286] Thus, a neural differential form based body-in-white assembly geometric accuracy posterior state inference model is constructed. The model can effectively learn the continuous differential form and metric tensor of the body-in-white assembly surface from sparse key point measurement data, and use exterior differentiation, Hodge star operator, learnable nonlinear transformation, nonlinear superposition, etc. operations to extract more rich geometric features, and finally obtain the posterior state of the assembly geometric accuracy, which can strongly represent the key factors of the body-in-white assembly geometric accuracy change caused by the welding process of the multi-surface part.

[0287] 32) A body-in-white assembly geometric accuracy generation mechanism, comprising: defining a unified state vector x (n) updating the displacement field and the physical field state through a multi-linear interaction layer

[0288] Specifically, in the actual welding process of the body-in-white, multiple surface parts jointly affect the geometric accuracy of the body-in-white assembly through the high-order interaction mechanism represented by formula (39). This process is essentially a generation process, and the high-order interaction between the multi-surface parts is the internal mechanism that dominates this generation process. The high-order interaction is essentially a multi-linear mapping. Compared with the layer-by-layer linear mapping and nonlinear activation in deep neural networks, the high-order interaction based on multi-linear mapping can provide more powerful feature interaction and combination ability, making the model better capture the complex coupling relationship between the multi-surface parts. Therefore, an important role of formula (39) is actually to introduce a new multi-linear layer for deep neural networks, which can explicitly model the high-order interaction between the surface parts, greatly supplementing and enhancing the expression ability of the traditional neural network layer. It is also noted that formula (39) only describes the increment of the body-in-white assembly geometric deviation at a single time step, so it is also necessary to stack multiple time steps to construct a deep neural network based on the multi-linear interaction layer, thereby realizing the deep generative prediction of the body-in-white assembly geometric accuracy under the multi-surface part.

[0289] ​​Specifically, assuming that the geometric accuracy of m curved parts can be observed before a welding process is performed, the deterministic posterior state of each curved part can be inferred where N m represents the number of grid nodes of the mth curved part, F m represents the feature dimension of each node, reflecting the initial material properties, surface quality, geometric features, and other information of the curved part. Therefore, s (m) can be regarded as an initial comprehensive state of the curved part m after stamping and before welding, which should determine the initial physical field state in equation (39) This can be achieved by constructing a learnable encoder network ε: :

[0290]

[0291] where is the initial physical field state of the mth curved part, P is the fixed number of welding points, k is the number of physical fields, and D is the feature dimension. It should be noted that the deterministic posterior state S (m) of the original curved part is defined on a graph structure, where each node corresponds to a feature vector. The encoder network ε disperses the graph structure and encodes the graph structure features of the entire curved part into a fixed-dimensional vector description, thereby achieving further abstraction and compression of the graph structure features. Meanwhile, the number of welding points P for each curved part is set to a fixed value, on the one hand, because the actual number of welding points on each curved part is not easily known; on the other hand, more importantly, the number and distribution of welding points on a particular curved part are fixed and unchanging, and the welding points that significantly contribute to the geometric deformation of the body-in-white are also limited. Therefore, setting the number of welding points to a fixed value can effectively reduce the number of model parameters, allowing the model to automatically learn the influence of the top P most important welding points, thereby implicitly learning the fixed structural information.

[0292] To better describe the changes in the body-in-white assembly and state vector caused by the interaction of multiple curved parts at each time step, a unified state vector x (n) is defined as:

[0293]

[0294] where is the geometric accuracy or other related features of the geometric accuracy of the body-in-white assembly at the nth generation step, and d is the feature dimension. is the physical field state of the mth panel. Thus, the update of the unified state vector at each time step can be expressed as:

[0295]

[0296] where is a nonlinear function defined as:

[0297]

[0298] where [·] denotes the index of the corresponding element in the vector x (n-1) , and correspond to the displacement field increment of the body-in-white assembly and the update of the mth panel physical field state, respectively. is calculated according to equation (5.39), while can be updated by:

[0299]

[0300] where is a fourth-order tensor that maps the intermediate features in R to P welds, each corresponding to k physical fields, each corresponding to a D-dimensional feature vector. is a learnable weight modulation vector. The state information of all welds on the mth panel in all physical fields is fused to obtain an R x P matrix. This matrix is multiplied by the output mapping matrix B m along the fourth dimension of B m to obtain a P x k x D tensor, which represents the update amount of each weld in each physical field. Finally, the tensor is Hadamard multiplied with the weight modulation vector Γ to finely adjust the update amount of each weld in each physical field, obtaining the final physical field state increment tensor

[0301] According to the form of equation (59), superimposing multiple such residual blocks can achieve the construction of the body-in-white assembly geometry precision generation model. Specifically, assuming that the generation process has a total of N discrete generation steps, the generation process can be expressed as:

[0302]

[0303] where denotes the residual block of the nth generation step, x (0) is the initial unified state vector, x (N) is the final state vector, and σ is a nonlinear activation function.

[0304] Thus, a white body assembly geometric precision generation mechanism based on high-order interaction of multi-surface parts is realized. The mechanism can effectively capture the high-order interaction relationship between multi-surface parts, thereby realizing the generative prediction of the geometric precision of the white body assembly. The white body assembly geometric precision posterior state inference based on the neural differential form can effectively learn the continuous differential form and metric tensor of the white body assembly surface from sparse key point measurement data, and effectively infer the key state of the white body assembly in the welding forming process, thereby helping the generation mechanism to predict the geometric precision of the assembly after welding.

[0305] Step four: constructing a white body assembly geometric precision random dynamic evolution prediction model.

[0306] The white body assembly is a complex assembly formed by connecting multiple stamping surface parts through welding process, and the assembly quality is directly related to the quality and performance of the whole vehicle. The geometric precision of the assembly is one of the important indicators for evaluating the quality of the assembly. However, in the actual mass production process, the geometric precision of the white body assembly is easily affected by many factors, such as the size deviation of the surface part itself, the alignment error between the surface parts, the fluctuation of the welding process parameters / state, etc. The complex interaction of these factors makes the geometric precision evolution process of the white body assembly extremely complex, making it very difficult to efficiently predict and actively control it.

[0307] In view of this, the embodiment combines the white body assembly geometric precision posterior state inference model based on neural differential form and the white body assembly geometric precision generation mechanism based on high-order interaction of multi-surface parts to construct a model that can efficiently predict the random dynamic evolution of the white body assembly geometric precision in the mass production process, including: calculating high-order Taylor coefficients through Taylor mode automatic differentiation Using the intermediate state prediction network Generating the Lagrange remainder of the p-order Taylor expansion, outputting the geometric precision prediction value at the target time

[0308] 41) White body assembly geometric precision random dynamic evolution overall framework: based on deep neural random process ζ t Modeling the random dynamic evolution of the white body assembly geometric precision in the time scale.

[0309] Consider a large-scale continuous stamping and welding production line that can observe M key surface parts and their corresponding white body assembly geometric precision. Let t be a beat point time in the continuous production process of the white body assembly, and at time t, the geometric precision of the white body assembly can be observed as where K is the number of key measurement points on the body-in-white assembly, and d is the dimension of each measurement point, which is generally 6-dimension (reference coordinates x, y, z and offset δx, δy, δz). Meanwhile, the geometric accuracy of each surface part m at time t can also be observed:

[0310]

[0311] where, V (m) is the vertex set of the graph |V (m) | = n (m) is the number of measurement points on the m-th surface part. The geometric accuracy static probability distribution of each surface part can be modeled and learned in advance using fractional order diffusion generation model, and thus for a given geometric accuracy of each surface part, a corresponding deterministic posterior state

[0312] To model the random dynamic evolution of the body-in-white assembly geometric accuracy U t in the time scale, a stochastic ordinary differential equation can be used to describe it:

[0313]

[0314] where, is the drift term, which describes the deterministic evolution trend of the body-in-white assembly geometric accuracy over time, which is influenced by the current assembly geometric accuracy, the posterior state of each surface part, and the time t. is the coefficient matrix of the diffusion term, which describes the random fluctuation degree of the body-in-white assembly geometric accuracy over time. dW t is the differential increment of the standard Wiener process, representing the unpredictable random factors in the body-in-white manufacturing system. α and β are parameters of the drift term and the diffusion term, respectively, which need to be learned from data.

