Digital twin modeling and parameter self-optimization system for steel pipe spraying process

Abstract

The application discloses a digital twin modeling and parameter self-optimization system for a steel pipe spraying process, and relates to the technical fields of computer simulation analysis and process parameter self-optimization; the technical points are as follows: the system comprises a data acquisition and digital twin initialization module, a gradient difference Bayesian significance inference module, a nonlinear neural operator gradient propagation mapping module and a risk perception double-closed coupled feedback optimization module; the Bayesian deep ensemble network is used to calculate the posterior significance probability of the gradient difference of each process parameter in the weld area and the general area, and the real spatial heterogeneity and random fluctuations are quantitatively distinguished; and the different areas are subjected to differential intervention strategies according to the posterior significance probability and the gradient sign, the gradient update direction is locked to the direction consistent with the requirement of the weld area and the update amplitude is reduced when the gradient directions conflict, so that the problems that the weld area and the general area cannot be subjected to differential treatment and the real gradient difference and random fluctuations cannot be distinguished are solved.

Description

Technical Field

[0001] This invention relates to the field of computer simulation analysis and process parameter self-optimization technology, specifically a digital twin modeling and parameter self-optimization system for steel pipe spraying process. Background Technology

[0002] Modeling and parameter optimization of steel pipe spraying process refers to establishing a mathematical model that describes the physicochemical changes of paint flow, atomization, deposition and curing during the spraying process. Based on this model, process parameters such as spray gun distance, posture, flow rate, pressure and moving speed are intelligently adjusted to form a uniform thickness and qualified anti-corrosion coating on the surface of the steel pipe. The core objective is to achieve precise control of coating quality while improving material utilization and production efficiency.

[0003] Existing digital twin modeling and parameter self-optimization technologies for steel pipe spraying processes typically involve constructing a digital twin that includes equations for fluid dynamics and heat transfer, receiving real-time data from production line sensors, and using optimization algorithms such as gradient descent and model predictive control to automatically adjust spraying parameters so that the predicted coating thickness distribution approaches the target value, thereby achieving closed-loop optimization control of coating uniformity.

[0004] In actual large-scale oil and gas pipeline coating production scenarios, there are axially extending weld seams on the surface of the steel pipe, and existing technologies still have some problems: First, because the weld area and the ordinary pipe area have significant differences in physical properties such as thermal conductivity and viscosity, the gradient directions calculated for the same set of process parameters in the weld area and the ordinary area to guide the improvement of thickness uniformity may be opposite during the optimization process. Existing technologies treat the steel pipe as a homogeneous body or use overall averaging, which cannot statistically distinguish whether this gradient difference is due to real physical heterogeneity or random fluctuations caused by sensor noise or model errors. If random fluctuations are intervened, it will destroy the converged optimization direction and reduce the control accuracy.

[0005] Secondly, the spraying process involves highly nonlinear phenomena such as droplet splashing, phase transitions, and severe temperature gradients. Especially near the weld area where physical properties change abruptly, the influence of process parameter changes on coating thickness is highly nonlinear. Existing technologies typically use gradient propagation models based on linearized approximations to estimate how corrections to regional parameters will affect other regions. Under highly nonlinear conditions, such linear models will seriously underestimate or mispredict the spatial propagation of parameter adjustment effects, causing correction commands issued from the weld area to be distorted during propagation to ordinary regions. This not only fails to reconcile contradictions between regions but may even cause divergence in the optimization process of ordinary regions. Summary of the Invention

[0006] To achieve the above objectives, the present invention provides the following technical solution: A digital twin modeling and parameter self-optimization system for steel pipe spraying processes includes a data acquisition and digital twin initialization module. This module acquires multi-physics data and process parameters during the steel pipe spraying process using a computer. It constructs a computer-generated virtual digital twin model based on three-locked coupled partial differential equations and initializes a Bayesian deep ensemble network and physically constrained Fourier neural operators. The system outputs a spatiotemporally aligned multi-physics dataset, the initialized model and network weights, and the current process parameter vector. A gradient difference Bayesian significance inference module, through forward computation and automatic differentiation of the digital twin model, obtains the average gradient contribution of each process parameter in the weld region and the ordinary region. It then uses the Bayesian deep ensemble network to calculate the spatial uncertainty field of the gradient difference and the subsequent... The system verifies the significance probability, generates an intervention intensity ranking sequence, and outputs the inference results. The nonlinear neural operator gradient propagation mapping module formulates a differentiated gradient correction strategy based on the posterior significance probability, obtains the gradient correction vector of the weld area, executes nonlinear gradient propagation mapping through a physical constraint Fourier neural operator, calculates the cross-layer excess compensation in the process parameters, and outputs the corrected gradient update vector. The risk-aware double-closed coupling feedback optimization module applies the corrected gradient update vector to the digital twin basic model to update the process parameters, re-triggers the gradient difference Bayesian significance inference module to form a regional consistency closed loop, forms a time-scale consistency closed loop based on the stability monitoring of the coupling matrix, and outputs the optimal combination of process parameters verified by the double loop.