[0315] Specifically, directly using equation (64) to model the body-in-white assembly geometric accuracy has some limitations. Mainly, equation (64) assumes that the evolution of the body-in-white assembly geometric accuracy is a Markov process, i.e. the current state U t is only influenced by the previous state U t-Δt (assuming Δt is the discrete time step of equation (64)), so it is difficult to capture the complex random behavior behind the long-range dependence and dynamic evolution of the body-in-white assembly geometric accuracy in the time scale. Therefore, based on the deep stochastic process theory, a deep neural stochastic process ζ t is proposed to model the random dynamic evolution of the body-in-white assembly geometric accuracy in the time scale:

[0316]

[0317] where t1< t2<... < t C ≤ t, is a continuous stochastic function at time t, which represents the comprehensive mechanism determining the stochastic evolution of the body-in-white assembly geometric accuracy in the time scale. is a nonlinear functional, which maps the observable context information (including the historical body-in-white assembly geometric accuracy and the surface part geometric accuracy) to a low-dimensional latent space ( The specific form of is determined by the specific structure of ). The output of the functional can be represented as where is used to represent a comprehensive latent space representation of the entire body-in-white manufacturing system, which contains the global uncertainty of the body-in-white assembly and the deterministic features at the target time. is a conditional information of ζ t , which actually represents the initial physical field state of the m stamped surface parts is a parameter of ζ t , which needs to be learned from data. t1< t2<... < t C ≤ t is the historical observation time point of the geometric accuracy, and C is the number of historical observations. v t is a Gaussian white noise, which is used to describe the random factors that cannot be explained in the model, such as environmental noise, sensor error, etc.

[0318] Equation (65) gives the overall representation of the stochastic dynamic evolution modeling of the body-in-white assembly geometric accuracy, which models the complex stochastic dynamic evolution mechanism of the body-in-white assembly geometric accuracy by introducing a conditional deep stochastic process ζ t . Among them is the core key of ζ t , since ζ t itself is a deep stochastic process, therefore substantially contains a posterior inference of and U t . Specifically, the posterior inference of a single surface part should include a posterior (Gaussian) distribution and a deterministic feature representation of the target time point t * , where the calculation process of the Gaussian posterior distribution is:

[0319]

[0320]

[0321] The calculation process of the deterministic representation of the target time point t * is:

[0322]

[0323] Since each surface component has already undergone a separate depth randomization process The training was performed on the above, therefore, all parameters φ, α, β, θ, W in equations (66) and (67) are... μ b μ W σ b σ All of these are pre-trained stochastic dynamic evolution models ζ based on the geometric precision of the body-in-white assembly. t During the training process, these parameters are fixed. Meanwhile, according to equation (57), the posterior inference of the m-th surface component should output the initial physical field state during the welding process. The initial physical field state of welding is actually determined by the material properties, residual stress, and geometric characteristics of each stamped curved surface part. Therefore, It can be considered as being caused by r and This is determined by [the specific context]. Specifically, in order to [determine the posterior Gaussian distribution of the curved surface component m]... and target time point t * The decisive feature indicates Transform into initial physical field state Here we introduce a new neural network:

[0324]

[0325] Where D r It is r and The feature dimension, ψ is ε (m) The parameters. The aggregated representation r and the decisive feature representation As input, it outputs another Gaussian distribution. The mean and variance vectors:

[0326]

[0327] in and Will By reshaping it into a tensor of (P×k×D), the Gaussian distribution can be obtained. The mean of the matrix is ​​given by the matrix, while the covariance matrix is ​​a diagonal matrix whose diagonal elements are composed of the matrix's mean. This is obtained after reshaping into a (P×k×D) tensor. Finally, the initial physical state... This was obtained by sampling from the Gaussian distribution:

[0328]

[0329] Equation (70) gives the posterior of m simple surface components. For the posterior aspect of the body-in-white assembly, based on the content of step 31), a deep neural network based on neural differential form needs to be constructed to perform posterior inference on the geometric accuracy of the body-in-white assembly. Specifically, given the historical geometric accuracy information of the body-in-white assembly... For each First, construct a coordinate mapping network φ based on MLP: The reference coordinates of each measurement point are mapped to a local coordinate system on the two-dimensional manifold space of the body-in-white assembly surface. Then, a multi-head self-attention network (Equation (44)) is constructed to map the local coordinate systems of all measurement points to a local coordinate system. Value in 0 form:

[0330]

[0331] Where x i The reference coordinates of the i-th measurement point are k. i Let be the local coordinates of the i-th measurement point on the two-dimensional manifold space of the body-in-white surface, and K be the set of local coordinates of all measurement points. MultiHead(·) is a multi-head self-attention network. The initial 0 form of the body-in-white assembly output by MultiHead(·) at K measurement points. Value sampling. Furthermore, according to equation (56), the original cause of the deformation of the body-in-white assembly after processing with l layers of neural differential form can be obtained. (i.e., the external source term S in the deformable constitutive equation (equation (40))). The calculation process of H(·,·) is given by equation (52). The calculation process for each layer is given by equation (56). Finally, Input into a new neural network In, and output

[0332]

[0333] current h A In reality, it implicitly represents a moment t. j Latent space representation of the body-in-white assembly geometry with high precision. To achieve this at all observation contexts and time points. Posterior inferences with commutative invariance on [the data], for We perform averaging and use an MLP to generate the mean and variance of the posterior Gaussian distribution:

[0334]

[0335] By sampling from this Gaussian distribution, we obtain

[0336]

[0337] in R represents the mode coefficient matrix representing the influence of the i-th physical field on the geometric accuracy of the body-in-white assembly, and R is the higher-order tensor of the deformation of the resistance spot-welded sheet. The rank of the CP decomposition.

[0338] Thus, the hidden space is obtained. All representations in , It simultaneously represents the overall uncertainty and decisive characteristics of the body-in-white assembly; it is a shape of... The tensor. It is ζ t The conditional information includes the initial physical field states of m stamped curved surface parts. Therefore, the entire hidden space... This can be interpreted as:

[0339]

[0340] Hidden Space It includes all the key influencing factors of the stochastic dynamic evolution of the geometric accuracy of the body-in-white assembly, among which The global uncertainty of the body-in-white assembly is characterized, including the randomness caused by factors such as fluctuations in welding process parameters and changes in equipment status. The influence mode coefficient matrix A of the physical field is used to describe the influence patterns of different physical fields on the geometric accuracy of the body-in-white assembly. This contains the initial physical field state information from each stamped curved surface part. It reflects the material properties, geometry, and inherent defects formed during the stamping process of each curved surface part, and can be considered as the initial conditions for the welding process of the body-in-white. The hidden space is thus obtained. After all the representations are completed, the body-in-white assembly at the target time point t can be generated according to the previous method. * The rate of change of geometric accuracy. Specifically, according to equation (58), firstly, a unified state vector x for the geometric accuracy generation process of the body-in-white assembly is defined. (n) :

[0341]

[0342] in It is the m-th surface component at the target time point t * The physical field states are given by n = 1, 2, ..., N, where n is the number of layers in the multi-layer residual network describing the geometric accuracy generation process of the body-in-white assembly. It is a posterior inference network of single-curved surface components The samples were obtained, while the initial time was... is a zero matrix According to equation (62), the generation process of the body-in-white assembly can be represented as:

[0343]

[0344] where denotes the residual block of the nth generation step. A is sampled by equation (75). The actual calculation process of is given by equation (61) and equation (39).

[0345] Finally, the body-in-white assembly at the target time point t * is predicted by the model. The geometric accuracy rate prediction of the body-in-white assembly at time point t (N) can be obtained by applying a selection operation s(·) to x

[0346]

[0347] where s(·) selects the first element in x (N) , that is, the geometric accuracy rate of the body-in-white assembly, as the output.