[0007] Furthermore, the multiphysics data includes temperature field, flow field, and thickness field data; the process parameters include millisecond-level fast-changing layer parameters and second-level medium-changing layer parameters; the three-locked coupled partial differential equations include the heat conduction equation for describing temperature diffusion, the Navier-Stokes equation for describing the motion of sprayed droplets, and the mass conservation equation for describing the accumulation of coating thickness, and are coupled through constitutive parameters.

[0008] Furthermore, the Bayesian deep ensemble network consists of multiple deep neural network copies with identical structures but randomly initialized weights; the physical constraint Fourier neural operator is based on the Fourier neural operator architecture, and the residual of the three-locked coupled partial differential equation is embedded at the network output as a physical constraint.

[0009] Furthermore, in the gradient difference Bayesian significance inference module, the process of obtaining the average gradient contribution includes: inputting the current process parameter vector into the digital twin basic model for forward calculation to obtain the predicted thickness field; constructing a loss function with the predicted thickness field and the measured thickness field and obtaining the partial derivative through automatic differentiation; and calculating the average gradient contribution and local average gradient difference index of the weld area and the ordinary area respectively. The calculation process of the posterior significance probability includes: calculating the mean and variance of the gradient difference based on the forward inference results of each replica of the Bayesian deep ensemble network to form the gradient difference spatial uncertainty field; for each process parameter, calculating the Z score based on the difference in the average gradient contribution between the weld area and the ordinary area and the posterior standard deviation, and obtaining the corresponding posterior significance probability according to the standard normal distribution.

[0010] Furthermore, the process of generating the intervention intensity ranking sequence includes: comparing the posterior significance probability with a preset confidence threshold; marking parameters with a posterior significance probability less than a first threshold as strongly significant conflict parameters; marking parameters with a posterior significance probability greater than or equal to the first threshold and less than or equal to a second threshold as weakly significant conflict parameters; and marking parameters with a posterior significance probability greater than the second threshold as insignificant parameters; and arranging all parameters marked as strongly significant conflict parameters or weakly significant conflict parameters in ascending order of their posterior significance probability values ​​to generate the intervention intensity ranking sequence.

[0011] Furthermore, the process of formulating a differentiated gradient correction strategy based on posterior significance probability includes: screening out process parameters marked as strongly significant conflict parameters or weakly significant conflict parameters; for each screened parameter, comparing the sign of its average gradient contribution in the weld region with that in the ordinary region; if the signs are opposite, locking the gradient update direction of the parameter to the same direction as the sign of the average gradient contribution in the weld region, and reducing the gradient update magnitude by a corresponding factor according to its marking type to obtain the gradient correction vector in the weld region.

[0012] Furthermore, the process of performing nonlinear gradient propagation mapping includes: encoding the gradient difference space uncertainty field into a condition vector; concatenating the weld region gradient correction vector with the condition vector and inputting it into the physical constraint Fourier neural operator; sequentially performing fast Fourier transform, frequency domain linear transform, and inverse Fourier transform; and then fusing it with the condition vector through a cross-attention mechanism to output the predicted value of the gradient change in the ordinary region. The process of calculating the cross-layer excess compensation amount in the process parameters includes: extracting the gradient correction amount of the rapidly changing layer parameters; applying a 0.5 mm perturbation to the spray gun distance and a 0.01 radian perturbation to the spray gun attitude angle; constructing a cross-scale propagation coefficient matrix by observing the change in the gradient of the intermediate changing layer; calculating the compensation intensity modulation factor based on the posterior significance probability and the gradient difference space uncertainty field; multiplying the gradient correction amount of the rapidly changing layer parameters, the cross-scale propagation coefficient matrix, and the compensation intensity modulation factor to obtain the cross-layer excess compensation amount; and superimposing the cross-layer excess compensation amount onto the intermediate changing layer gradient to form a corrected gradient update vector.

[0013] Furthermore, the process of re-triggering the gradient difference Bayesian significance inference module to form a regional consistency closed loop includes: inputting the updated process parameters into the digital twin base model, and re-executing the gradient difference Bayesian significance inference module to obtain a new posterior significance probability and a new uncertainty field; statistically analyzing the proportion of parameters whose posterior significance probability decreases as the improvement proportion, and combining the changing trend of the new uncertainty field in the weld area's average value, selecting one of the three mutually exclusive decision branches: confirmation update, extended observation, and automatic rollback.

[0014] Furthermore, the risk-aware dual-closed-loop coupling feedback optimization module uses the gradient difference spatial uncertainty field as the global modulation signal to dynamically adjust at least one of the following control boundaries: the differentiated step size decay factor for the gradient components of the weld area and the ordinary area, the random inactivation rate of the physical constraint Fourier neural operator, and the thickness variance threshold in the system convergence determination; wherein, the step size decay factor is negatively correlated with the average uncertainty field of the corresponding area, the random inactivation rate is positively correlated with the average uncertainty field of the weld area, and the thickness variance threshold is positively correlated with the average uncertainty field of the weld area.

[0015] Furthermore, the process of forming a time-scale consistent closed loop based on the stability monitoring of the coupling matrix includes: performing singular value decomposition on the coupling matrix constructed from the intermediate variation layer parameters, calculating the singular value change weighted by the uncertainty field of the gradient difference space; if the weighted change exceeds a preset change threshold of 0.15, it is determined that the coupling matrix has changed significantly, and a feedback signal is sent to the nonlinear neural operator gradient propagation mapping module and the gradient difference Bayesian significance inference module.

[0016] This invention provides a digital twin modeling and parameter self-optimization system for steel pipe spraying processes, which has the following beneficial effects: 1. This invention uses a Bayesian deep ensemble network to calculate the posterior significance probability of the gradient difference between the weld area and the ordinary area for each process parameter, quantitatively distinguishing between real spatial heterogeneity and random fluctuations caused by sensor noise and model uncertainty. Furthermore, it implements a spatial differentiation intervention strategy for different areas based on the posterior significance probability and gradient sign. Only when the gradient difference reaches statistical significance and the signs are opposite, the gradient update direction is locked to the direction consistent with the requirements of the weld area, and the update amplitude is reduced. This solves the technical problems of existing technologies that treat gradient conflicts as a global phenomenon, cannot differentiate between different areas, and mistakenly treat random fluctuations as real differences for intervention. It achieves that the coating thickness uniformity error in both the weld area and the ordinary area simultaneously meets the target value requirement.

[0017] 2. This invention embeds a physically constrained Fourier neural operator into the gradient propagation mapping process to learn the strong nonlinear coupling law between partial differential equations in an end-to-end manner. It uses the uncertainty field of the gradient difference space as a conditional input to accurately predict the propagation process of the gradient correction effect from the weld region to the ordinary region. This effectively overcomes the technical defect of linear approximation in the case of propagation distortion under the condition of sudden change in physical properties, which leads to the destruction of the optimization direction in the ordinary region. It provides an accurate propagation tensor basis for the coordinated convergence of the two regions.

[0018] 3. This invention constructs a dual-closed-loop coupled feedback structure with regional consistency and temporal consistency. The regional consistency loop verifies the gradient correction effect by re-triggering saliency inference and performs confirmation updates, extended observation, or automatic rollback based on the saliency improvement ratio and the trend of uncertainty field changes. The temporal consistency loop monitors system stability by performing uncertainty-weighted singular value decomposition on the coupling matrix of the intermediate-variable layer and triggers online fine-tuning when there are significant changes. At the same time, the uncertainty field of the gradient difference space is used as the global modulation signal to dynamically adjust the step size attenuation factor, the random inactivation rate of the physical constraint Fourier neural operator, and the thickness variance threshold in the convergence condition to achieve risk-aware adaptive control. This fundamentally solves the technical problems of fixed threshold closed-loop feedback easily causing false rollback and insufficient stability due to over-optimization, ensuring robust convergence of the entire parameter self-optimization process under strong nonlinear and high uncertainty conditions. Attached Figure Description

[0019] Figure 1 This is a system overall framework diagram of Embodiment 1 of the present invention.

[0020] Figure 2 This is a flowchart of the gradient difference inference and nonlinear propagation mapping process of the present invention.