[0348] The training target of the model is to make the predicted geometric accuracy rate of the body-in-white assembly as close as possible to the real geometric accuracy rate dU t / dt, and at the same time ensure that the initial physical field state distribution of each surface part is as close as possible to the prior distribution p(z), and the distribution of the physical field influence mode coefficient matrix A of the body-in-white assembly is as close as possible to the prior distribution p(z A ). Therefore, the loss function here can be defined as:

[0349]

[0350] where, is the geometric accuracy rate of the body-in-white assembly at time point t predicted by the model; is the real geometric accuracy rate of the body-in-white assembly at time point t * ; is the initial physical field state distribution of the mth surface part; is the Gaussian distribution output by the body-in-white assembly posterior inference network; p(z A ) is the hidden state variable z Aprior distribution, usually standard normal distribution; and λ1 and λ2 are control parameters to balance the importance of different loss terms. Equation (80) mainly contains three parts, in which the body-in-white assembly reconstruction loss is used to measure the difference between the model predicted body-in-white assembly geometry accuracy change rate and the true change rate; the stamping surface initial state KL divergence loss is used to ensure the closeness between the initial physical field state distribution of each surface and the prior distribution; and the welding process high-order interaction factor matrix KL divergence loss is used to ensure the closeness between the physical field influence mode coefficient matrix distribution of the body-in-white assembly and the prior distribution. By minimizing equation (80), the body-in-white assembly geometry accuracy stochastic dynamic evolution model ζ t can be trained, which can accurately predict the geometry accuracy change rate of the body-in-white assembly at the future time, and by further solving equation (65), the stochastic dynamic evolution process of the body-in-white assembly geometry accuracy can be obtained.

[0351] 42) Fast and reliable prediction of body-in-white assembly geometry accuracy based on neural numerical integration: Based on the neural network numerical integration method, the high-order Taylor coefficients are calculated by Taylor mode automatic differentiation Use the intermediate state prediction network Generate the Lagrange remainder of the p-order Taylor expansion, and output the geometry accuracy prediction value U at the target time t* .

[0352] Specifically, equation (65) gives the overall representation of the body-in-white assembly geometry accuracy stochastic dynamic evolution model, and by introducing a deep random process ζ t , the complex stochastic dynamic evolution mechanism of the body-in-white assembly geometry accuracy is modeled. However, it is very difficult to directly solve the numerical solution of equation (65). Specifically, since equation (65) is essentially a continuous stochastic differential equation, it usually needs to use numerical integration methods (Euler, implicit Euler, Runge-Kutta, etc.) to solve. However, these traditional numerical integration methods usually require a small time step to ensure the stability and accuracy of the numerical solution. The key here is that ζ t essentially represents the fluctuations and degradation of the entire body-in-white assembly manufacturing system, which may occur at multiple time scales, from a few hours to several weeks or even months. Therefore, if a larger time step is used, the subtle fluctuations and degradation will be ignored, resulting in a large deviation in the prediction result; and if a smaller time step is used, the calculation amount will increase sharply, thereby affecting the real-time management and control ability of the system.

[0353] In order to solve the above problems, the embodiment introduces a neural network-based numerical integration method, the core idea of which is to combine Taylor expansion and neural network to approximate the solution of ODE. Specifically, neural numerical integration uses a fixed-order Taylor expansion to approximate the body-in-white geometry accuracy Ut At the future time point, and learn to predict the Lagrange remainder of Taylor expansion using neural network, so as to avoid the dependence of traditional numerical integration method on time step while ensuring accuracy. Specifically, at the current time point t, assuming the value of body-in-white assembly geometric accuracy U t is known, in order to predict the geometric accuracy of at some future time t * , it is necessary to perform numerical integration on formula (65):

[0354]

[0355] Where the first term U t is the geometric accuracy of body-in-white assembly at the current time, the second term is the integral of depth random process over the time interval [t, t * ], and the third term is the integral of random noise v t over the time interval [t, t * ].

[0356] For the random noise term v t , which represents the random error of body-in-white geometric accuracy in the measurement process, it can be considered as a fixed Brownian motion, and its integral is a Wiener process, which means that its increment at each time point follows a normal distribution and is independent of each other. Since the analytical solution of Wiener process is known, the value of can be directly calculated, so the influence of in formula (81) can be considered separately, and it can be directly added to the final prediction result as a random variable following a normal distribution:

[0357]

[0358] Where, represents the increment of Wiener process over the time interval [t, t * ].

[0359] For the integral , the traditional numerical integration method needs to divide the integral interval [t, t * ] into multiple small intervals Δt and perform approximate calculation on each small interval. However, as mentioned earlier, this method depends on the choice of time step, and the calculation amount is large. In order to overcome these problems, first consider the p-order Taylor expansion of U t at t:

[0360]

[0361] Where R p (t * ​​) is the p-th order Lagrange remainder term, which can be expressed as:

[0362]

[0363] Here, η∈[t,t * [ ] represents an unknown time point within the integration interval.

[0364] Due to U t It is generally quite complex, and U is calculated according to the higher-order chain rule. t Higher-order derivatives involve a large number of combinatorial terms, leading to the combinatorial explosion problem. Furthermore, since η is unknown, the Lagrange remainder R... p (t * η is also unknown. A direct solution is to consider introducing a neural network to predict η; however, directly predicting η to obtain a value from R is not feasible. p (t * There are two problems with estimating η: First, η is a latent variable whose value depends on U. t The high-order behavior over the entire time interval makes it difficult to predict accurately; secondly, η is very sensitive to higher-order derivatives, and small prediction errors can lead to large differences in higher-order derivatives, affecting numerical stability, and the prediction errors of neural networks may introduce this instability. To address these issues, we first introduce a Taylor mode automatic differentiation method to efficiently calculate higher-order Taylor coefficients recursively. Specifically, for U... t The p-th order Taylor expansion can be expressed as:

[0365]

[0366] Where Γ is U t The value at t = η is called the intermediate state. Represents ζ t The l-th order Taylor coefficient at time t can be calculated using the following recursive formula:

[0367]

[0368] in, express About U t The derivative of This indicates that the Jacobian matrix will be used. with vector Multiplication. U can be efficiently calculated using the recursive formula (5.87). t The p-th order Taylor coefficients.

[0369] According to equation (85), in order to predict the white body assembly at a certain target point t in the future... *the midpoint η in the Lagrangian remainder is no longer needed, but only the intermediate state Γ. Directly predicting the intermediate state Γ avoids both the instability introduced by η and the difficulty of computing high-order derivatives by Taylor mode automatic differentiation. For the intermediate state Γ, a parameterized neural network is introduced to predict it. Specifically, with the current geometry U t and the time step Δt = t * -t as input, and outputs an estimate of the intermediate state Γ. For the Since the output space of the neural network is too free, directly outputting the value of Γ cannot guarantee reliability. Once the predicted Γ deviates greatly from the true value, it will lead to inaccurate prediction results, especially in high-dimensional space and large prediction span t * -t. To make the calculation process more controllable, a linear integration tensor is introduced here, such that:

[0370]

[0371] where ζ t (U t )[k, : ] represents the result of the dynamics function ζ t at the kth measurement point, which is a d-dimensional vector, denotes the d × d matrix corresponding to the kth measurement point in Based on (85), it can be rewritten as:

[0372]

[0373] where, denotes the p-order Taylor coefficient calculated at the predicted intermediate state .

[0374] Through (89), even if a larger time step Δt is used, the prediction accuracy can be guaranteed. Specifically, define the error of predicting t * at t

[0375]

[0376] Decompose the total error E into two parts:

[0377]

[0378] is the error caused by the truncation of the Taylor expansion to p order, and:

[0379]

[0380] is due to the prediction error of the neural network .