[0021] Figure 3This is a flowchart of the risk perception dual closed-loop feedback optimization process of the present invention.

[0022] Figure 4 This is a flowchart of the method in Embodiment 2 of the present invention. Detailed Implementation

[0023] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0024] Example 1: Please see Figures 1 to 3 This embodiment provides a digital twin modeling and parameter self-optimization system for steel pipe spraying processes. Deployed on a large-scale automated production line for anti-corrosion coating of oil and gas transport steel pipes, the production line processes straight-seam welded pipes with weld seams on their surfaces. The spraying employs a high-pressure airless spraying process. The core adjustable process parameters are divided into millisecond-level fast-change layers and second-level medium-change layers. The millisecond-level fast-change layer parameters include the spray gun distance and spray gun attitude angle, while the second-level medium-change layer parameters include the spray gun flow rate, spraying pressure, and spray gun movement speed. This system uses digital twin technology to predict the coating thickness distribution in real time and adjusts these parameters using a parameter self-optimization algorithm to ensure that the coating thickness in both the weld seam area and the normal area simultaneously reaches the target value, precisely controlling uniformity errors.

[0025] This system consists of four modules: Module 1 is the data acquisition and digital twin initialization module; Module 2 is the gradient difference Bayesian significance inference module; Module 3 is the nonlinear neural operator gradient propagation mapping module; and Module 4 is the risk perception double closed coupling feedback optimization module.

[0026] Module 1 is responsible for real-time acquisition of temperature field, flow field, thickness field data, and process parameter data during the steel pipe spraying process. It constructs a discretized model of a three-locked coupled partial differential equation system through computer simulation and pre-trains a Bayesian deep ensemble network and a physically constrained Fourier neural operator using machine learning methods to achieve the initialization of the digital twin in computer-aided engineering analysis. Module 1 outputs a spatiotemporally aligned multiphysics dataset and an initialized network model, providing a data foundation and model carrier for subsequent gradient calculation and uncertainty inference.

[0027] Module 2 receives the digital twin state and process parameters output by Module 1, and obtains the average gradient contribution of each process parameter in the weld area and the ordinary area through forward computation and automatic differentiation. Then, it uses a Bayesian deep ensemble network to replace the traditional Monte Carlo random sampling, directly outputs the spatial uncertainty field of gradient difference, and calculates the posterior significance probability. The output of Module 2 includes the posterior significance probability of each process parameter, the intervention intensity ranking sequence, and the global uncertainty field. These results are passed to Modules 3 and 4.

[0028] Module 3 uses the uncertainty field output from Module 2 as a conditional input and replaces the traditional linear Moore-Penrose generalized inverse with a physically constrained Fourier neural operator. First, it formulates a differentiated intervention strategy based on the posterior significance probability and gradient sign. Then, it inputs the gradient correction amount of the weld region into the physically constrained Fourier neural operator and outputs the adaptive nonlinear gradient propagation mapping result of the ordinary region. At the same time, it calculates the cross-layer overcompensation amount of the fast-changing layer to the medium-changing layer. The compensation intensity is jointly modulated by the posterior significance probability and the uncertainty field. Module 3 outputs the finally determined gradient update vector and overcompensation vector and passes them to Module 4.

[0029] Module 4 receives the gradient update from Module 3, applies it to the digital twin to update process parameters, and then re-triggers the saliency inference of Module 2, forming a region-consistent closed loop. Simultaneously, it performs uncertainty-weighted singular value decomposition on the coupling matrix of the intermediate-variable layer, monitors stability changes, and feeds back to adjust the parameters of the fast-variable layer, forming a time-scale-consistent closed loop. Module 4 uses the uncertainty field as a global modulation signal to dynamically adjust the rollback threshold, step size scaling factor, and convergence conditions. Finally, it outputs the optimal combination of process parameters verified by the double closed loop and determines whether to trigger the next iteration or terminate the optimization.

[0030] Module 1 provides the initial model foundation for the entire system; Module 2 performs statistical inference on the gradient data output by Module 1 to determine whether gradient differences actually exist; Module 3 performs nonlinear propagation mapping based on the judgment result of Module 2 to determine how to correct and how to propagate after correction; Module 4 verifies the correction effect of Module 3 to determine whether the correction is correct and whether rollback or further optimization is needed; the verification result of Module 4 is fed back to Module 2 and Module 3 to form a closed loop; the entire system executes in a loop with a period of 50 milliseconds until the convergence condition is met.

[0031] In the weld region, due to abrupt changes in material properties and drastic temperature gradients, the gradient directions corresponding to the heat conduction equation and the Navier-Stokes equation are often completely opposite. That is, increasing the spray gun flow rate will improve the thickness uniformity of the weld region but worsen the physical consistency of heat conduction. However, in ordinary regions, the gradient directions of the same set of equations are highly consistent. Traditional gradient management methods treat gradient conflicts as a global phenomenon and cannot differentiate between different spatial regions, resulting in low accuracy of optimization strategies. This system distinguishes the degree of gradient conflict between the weld region and ordinary regions through Bayesian saliency inference and implements a spatially differentiated intervention strategy to achieve independent correction by region.

[0032] If there is a difference in gradient values ​​between the weld area and the normal area, this difference may be due to real spatial heterogeneity or random fluctuations caused by sensor noise and model uncertainty. If the difference is only caused by random fluctuations, intervening in it will actually harm the optimization quality. The differentiation strategy should only be triggered when the difference reaches statistical significance. This system introduces Bayesian hypothesis testing to calculate the posterior significance probability in order to quantitatively answer the probability that the gradient difference actually exists, thereby avoiding ineffective intervention in random noise.

[0033] Existing technologies generally use the Moore-Penrose generalized inverse of a three-locked coupled partial differential equation system as the spatial propagation model for gradient correction. This model is essentially a linear approximation. Under the strongly nonlinear conditions of steel pipe spraying, especially when the molten pool fluctuates violently and physical parameters change abruptly, the linear generalized inverse will seriously underestimate the actual coupling effect of the partial differential equations, resulting in a distortion of the gradient correction amount propagating from the weld region to the ordinary region, which in turn destroys the originally convergent optimization direction in the ordinary region. This system uses a physically constrained Fourier neural operator to replace the linear generalized inverse, learns the strong nonlinear coupling law end-to-end, generates an adaptive nonlinear propagation tensor, and accurately predicts the spatial propagation of the correction effect.

[0034] In regions with low model prediction confidence, traditional closed-loop feedback uses a fixed threshold method, which can easily lead to erroneous rollback or over-optimization, resulting in optimization oscillations or insufficient stability. This system constructs a dual closed loop with regional consistency and time-scale consistency, and dynamically modulates the rollback threshold, step size factor and convergence conditions with an uncertainty field to achieve risk-aware adaptive control.

[0035] Taking one production scenario as an example: the production line processes one 12-meter-long, 500-millimeter-diameter straight seam welded pipe per minute. The steel pipe surface has an axially extending weld seam area with a width of 15 mm. Due to the heat of welding, the thermal conductivity of the material in the weld seam area is 40% lower than that in the ordinary area, and the viscosity change rate is about 3 times higher. Furthermore, the peak temperature gradient during the spraying heating process can reach 80°C per millimeter. The spraying uses a high-pressure, airless process, with core adjustable process parameters divided into millisecond-level fast-change layers and second-level medium-change layers. The millisecond-level fast-change layer parameters include the spray gun distance and spray gun attitude angle, while the second-level medium-change layer parameters include the spraying flow rate, spraying pressure, and spray gun movement speed. The system uses digital twin technology to predict and optimize these parameters in real time, aiming to control the coating thickness uniformity error between the weld seam area and the ordinary area within ±0.1 mm.

[0036] Furthermore, for Module 1: Data Acquisition and Digital Twin Initialization Module, its working process includes three stages.

[0037] Step 1.1: Synchronization and preprocessing of multi-source heterogeneous data.

[0038] Using a sampling period of 50 milliseconds, the physical field data collected in real time by the sensor and the process parameter data read by the programmable logic controller are aligned according to the timestamp. The physical field data includes the surface temperature field distribution of the steel pipe obtained by the infrared thermal imaging sensor, the velocity distribution of the sprayed droplet flow field obtained by the laser Doppler velocimeter, and the coating thickness distribution obtained by the laser thickness gauge. The process parameter data includes the spray gun distance and spray gun attitude angle in the fast-changing layer parameters, and the spray gun flow rate, spraying pressure, and spray gun movement speed in the medium-changing layer parameters. The dynamic response time constant of the spray gun posture and flow rate in the high-pressure airless spraying process is usually between 20 milliseconds and 80 milliseconds. The 50-millisecond sampling period can completely capture the evolution trajectory of the parameters of the fast-changing layer and the medium-changing layer, while meeting the real-time computing power constraints of the industrial control system.