[0381] According to the Lagrange remainder form of Taylor's theorem, the truncation error R can be expressed as:

[0382]

[0383] where η ∈ [t, t * ]. Suppose the norm on the interval [t, t * ] has an upper bound M: then:

[0384]

[0385] Let the Lipschitz constant of Γ be L, then:

[0386]

[0387] Let the error of the neural network in predicting the intermediate state be δ, then the norm of the intermediate state prediction error satisfies Adding the truncation error and the intermediate state prediction error, the upper bound of the total error is obtained:

[0388]

[0389] From the upper bound of the total error, it can be seen that although a smaller Δt can effectively reduce the truncation error by appropriate selection, increasing p can make the Δt p and Δt p+1 terms rapidly decrease, so as to still maintain a small error at a larger time step. At the same time, ζ [p-1] the Lipschitz constant L of Γ and the norm M on the interval [t, t * ] are also limited, so as long as the order p is large enough, the total error can be kept in a small range, even if a larger time step Δt is used. By setting appropriate p and Δt, the calculation amount can be greatly reduced while ensuring the prediction accuracy, and the rapid and reliable prediction of the body-in-white assembly geometric accuracy can be realized.

[0390] 43) White body assembly geometric accuracy random dynamic evolution model construction and training

[0391] Based on the foregoing theoretical analysis and model design, the main structure of the white body assembly geometric accuracy random dynamic evolution model is as shown in Figure 2 , and the training optimization process of the model is as shown in Table 1.

[0392] Table 1 White body assembly geometric accuracy random dynamic evolution prediction model training optimization process

[0393]

[0394] (1) Single-surface part welding initial state inference module. According to the geometric accuracy measurement data of each of the M single-surface parts, the initial state characteristics before welding are extracted, which provides input for subsequent multi-component coupled modeling. Specifically, for each of the M single-surface parts, the initial state characterization process is as follows: first, the geometric accuracy data of the mth single-surface part at time t is obtained The data contains point cloud of three-dimensional coordinates and deviation information; then, the geometric accuracy data is input into a pre-set feature extraction network, which maps high-dimensional, unstructured point cloud data into a fixed-dimensional initial state vector The vector summarizes the key information that determines the subsequent welding deformation behavior of the surface part, such as material properties, geometric shape, and inherent deviation formed during the stamping process. Further, in order to generate a physical field state that can be used for subsequent multi-component high-order interaction modeling, the initial state vector is input into a state generation network, which outputs a Gaussian distribution parameter and samples the physical field state of the surface part at the initial time of the welding process The feature extraction network and state generation network can be implemented using multi-layer perception (MLP) or graph neural network (GNN) (the specific implementation can be adjusted according to actual conditions), and the mapping from the original geometric data to the abstract state representation is learned through end-to-end training. Through this step, the initial physical field state is output, which provides prior information containing material properties, mechanical state, and geometric characteristics of each single-surface part for the subsequent white body assembly geometric accuracy generation mechanism.

[0395] (2) White body assembly posterior inference module. According to the geometric accuracy data of the white body assembly at historical context time points, the state of the current welding process system is inferred, which can also be considered as a welding process state inference module. This module takes the geometric accuracy of the white body assembly at each context time point t as input, first maps the three-dimensional space coordinates of the white body assembly to the local coordinate system of the two-dimensional manifold space through the coordinate mapping network φ, then learns the interaction relationship between the key measurement points on the surface of the white body assembly using the multi-head self-attention network, and generates the initial differential 0 form V, which represents the multi-physical field state of the surface of the white body assembly. Further, V extracts the high-order differential form features of the surface of the white body assembly through the neural differential form network , and finally generates the hidden space representation of the white body assembly by the feature mapping network ​Finally, the average of all the context time points is taken and used as the MLP network input to generate the Gaussian distribution parameters of the body-in-white assembly posterior state and from which the welding coupling factor matrix A is sampled (i) .

[0396] (3) Body-in-white geometry accuracy generation module. Based on the single-surface-piece welding initial state and welding process state (welding coupling factor matrix), the body-in-white assembly geometry accuracy dynamics state at the target time, i.e., the geometry accuracy change rate, is predicted. The core of this module is the generation mechanism based on the high-order interaction of multi-surface-pieces. First, the initial physical field state of each surface piece and the initial geometry accuracy change rate of the body-in-white assembly (usually a zero matrix) are integrated into a unified state vector x (1) (Eq. (58)); then, the unified state vector is updated using a multi-linear interaction layer , which realizes the high-order interaction between surface pieces and physical fields through factor matrices and Hadamard product, and outputs the body-in-white assembly displacement field increment and the update of each surface piece physical field state, such as (Eq. (59)); further, the multi-linear interaction layer is embedded into a deep residual network, and through multi-layer superposition and nonlinear activation function, the expression ability of the model is enhanced, and the unified state vector x (N) at the target time is output (Eq. (62)); finally, the predicted value of the body-in-white assembly geometry accuracy change rate (N) is extracted from the unified state vector x t (Eq. (63)).

[0397] (4) Body-in-white assembly geometry accuracy random dynamic evolution module. Capture the long-range dependence and dynamic evolution law of the body-in-white assembly geometry accuracy in the time scale. Its input is the geometry accuracy data of the body-in-white assembly and each surface piece at the historical context time points, as well as the hidden space representation obtained by the single-surface-piece welding initial state inference module and the body-in-white assembly posterior inference module The core of this module is a deep random process ζ * , which integrates the geometry accuracy information of the body-in-white assembly and each surface piece at the historical time points into a low-dimensional hidden space (Eq. (76)) through nonlinear functional mapping of historical observation data, and predicts the geometry accuracy change rate of the body-in-white assembly at t * according to the hidden space representation and the target time t t (Eq. (77)).

[0398] (5) White body assembly geometry accuracy fast and reliable prediction module. Based on the predicted geometry accuracy change rate of the white body assembly geometry accuracy random dynamic evolution module, combined with the neural numerical integration method, the future time point of the white body assembly geometry accuracy is realized quickly and reliably, and the confidence interval of the prediction result is given, which is used to quantify the reliability of the prediction result. Specifically, this module first uses formula (89) to predict the white body assembly geometry accuracy U t p-order Taylor expansion is performed, where high-order Taylor coefficients are calculated efficiently by Taylor mode automatic differentiation method (formula (87)), and Lagrange remainder is calculated by neural network The intermediate state Γ is predicted, so as to obtain the geometry accuracy prediction value of the target time point t * In the meantime, the calculation amount is greatly reduced, and the fast and reliable prediction of the white body assembly geometry accuracy is realized.

[0399] Step five: experimental verification: white body assembly geometry accuracy random dynamic evolution prediction experiment

[0400] 51) Experiment introduction

[0401] In order to fully verify the effectiveness of the white body assembly geometry accuracy random dynamic evolution model constructed in this paper, this embodiment is based on the white body manufacturing process of a certain mass-produced vehicle model of a leading domestic new energy vehicle manufacturer. The main factory has complete punching, welding, coating and total automatic production line, and is committed to the research and manufacture of intelligent and high-end new energy vehicles, with an annual output of more than 100,000 units, belonging to the intelligent manufacturing lighthouse factory. Based on the actual data of the white body manufacturing process of the main factory, this embodiment will evaluate the performance indicators of the proposed model in the white body assembly geometry accuracy prediction problem, display the model prediction results, and analyze the feature representation learned by the model, in order to verify its effectiveness and practicality in solving the white body assembly geometry accuracy modeling and prediction problem.

[0402] Figure 3 ​The general process of the new energy vehicle body-in-white assembly manufacturing is demonstrated, including various key links from raw material stamping to online three-coordinate measurement. First, in the stamping process link, the stamping sheet raw material is processed into various shaped curved parts, such as the left side wall, the right side wall, the top cover, the front cover inner plate, the left and right rear door outer plate, and the back door inner plate, etc. These curved parts will be used to form the body-in-white assembly in the subsequent welding process link. The geometric accuracy of the stamped curved parts not only directly affects the quality of the subsequent welding and assembly, but also has complex dynamic evolution characteristics in the time scale, which will further amplify the influence on the geometric accuracy of the body-in-white assembly. Therefore, it is crucial to infer the material properties and mechanical properties of the curved parts based on geometric accuracy, and to model and predict the random dynamic evolution law of the geometric accuracy of the stamped curved parts, for the final accurate prediction of the geometric accuracy of the body-in-white assembly. Subsequently, these stamped curved parts are sent to the welding line, and through the collaborative work of multiple robots, each curved part is welded and assembled into a body-in-white assembly. In the welding process, due to the complex coupling of various factors such as thermal distortion caused by welding, assembly gap between curved parts, process parameter fluctuation, etc., the geometric accuracy of the body-in-white assembly often presents significant nonlinear dynamic evolution characteristics. The lower right side of the figure demonstrates the online three-coordinate measurement link. After welding is completed, the online three-coordinate measuring machine is used to automatically measure the key measurement points on the assembly to obtain the three-dimensional coordinate deviations of each key measurement point. These measurement data contain high-value information about the geometric accuracy quality of the body-in-white assembly and the state of the stamping and welding manufacturing process, which will be used to evaluate the performance of the model proposed in this embodiment in the actual production environment.