[0039] A cubic spline interpolation algorithm was used to unify all data into a structured grid coordinate system of the digital twin. This grid coordinate system was divided into 200 nodes in the circumferential direction and 500 nodes in the axial direction, for a total of 100,000 grid points. This division was based on the balance between spatial resolution requirements and industrial control computer computing resources. Median filtering was used to remove salt-and-pepper noise from the temperature data, with a filtering window size of 3×3. Outlier detection was performed on the thickness data. The mean and standard deviation of each grid point over the past 20 sampling periods were calculated. Data points exceeding three times the standard deviation were marked as outliers and replaced with the mean of neighboring points.

[0040] The weld location markers in the prior information of the steel pipe are used to generate weld area masks and ordinary area masks, the three-dimensional geometric model is used to determine the boundary of the mesh coordinate system, and the material property parameters are used to set the initial constitutive parameters of the digital twin model.

[0041] Step 1.2: Constructing a basic model of a three-locked coupled digital twin.

[0042] The three-locked coupled digital twin basic model has three built-in partial differential equations: Heat conduction equation: used to describe the diffusion behavior of temperature in the direction of steel pipe wall thickness and surface direction. The equation is in the form of density multiplied by specific heat capacity multiplied by temperature with respect to time equal to thermal conductivity multiplied by temperature with respect to space second partial derivative plus heat source term. The Navier-Stokes equations describe the trajectory and velocity distribution of sprayed droplets in an airflow field. The equations are expressed as: the partial derivative of density multiplied by velocity with respect to time equals the negative pressure gradient plus the second partial derivative of viscosity multiplied by velocity with respect to space plus the body force. Mass conservation equation: used to describe the cumulative process of coating thickness, the equation is in the form that the partial derivative of thickness with respect to time is equal to the spraying flux minus the evaporation loss.

[0043] The three equations are strongly coupled through constitutive parameters, namely thermal conductivity, viscosity, and surface tension. The finite volume method is used to discretize the above equations, generating a computational mesh containing 100,000 mesh nodes. The initial material properties of the weld region and the ordinary region are set: the initial value of thermal conductivity in the weld region is set to 35 W / m Kelvin, and the initial value of thermal conductivity in the ordinary region is set to 58 W / m Kelvin; the initial value of viscosity in the weld region is set to 0.08 Pascal-second, and the initial value of viscosity in the ordinary region is set to 0.03 Pascal-second. The above material property values ​​are based on the typical physical property measured data of the heat-affected zone and the base material of the straight seam welded pipe.

[0044] Step 1.3: Initialize Bayesian deep ensemble network and physically constrained Fourier neural operators.

[0045] The Bayesian deep ensemble network consists of five independent deep neural network replicas, each with a three-layer fully connected structure and 128 neurons per layer. The activation function is the hyperbolic tangent function. The five replicas use different randomly initialized weights, and the weight initialization uses the Glorot uniform initialization method.

[0046] Physically Constrained Fourier Neural Operator: Based on the Fourier neural operator structure, it contains four Fourier layers. Each layer retains 32 Fourier modes in the frequency domain for learnable linear transformation, and embeds the residuals of three-locked coupled partial differential equations at the network output as physical constraints.

[0047] Two networks were pre-trained using 1000 samples extracted from historical production data. The pre-training employed a mean squared error loss function, an Adam optimizer, a learning rate of 0.001, and 100 training epochs. The learning rate and number of training epochs were selected based on common deep learning pre-training experience values, and grid search verification confirmed stable convergence on this dataset. An additional regularization term was added during pre-training: the solution of the traditional linear Moore-Penrose generalized inverse was used as a regularization constraint. The weight coefficient of this constraint was initially set to 0.5, decaying by 5% every 20 training epochs, eventually decreasing to 0.05. This weight decay setting aims to ensure the physical rationality of the solution through strong constraints in the early stages of training, while gradually releasing the constraints later to allow the operator to fully learn the nonlinear features in the data. After pre-training, the weight parameters of the five replicas of the Bayesian deep ensemble network and the weight parameters of the physically constrained Fourier neural operator were saved.