[0403] 52) Data preprocessing and model parameter setting

[0404] The experimental data is derived from the blue light scanning and online three-coordinate measurement links of the stamping and welding workshop shown in Figure 3 , including the geometric accuracy measurement data of 6 key curved parts (left / right side wall, top cover, back door inner plate, left / right rear door outer plate) and body-in-white assembly. The experiment covers 10 consecutive weeks of production data, a total of 6820 samples. Among them, the body-in-white assembly contains 94 key measurement points, the left side wall contains 241 key measurement points, the right side wall contains 248 key measurement points, the top cover contains 265 key measurement points, the back door inner plate contains 785 key measurement points, the left rear door outer plate contains 79 key measurement points, and the right rear door outer plate contains 65 key measurement points. Each measurement point contains a 6-dimensional feature vector (x, y, z, δ x , δ y , δ z ), where (x, y, z) is the reference coordinate of the measurement point, and (δ x , δ y , δ z ) is the coordinate deviation of the point in the x, y, z directions.

[0405] In the data preprocessing stage, the geometric accuracy data of the curved surface parts is first constructed into a graph structure. Specifically, as shown in FIG. 1, each "sheet" stacked in the graph represents a sample, and each sample contains the geometric accuracy data of six key curved surface parts (left / right side walls, roof, back door inner panel, left / right rear door outer panel) and the geometric accuracy data of the assembled body-in-white assembly. For each curved surface part, the conversion process from the original point cloud data to the triangular mesh is shown, and a targeted mesh construction algorithm is used to automatically generate the triangular mesh (this embodiment is based on the Pyvista library), supplemented by a small amount of manual modification to ensure mesh quality. The final triangular mesh is the graph structure representation of the curved surface part, which is used as the input of the single curved surface part posterior state inference module, and serves as the condition for the assembly geometric accuracy prediction. Figure 4

[0406] Then, the data is processed by using a sliding time window mechanism to convert the measurement data in continuous time into the input sequence of the model. The time window size is set to 200 (empirically set according to the specific production line), and the sliding step size is set to 1, so that each sample contains the geometric accuracy data of 200 consecutive body-in-white assemblies and the geometric accuracy data of the corresponding six curved surface parts. Through the sliding window mechanism, training samples containing time-dependent relationships can be constructed for model training and verification. After sliding window processing, the final data set is represented as:

[0407]

[0408] where i represents the i-th sample sequence, and the value range is i = 1, 2, …, 6621. represents the body-in-white assembly geometric accuracy data of the i-th sample sequence at time t, which contains a 6-dimensional feature vector of 94 measurement points. represents the geometric accuracy data of the m-th curved surface part of the i-th sample sequence at time t, N m N1 = 241 (left side wall), N2 = 248 (right side wall), N3 = 265 (roof), N4 = 785 (back door inner panel), N5 = 79 (left rear door outer panel), and N6 = 65 (right rear door outer panel). The value range of the time index t is t = i, i+1, …, i+199, i.e., from t = i to t = i+199, a total of 200 consecutive time points. Further, the data set is divided into a training set, a validation set and a test set, and the division is shown in Table 2.

[0409] Table 2 Detailed information of data set division

[0410] ​

[0411] Table 2 summarizes the parameter settings of the model in this example in the experiment, including the single curved surface posterior state inference module, the body-in-white assembly posterior inference module, the body-in-white assembly geometric accuracy generation module, the neural numerical integration module, and the related parameter settings of the model training.

[0412] Table 3 Model parameter settings

[0413]

[0414] 53) Experimental results and analysis

[0415] Based on the parameter settings in Table 3, the proposed body-in-white assembly geometric accuracy random dynamic evolution model is trained on the preprocessed data set. During the model training process, the Adam optimizer and the learning rate decay strategy are used to minimize the loss function defined in equation (80). After 500 rounds of training, the loss function value of the model on the validation set tends to be stable, indicating that the model training has converged.

[0416] In order to comprehensively evaluate the performance of the model, this example will analyze the model from multiple angles. First, the prediction ability of the model for the future time body-in-white assembly geometric accuracy is investigated. Figure 5 The prediction results of the model for the absolute deviation of the geometric accuracy of the key measurement points of the body-in-white assembly in the future 9 days are shown. Each subgraph corresponds to the prediction results of one day, the horizontal axis represents the actual measurement value, the vertical axis represents the model prediction value, the color depth represents the data point density, and the red dotted line represents the ideal case where the predicted value and the actual value completely coincide.

[0417] From Figure 5As can be seen, the model has high prediction accuracy for the next few days. Especially in the early prediction period (e.g. 1 to 3 days in the future), most of the data points are closely clustered around the red dashed line, indicating that the model's prediction results are highly consistent with the actual measurement results, and can accurately capture the short-term random fluctuation trend of the body-in-white assembly geometric accuracy. This shows that the model has successfully learned the main influencing factors and dynamic evolution laws in the short term during the manufacturing process, and can make relatively accurate predictions of future geometric accuracy based on these laws. The high accuracy performance of the model in the early prediction period is due to the strong representation ability of the deep attention neural random process for modeling the time series of the body-in-white assembly geometric accuracy, and also reflects the relatively stable dynamic characteristics of the body-in-white assembly manufacturing system in the short term. As the prediction time span increases (e.g. from 6 days in the future), it can be observed that the deviation of some data points from the red dashed line increases, indicating that the long-term prediction performance of the model has decreased slightly. This phenomenon may be caused by the gradual accumulation of the complexity and uncertainty of the manufacturing system over time, making it difficult for the model to fully capture the dynamic evolution laws of the system in long-term prediction. However, even in the later prediction period (e.g. 9 days in the future), the data points are still distributed around the red dashed line on the whole, indicating that the model's prediction results still have high reference value and can reflect the long-term evolution trend of the body-in-white assembly geometric accuracy. This shows that the model has overcome the challenges posed by the complexity and uncertainty accumulated by the manufacturing system over time to some extent, and can still capture the main trend of geometric accuracy evolution.

[0418] To evaluate the model's ability to generalize and distinguish different quality levels of body-in-white assembly samples, and the impact of different error calculation methods on the evaluation results, Figure 6The evaluation results of the model under different Cpk levels, different distribution distance metrics and different error calculation methods are shown. In the body-in-white manufacturing process, engineers usually judge the quality level of the product according to the Cpk level, and the higher the Cpk value represents the better product quality and the more stable manufacturing process. Therefore, the performance of the evaluation model on samples of different Cpk levels can directly reflect the prediction ability and discrimination ability of the model for white body assemblies of different quality levels. Four different distribution distance metrics are used in the experiment: OT distance, Wasserstein distance, MMD distance and manifold coverage, which are used to measure the difference between the geometric accuracy distribution predicted by the model and the geometric accuracy distribution obtained by actual measurement. Among them, the OT distance can be understood as the minimum "cost" required to "transport" one distribution to another, the Wasserstein distance focuses more on the overall shape difference of the distribution, the MMD distance measures the distance between two distributions by comparing their differences in high-dimensional feature space, and the manifold coverage measures the extent to which the model-generated samples cover the distribution range of the actual measurement samples. In the experiment, the distance metric of the sample space is defined based on MAE, RMSE and MaxAE, which are used to calculate the four indicators. The three error calculation methods reflect the overall deviation, stability and worst case of the model prediction results.

[0419] As can be seen from the figure, as the Cpk level decreases, the distribution difference between the model prediction results and the actual measurement results generally shows a gradually increasing trend, which shows that although the model is evaluated based on distribution distance metrics, it still has strong correlation with the classic Cpk evaluation method, which preliminarily verifies the effectiveness of the model in basic manufacturing capability index evaluation. On the other hand, by comparing the three different error calculation methods, it can be seen that the evaluation results based on RMSE and MaxAE are generally higher than those based on MAE. This is because RMSE amplifies the influence of large deviations by squaring the error, and MaxAE directly investigates the maximum deviation of the prediction results, so these two indicators are more sensitive to outliers or extreme errors in model prediction and are more easily affected by individual abnormal samples. In contrast, MAE, as an error measurement method based on absolute value, is less affected by extreme error values and can more stably reflect the overall prediction performance of the model. However, overall, the influence of different error calculation methods on the model evaluation results is not significant, which shows that the prediction results of the model have good stability and consistency under different error calculation methods. These results show that the proposed model has good generalization ability and discrimination ability on white body assembly geometry accuracy samples of different quality levels, and the evaluation results of different distribution distance metrics and error calculation methods are basically consistent, reflecting the consistency between the white body assembly geometry accuracy feature distribution learned by the model and the true distribution in terms of quality semantics.

[0420] In the process of BIW assembly welding, the state of each curved surface part has an important influence on the geometric accuracy of the final assembly. And the state of single curved surface part is obtained by the deterministic posterior state inference module of the pre-trained fractional order diffusion generative model in this model. Therefore, in order to further explore the mechanism of the fractional order diffusion generative model in the prediction of the geometric accuracy of the BIW assembly, this embodiment designs a set of ablation experiments, focusing on the influence of the fractional order diffusion generative model on the prediction performance of the geometric accuracy of the BIW assembly. Specifically, the original geometric accuracy data of the six key curved surface parts (left side wall, right side wall, roof, back door inner panel, left rear door outer panel and right rear door outer panel) is kept unchanged, and only the pre-trained fractional order diffusion generative model originally used to infer the deterministic state of these curved surface parts is replaced by a simple graph neural network (GNN). By comparing the performance difference of the two model configurations using fractional order diffusion generative model and GNN for state inference in the prediction task of the geometric accuracy of the BIW assembly, the importance of the fractional order diffusion generative model in improving the prediction accuracy is quantitatively evaluated. Other model components, such as deep stochastic neural process, neural differential form and neural numerical integration module, are kept unchanged to isolate the influence of the fractional order diffusion generative model. The experimental results are shown in Figure 7 The figure contains six subgraphs, corresponding to the prediction results of 1 to 6 days in the future. The horizontal axis of each subgraph represents different curved surface parts, including left side wall, right side wall, roof, back door inner panel, left rear door and right rear door. The vertical axis represents the mean absolute error (MAE). For each curved surface part, the mean absolute error of the model using fractional order diffusion generative model and GNN for state inference is calculated on 10 different randomly selected data segments. The box plot in the figure clearly shows the distribution of MAE values of the 10 experiments, where the horizontal line in the box represents the median, the upper and lower boundaries of the box represent the upper quartile and the lower quartile, and the red dotted line (baseline MAE) corresponds to the average MAE of the pre-trained fractional order diffusion generative model.

[0421] As can be seen from the figures, after replacing the pre-trained fractional-order diffusion generative model with GNN, the MAE values of the predictions for all six curved parts and all predicted days are significantly increased, and the distribution range of the box plot is also obviously widened. This clearly shows the important role of the pre-trained fractional-order diffusion generative model in accurately predicting the body-in-white assembly geometric precision. The pre-trained model can extract more representative and more robust feature representations by learning the probability distribution of the geometric precision of the curved part, thereby effectively reducing the error of the assembly geometric precision prediction and improving the stability of the prediction results. On the contrary, although the expression ability of GNN is strong enough, it shows a high inductive bias in this task, tends to learn false correlations in the training data, and leads the model to fail to effectively generalize to the test set, showing high prediction error and large error volatility. Further, the dependence of different curved parts on the pre-trained model varies. For example, for the left side wall, the right side wall and the back door inner plate, which have a greater impact on the body-in-white assembly geometric precision, the increase in MAE value is significantly greater after replacing the pre-trained model with GNN. This shows that the feature extraction effect of the pre-trained model for these key curved parts is particularly significant, and it can more effectively capture their impact on the assembly precision. For the left and right rear doors, which have a relatively small impact, although the MAE value also increases, the increase is relatively small. This may be because the geometric shape of these curved parts is relatively simple, or their impact on the assembly precision is small, so even if GNN is used, some effective features can be extracted. In addition, as the prediction days increase, the MAE values of all model configurations increase, which is as expected, because long-term prediction is more difficult and errors are more likely to accumulate. However, even when predicting 6 days in the future, the complete model using the pre-trained model still maintains the lowest MAE value and the smallest error volatility, which again demonstrates the significant advantage of the pre-trained model in improving long-term prediction accuracy and stability. However, for the GNN model, as the prediction days increase, the MAE value increases more and the error volatility is greater, which shows that the performance of the GNN model is more unstable in the long-term prediction task, and the fundamental reason is that it has not captured the correct distribution of the single curved part manufacturing state, but has introduced some false inductive bias. These results further reflect the effectiveness of the pre-trained fractional-order diffusion generative model in handling the single curved part manufacturing state inference task and its importance in improving the body-in-white assembly geometric precision prediction performance, and prove its key role in the overall performance of the model.

[0422] To further verify the effectiveness of the proposed white body assembly geometric accuracy stochastic dynamic evolution model in this chapter, this embodiment compares the performance of the full model and other model variants at different prediction time spans, and evaluates the impact of key modules on the overall performance. In the experiment, the compared models include: the full model (Full), and model variants that replace or remove key components. Among them, the models that replace the neural differential form module (NDF) include replacing the GWNN constructed in this paper with MoNet, GCN and PointNet++ models, respectively denoted as "MoNet GWNN", "GCN GWNN" and "PN++ GWNN". In addition, there are also models without neural differential form module (w / oNDF) and models without dynamic evolution module based on neural numerical integration (w / oDyn.). The design of these models aims to evaluate the impact of different components on the overall performance, and thus verify the importance of each module in the model. The experiment uses mean absolute error (MAE), root mean square error (RMSE), coefficient of determination (R 2 ), prediction interval coverage rate (PICP), prediction interval average width (PIAW) and coverage width criterion (CWC) indicators to comprehensively evaluate the performance of each model in predicting the geometric accuracy of the white body assembly in the future 1 to 9 days. These indicators measure the prediction accuracy, stability and uncertainty quantification ability of the model from different angles. Figure 8 The experimental results of each model on the above indicators are shown.

[0423] As can be seen from Figure 8 , the full model achieves the best performance on all indicators, especially in prediction accuracy and stability. For example, in terms of MAE and RMSE, the full model shows the lowest error value in all prediction days, and the error grows slowly, which indicates that the full model can more accurately predict the future geometric accuracy change and has better warning ability for potential problems in the white body manufacturing process. In contrast, the error values of other models are significantly higher than those of the full model, and the error grows more rapidly with the increase of prediction days. This shows that the organic combination of modules in the full model can effectively improve the prediction ability of the model. In terms of the coefficient of determination (R 2 ), the full model maintains a high goodness of fit in each prediction period, indicating that the full model can better explain the trend of geometric accuracy change and has important reference value for predicting and solving potential problems in the manufacturing process. In contrast, the R 2 values of other models are significantly lower than those of the full model, especially when the neural differential form module and the dynamic evolution module are removed, the R 2The value decreases more obviously, indicating that these modules play a key role in improving the interpretability and long-term prediction performance of the model. In terms of uncertainty quantification indicators, the complete model also achieves the best performance. In terms of prediction interval coverage probability (PICP), the PICP value of the complete model is about 0.96 when predicting 1 day in the future. As the prediction days increase, the PICP value gradually decreases to about 0.72, but is still higher than other models. This shows that the prediction interval provided by the complete model has high reliability and can effectively cover the actual measurement value, helping engineers more accurately assess product quality during the manufacturing process. The prediction interval average width (PIAW) indicator reflects the width of the prediction interval. The PIAW value of the complete model remains low for all prediction days, meaning that while providing high coverage, the prediction interval is not overly broad and has good practicality. Considering PICP and PIAW, the coverage width criterion (CWC) indicator can more comprehensively evaluate the performance of the model in uncertainty quantification. The CWC value of the complete model is the lowest for all prediction days, indicating that it has a clear advantage in uncertainty quantification and can provide more reliable support for quality control and risk management in the manufacturing process. In contrast, the CWC values of other models are relatively high, especially when the key modules are removed, the CWC value increases significantly, indicating that these models have deficiencies in uncertainty quantification and may not meet the high requirements of prediction accuracy and reliability in actual production. Therefore, the experimental results fully demonstrate the importance and irreplaceability of each key module in the model in improving prediction accuracy, stability, and uncertainty quantification ability, and also reflect that they are not simply component-wise superposition, but mutual synergy, with each module playing a key role in different aspects of the white body assembly geometry precision prediction.