[0048] Module 1 outputs the following results: The spatiotemporally aligned multiphysics dataset contains the temperature field matrix at the current moment, with a size of 200×500, the flow velocity field matrix, with a size of 200×500, and the measured thickness field matrix, with a size of 200×500. This result is transferred to Module 2 for gradient calculation. An initialized three-locked coupled digital twin model, including discretized mesh coordinates, material property parameter tables, and a partial differential equation solver, is transferred to Modules 2, 3, and 4 as a carrier for forward computation, automatic differentiation, and closed-loop verification. The pre-trained set of weight parameters for the Bayesian deep ensemble network and the set of weight parameters for the physically constrained Fourier neural operators are transmitted to Module 2 and Module 3, respectively, for subsequent uncertainty quantization and nonlinear propagation mapping. The current process parameter vector includes five dimensions: spray gun distance, spray gun attitude angle, spray gun flow rate, spraying pressure, and spray gun movement speed. The initial value of this vector is read by the programmable logic controller and updated by module four after each iteration.

[0049] Furthermore, for Module 2: Gradient Difference Bayesian Significance Inference Module, its working process includes 4 steps.

[0050] Step 2.1: Calculation of process parameter gradient data.

[0051] The current process parameter vector is input into the three-locked coupled digital twin model, and the finite volume method solver is called to perform forward calculation to obtain the predicted thickness field distribution. The loss function is the mean square error between the predicted thickness field and the measured thickness field output by module one. The partial derivative of the loss function with respect to each process parameter is calculated by automatic differentiation. For the five parameters of spray gun distance, attitude angle, flow rate, pressure and speed, the arithmetic mean of their partial derivatives at all grid points in the weld region is calculated to obtain the average gradient contribution of the weld region. Similarly, the arithmetic mean of the partial derivatives at all grid points in the ordinary region is calculated to obtain the average gradient contribution of the ordinary region. The difference between the two is calculated as the local average gradient difference index.

[0052] Section 2.2: Quantification of spatial uncertainty of gradient difference.

[0053] The current process parameter vector, sensor data, and spatial coordinates are input into a Bayesian deep ensemble network. Each of the network's five replicas performs an independent forward inference, outputting a predicted gradient contribution for that spatial location. These five predictions form a sample set. For each grid point, the arithmetic mean of the five predictions is calculated as the gradient difference mean, and the sample variance of the five predictions is calculated as the gradient difference variance. The mean and variance of all grid points are combined to form a gradient difference mean matrix and a gradient difference variance matrix. These two matrices together constitute the gradient difference spatial uncertainty field. The square root of the gradient difference variance matrix reflects the reliability of the model's predictions in that region: the larger the variance, the higher the cognitive uncertainty.

[0054] Step 2.3: Inference of the significance of Bayesian posterior probability.

[0055] Based on the gradient difference mean matrix and variance matrix, hypothesis testing is performed independently for each process parameter. The null hypothesis is set as the true gradient difference of the parameter between the weld region and the ordinary region being 0. The Z-score is calculated as the local average gradient difference index divided by the posterior standard deviation, which is obtained by joint propagation of the variance matrix in the weld region and the ordinary region. Assuming that the posterior distribution follows a standard normal distribution, the posterior significance probability is calculated as the two-tailed probability corresponding to the absolute value of the Z-score, i.e., the posterior significance probability is equal to 1 minus twice the function value of the standard normal cumulative distribution function at the absolute value of the Z-score. This probability value ranges from 0 to 1, and the closer it is to 0, the more significant the gradient difference.

[0056] Step 2.4: Prioritization of parameter intervention.

[0057] The posterior significance probability of each parameter is compared with preset confidence thresholds of 0.05 and 0.30. 0.05 is a commonly used significance level in statistics, representing a 95% confidence level. 0.30 is an engineering experience threshold used to distinguish between weak significance and insignificance. When the significance probability of the difference is less than 70%, it is considered that random fluctuation is dominant.

[0058] If the posterior significance probability is <0.05, that is, the significance probability of the difference is >95%, then the parameter is marked as a strongly significant conflict parameter, and subsequent fully differentiated intervention will be accepted. If 0.05 ≤ posterior significance probability ≤ 0.30, it is marked as a weakly significant conflict parameter, and a reduced differential intervention is subsequently accepted, with the intervention intensity linearly scaled to between 0.3 and 0.8 times. If the posterior significance probability is >0.30, it is marked as an insignificant parameter and no differential intervention is performed. The intervention intensity is generated by sorting the parameters in ascending order of their posterior significance probability values. Parameters with lower posterior significance probabilities are ranked higher and have higher intervention intensity.