[0424] The formation of the white body assembly geometry precision is largely influenced by the state of each single surface part. In the model of this embodiment, the state of each single surface part is essentially dependent on the wavelet filter GWNN on the surface manifold. To further investigate the mechanism of the wavelet filter in predicting the white body assembly geometry precision and verify its key role, for each stamped surface part, a portion of the wavelet filter is randomly recovered from its wavelet operator matrix, and the gradient of each recovered wavelet filter with respect to the white body assembly geometry precision is calculated to quantitatively evaluate the contribution of different wavelet filters to the assembly precision prediction. The relevant experimental results are as follows: Figure 9The wavelet filter weight distribution and its gradient contribution value to the body-in-white assembly geometry accuracy prediction of 6 key stamping panel (left side wall, right side wall, roof panel, back door inner panel, left rear door outer panel and right rear door outer panel) are shown in the figure. The size of the wavelet filter weight and its comprehensive gradient contribution value to the body-in-white assembly geometry accuracy are indicated by the two color scales below the figure, the larger the value, the more red and yellow the color; the smaller the value, the more blue and green the color.

[0425] As can be seen from the figures, the distribution of the wavelet filter weights of each curved surface part presents obvious differences, among which the wavelet filter weight distribution of key curved surface parts such as the inner panel of the back door, the left and right side walls and the roof is more concentrated, while the wavelet filter weight distribution of secondary curved surface parts such as the left and right outer panels of the back door is relatively dispersed. This shows that the influence degree of different curved surface parts on the assembly geometric precision is different, and the state of the key curved surface parts has a more significant influence on the assembly precision. This conforms to the experience in actual production and also shows that the feature representation learned by the model has good interpretability. At the same time, the wavelet filter weight and its corresponding gradient contribution value have a certain correlation in spatial distribution. The area with higher weight often corresponds to a larger gradient contribution value. For example, in the first group of data, the wavelet filter weights of the A-pillar area of the left side wall and the lower right area of the inner panel of the back door are obviously higher than those of other areas, showing a high-light distribution of red and yellow. Correspondingly, in the gradient contribution value graph of this group of data, these two areas also show a higher gradient contribution value, with a color obviously leaning towards red. This shows that the geometric shape of the A-pillar of the left side wall and the lower right part of the inner panel of the back door plays a crucial role in the prediction of the body-in-white assembly precision, and the model effectively captures these key geometric features by assigning higher wavelet filter weights to these areas; a similar pattern is also presented in the second group of data. The wavelet filter weights of the lower part of the B-pillar of the left side wall and the bottom beam of the right side wall are higher than those of the surrounding areas, showing bright red and yellow areas in the weight distribution graph. In the gradient contribution value graph corresponding to this group of data, the surrounding areas of these two areas also show obvious red high-light areas, indicating that they have a larger contribution to the prediction of the body-in-white assembly geometric precision. By comparing the weight distribution and gradient contribution value of each curved surface part in different groups of data, it can be found that although there is a certain correlation between them, there are also some differences. This reflects the individual differences of different body-in-white samples in the actual production process. Due to the influence of factors such as incoming material quality, process parameters, assembly conditions, especially the uncertainty and randomness in the welding process, the state and interaction mode of the curved surface parts of different samples may have certain differences. This also reflects that it is very necessary and meaningful to introduce the welding factor matrix inferred based on the historical body-in-white assembly geometric precision data in the model. The experimental results further verify the importance of the wavelet filter of the stamped curved surface part in the prediction of the body-in-white assembly geometric precision, prove its contribution to the prediction of the assembly geometric precision, and also provide an important reference for quality traceability and problem positioning in the actual manufacturing process.

[0426] The above experimental results fully demonstrate the effectiveness and superiority of the white body assembly geometric precision random dynamic evolution model proposed in this embodiment. The model shows good generalization ability and discrimination ability for white body assembly samples of different Cpk quality levels, and the evaluation indicators also show high consistency and stability under different distribution distance metrics and error calculation methods. In addition, the ablation experiment further verifies the importance and irreplaceability of each key component in the model, especially the key role of the pre-trained fractional order diffusion generation model in accurately inferring the manufacturing state of single curved surface parts. At the same time, the visualization analysis of wavelet filter weights and gradient contribution values intuitively shows the influence of geometric features in different regions on the assembly precision prediction, highlighting the interpretability of the feature representation learned by the model. These results fully demonstrate the feasibility and effectiveness of the white body assembly geometric precision random dynamic evolution model proposed in this embodiment in actual production, providing important theoretical and methodological support for quality control, predictive maintenance and intelligent decision-making in white body manufacturing processes.

[0427] The above-described embodiments are merely preferred embodiments of the present application and are not intended to limit the protection scope of the present application. Equivalent substitutions or transformations made by those skilled in the art based on the present application are within the protection scope of the present application. The protection scope of the present application is defined by the claims.

Claims

1. A method for multiphysics-based depth modeling and geometrically accurate generative prediction of vehicle body welding, characterized in that: Includes the following steps: Step 1: Analysis and representation of the deformation mechanism of thin plates in spot welding connection process: 11) The total strain ε of the thin plate is determined through coupled iteration of electric field analysis, thermal field analysis, phase transformation analysis, and mechanical analysis. (n) ; 12) Represent the strain mechanism as a 7th-order tensor And based on 7th order tensor The total strain ε is obtained (n) The 7th order tensor The state vector set consists of 6 state vector dimensions and 1 output dimension. It contains key physical field information including stress, temperature, and solid fraction; Step 2: High-order interactive representation of geometric accuracy in the welding process of multi-curved parts: 21) Low-rank characterization of strain in monosurface components based on CP tensor decomposition, including: For higher-order tensors Perform CP decomposition to generate factor matrix A (i) and output weight matrix W; Nonlinear coupling of physical field increments is achieved through the Hadamard product (⊙). 22) High-order interactive characterization of geometric deformation during multi-spot welding of multi-curved parts, including: Define the generalized Green's function matrix Establish displacement increment Δu (n) With strain increment Δε (n) The mapping relationship; Introducing higher-order tensors Describe M surface parts and P for each surface part m. m Interaction between solder joints, and the use of a solder joint interaction mode matrix. Describe the coupling relationship between weld points within the same curved surface component; Step 3: A posteriori inference and generation mechanism of geometric accuracy of body-in-white assembly: 31) Construct a posterior state inference model for the geometric accuracy of the body-in-white assembly, including: A coordinate mapping network φ maps a three-dimensional point on the white car body to a point in a two-dimensional parameter space. Construct an attention-based neural network f to learn the zero form v and the metric tensor g; By combining the external differential operator d and the Hodge star operator to extract higher-order geometric features, a posterior state inference model for the geometric accuracy of the body-in-white assembly is constructed. 32) Mechanism for generating geometric accuracy of body-in-white assembly, including: Define a unified state vector x (n) Through multi-linear interaction layers Update the displacement field and physical field state; Step 4: Construct a geometric accuracy stochastic dynamic evolution prediction model for the body-in-white assembly. Based on deep neural stochastic processes ζ t Modeling the stochastic dynamic evolution of the geometric accuracy of the body-in-white assembly over time; A neural network-based numerical integration method automatically calculates higher-order Taylor coefficients through Taylor mode differentiation. Using intermediate state prediction network Generate the Lagrange remainder of the p-th order Taylor expansion and output the geometrically accurate predicted value of the target time.

2. The method for multiphysics depth modeling and geometric accuracy generative prediction of vehicle body welding according to claim 1, characterized in that: In step 11), the methods for electric field analysis, thermal field analysis, phase transition analysis, and mechanical analysis are as follows: Electric field analysis updates the conductivity σ using the time-varying Poisson equation of the electric potential field. (n) Generates Joule heat source Q (n) ; Thermal field analysis updates the temperature field T based on the heat conduction equation. (n) And comprehensively consider the effects of Joule heating, latent heat of phase change, and heat conduction on the temperature field T (n) The impact; Phase transition analysis utilizes the kinetic equations to determine the solid-liquid phase transition process and solves for the solid fraction f. s (n) and volume change ΔV (n) ; The mechanical analysis uses the back-mapping algorithm to solve for the plastic strain tensor ε. (p,n) .

3. The method for multiphysics depth modeling and geometric accuracy generative prediction of vehicle body welding according to claim 1, characterized in that: In step 12), the total strain ε (n) Represented as: v1=vec(σ (n) ),v2=vec(σ (n-1) ),v3=ΔT (n) ,v4=Δf s (n) ,v5=f s (n-1) ,v6=f l (n-1) in: For 7th order tensors × i v represents the tensor-vector product along the i-th mode; i σ represents the i-th state vector; (n) σ represents the conductivity at the current time step. (n-1) ΔT represents the conductivity at the previous time step. (n) Δf represents the temperature field at the current time step. s (n) f represents the solid fraction at the current time step; s (n-1) f represents the solid fraction at the previous time step; l (n -1) Indicates the liquid phase fraction at the previous time step; i = 1, 2, 3, 4, 5, 6.

4. The method for multiphysics depth modeling and geometric accuracy generative prediction of vehicle body welding according to claim 1, characterized in that: In step 21), the higher-order tensor Performing CP decomposition, it can be represented as the sum of several rank-1 tensors: in: It is the factor vector of the i-th pattern; It is the factor vector of the output pattern; ° represents the vector outer product; R is the rank of the CP decomposition, which determines the number of rank-1 tensors after decomposition; N i The feature dimension of the i-th state physical quantity is represented by k; k is the number of state physical quantities. Factor matrix A (i) Represented as: Factor matrix A (i) The element in the j-th row (A) (i) ) j,: This indicates the contribution of the j-th physical quantity in the i-th mode to the formation of R different rank-1 tensors; The weight matrix W is represented as: Among them: A (k+1) Corresponding to the output layer, it is used to map the low-dimensional feature space to the final output space; Nonlinear coupling of physical field increments is achieved through the Hadamard product (⊙), expressed as: Where: ⊙ represents the Hadamard volume; A represents the i-th state vector; (i) This represents the contribution level of the i-th physical quantity.

5. The method for multiphysics depth modeling and geometric accuracy generative prediction of vehicle body welding according to claim 1, characterized in that: In step 22), the displacement increment Δu (n) With strain increment Δε (n) The mapping relationship is as follows: higher-order tensors Represented as: Where: R is the rank of the CP decomposition, which determines the complexity and expressive power of the model; λ r These are the weighting coefficients of the decomposition; Indicates the contribution of different curved surface components; This represents the contribution of the solder joint on the m-th curved surface component; These represent the contributions of different physical fields; This indicates the contribution of the output strain increment; This represents the direct sum operation, used to connect the solder point vectors of different curved surfaces, so as to integrate the solder point contributions on different curved surfaces into a single tensor; The total strain increment of a multi-spot welding system for multi-curved parts is expressed as: Where: e m It is an indicator vector for surface component m, that is, a vector in which only the m-th element is 1 and the other elements are 0, used to select the corresponding surface component; It is another indicator vector used to select the p-th solder point on the surface part m; This represents the increment of the i-th physical quantity caused by the p-th solder joint on the m-th curved surface at the n-th time step; Displacement field increment Δu (n) The expression is: in: and Indicates the input pattern of the Green's function; It is a factor matrix; This is the solder joint interaction mode matrix; denoted as , indicating the increment of the i-th physical quantity at the n-th time step caused by the p-th solder joint on the m-th surface component; ⊙ represents the Hadamard product of multiple matrices or vectors; Λ is the weight coefficient vector of the CP decomposition.

6. The method for multiphysics depth modeling and geometric accuracy generative prediction of vehicle body welding according to claim 1, characterized in that: In step 31), the method for constructing the a posteriori state inference model of the geometric accuracy of the body-in-white assembly is as follows: 311) Construct a coordinate mapping network φ to map a 3D point on the white car body to a point in the 2D parameter space, represented as: Where: (x,y,z) represents a three-dimensional coordinate system; (x 1 ,x 2 () represents a local coordinate system; 312) Construct a neural network f to learn the form of 0 and the metric tensor, represented as: Where: f is a multi-head attention network; It is The value of 0 is in the form v; It is a metric tensor; 313) Applying the form v to the external differential operator d, we obtain a... Value 1 in the form dv; 314) Using Hodge's star operation, the form dv of type 1 is transformed into another form ★dv of type 1; 315) Applying the exterior differential operator d to the form ★dv of form 1 yields a form d★dv of form 2; 316) Applying the Hodge star operator again to the form d★dv, we obtain a form ★d★dv of 0; 317) Apply a non-linear activation function σ after ★d★dv and embed it into a deep neural network to construct a feature learning model.

7. The method for multiphysics depth modeling and geometric accuracy generative prediction of vehicle body welding according to claim 1, characterized in that: In step 32), the method for updating the displacement field and physical field state is as follows: Define a unified state vector x (n) , is represented as: in: d represents the geometric accuracy or other related features of the body-in-white assembly in the nth generation step; d is the feature dimension. It is the physical field state of the m-th curved surface component; At each time step, the update of the unified state vector is represented as: in: It is a non-linear function, defined as: Where: [·] represents the vector x (n-1) The index of the corresponding element in the middle; and These correspond to the displacement field increment of the body-in-white assembly and the update of the physical field state of the m curved parts, respectively, and: in: It is a fourth-order tensor that maps R-dimensional intermediate features to P solder joints, each solder joint corresponding to k physical fields, and each physical field corresponding to a D-dimensional feature vector. It is a learnable weight modulation vector; The state information of all weld points on the m-th curved surface part in all physical fields is fused to obtain an R×P matrix; This is the final physical field state increment tensor; By overlaying multiple residual blocks, a geometric accuracy generation model of the body-in-white assembly is constructed: Where: x (N) It is the final state vector; This represents the residual block of the nth generation step; x (0) σ is the initial unified state vector; σ is a nonlinear activation function; N is the number of discrete generation steps; This represents the outer product of vectors.

8. The method for multiphysics depth modeling and geometric accuracy generative prediction of vehicle body welding according to claim 1, characterized in that: In step four, the target time t * Geometric accuracy prediction for: Among them: U t The geometric accuracy of the body-in-white assembly at the current moment; t * t and t represent the target time and the current time, respectively; For ζ t The (l-1)th order Taylor coefficients at time t; For ζ t The (p-1)th order Taylor coefficient at time t; ζ t For deep neural stochastic processes; Γ is U t The value at t = η is called the intermediate state; η ∈ [t, t * [ ] represents an unknown time point within the integration interval.

9. The method for multiphysics depth modeling and geometric accuracy generative prediction of vehicle body welding according to claim 1, characterized in that: In step four, the loss function for training the stochastic dynamic evolution prediction model for the geometric accuracy of the body-in-white assembly is: in: It is the rate of change of geometric accuracy of the body-in-white assembly predicted by the model at time point t; It is the white body assembly at time point t * The rate of change of the true geometric accuracy; It is the initial physical field state distribution of the m-th curved surface component; and Gaussian distribution The mean and variance vectors; It is a Gaussian distribution output by the posterior inference network of the body-in-white assembly; μ A and σ A It is a Gaussian distribution The mean and variance vector of p(z); A ) is the hidden state variable z of the body-in-white assembly. A The prior distribution of λ1 and λ2 are control parameters.

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