[0059] Module 2 outputs the following results: The posterior significance probability and corresponding label classification of each process parameter are passed to Module 3 for in-depth processing. For parameters labeled as strongly significant conflict parameters or weakly significant conflict parameters, the information includes the posterior significance probability, label classification, gradient sign, and gradient magnitude. For parameters labeled as insignificant parameters, Module 3 only receives the non-intervention label and does not perform any gradient correction on such parameters, directly retaining the original gradient. The intervention intensity ranking sequence was transmitted to Module 3 to determine the processing priority in case of multi-parameter conflicts. The gradient difference space uncertainty field, containing the mean matrix and variance matrix, is simultaneously transmitted to both Module 3 and Module 4. In Module 3, the gradient difference space uncertainty field serves as the conditional input to the physically constrained Fourier neural operator. In Module 4, the gradient difference space uncertainty field is used to modulate the rollback threshold and step size scaling factor. If the average variance of the weld region in the gradient difference space uncertainty field is >0.5, indicating that the model is highly uncertain in this region, Module 4 triggers a conservative control mode, reducing the step size and relaxing the tolerance. If the average variance is <0.1, Module 4 triggers an aggressive control mode, allowing for larger step sizes and stricter convergence conditions. The average gradient contribution of each parameter in the weld area and the average gradient contribution in the ordinary area are transmitted to module 3 for subsequent comparison of the gradient signs of the two areas.

[0060] Furthermore, for Module 3: Nonlinear Neural Operator Gradient Propagation Mapping Module, its working process includes 3 steps.

[0061] Module 3 receives five types of data from Module 2: the posterior significance probability and label classification of each parameter, the intervention intensity ranking sequence, the gradient difference space uncertainty field, the average gradient contribution of the weld area, and the average gradient contribution of the ordinary area. It also receives the process parameter vector of the current iteration cycle.

[0062] Step 3.1: Formulating a differentiated gradient correction strategy.

[0063] First, process parameters marked as strongly and weakly significantly conflicting parameters are selected. For each selected parameter, they are processed sequentially according to the priority of the intervention intensity ranking sequence. For each processed parameter, the signs of the average gradient contribution in the weld region and the average gradient contribution in the ordinary region are compared. If the signs are the same, it indicates that the parameter has a positive synergistic effect in the two regions, and the original gradient update is retained without any modification. If the signs are opposite, it indicates that the parameter has a fundamental demand conflict in the two regions, and intervention is performed: the gradient update direction is locked to the direction with the same sign as the average gradient contribution in the weld region, and the update amplitude is reduced. The reduction factor of the update amplitude is determined according to the label classification: for strongly significantly conflicting parameters, the reduction factor is 0.3, which is determined based on the empirical range of ensuring the convergence of the coating thickness in the weld region without causing the divergence of the coating thickness in the ordinary region in the closed-loop test of historical data; for weakly significantly conflicting parameters, the reduction factor is 0.6, which is based on the fact that when the gradient direction deviates partially, a moderate reduction in amplitude can maintain the synergistic optimization of the two regions. The dimension of the corrected gradient correction vector in the weld region is equal to the number of process parameters.

[0064] Step 3.2: Calculation of nonlinear propagation mapping under uncertainty conditions.

[0065] The gradient difference spatial uncertainty field is encoded into a 128-dimensional conditional vector. Specifically, the sampled values ​​of the gradient difference spatial uncertainty field at the boundary between the weld region and the ordinary region are mapped into a fixed-length vector through a small encoder network. This encoder network is a three-layer fully connected network, with each layer having 64 dimensions. The corrected gradient correction vector of the weld region is concatenated with the conditional vector and input into a pre-trained physical constraint Fourier neural operator. The operator performs forward inference, performs a fast Fourier transform on the input to transform it to the frequency domain, applies a learnable linear transformation matrix in the frequency domain, and then performs an inverse Fourier transform back to the time domain. Finally, it is fused with the conditional vector through a cross-attention mechanism. The cross-attention mechanism is calculated by the query vector coming from the time domain features and the key and value vectors coming from the conditional vector. The operator outputs the predicted gradient change values ​​at each grid point in the ordinary region. Its data structure is the same as the original gradient field, which is a 200×500 matrix. This output can capture the real coupling effect under strongly nonlinear conditions.

[0066] Step 3.3: Calculation of cross-layer excess compensation for uncertainty modulation.

[0067] The process parameters are divided into fast-changing layer parameters and medium-changing layer parameters. Fast-changing layer parameters include spray gun distance and spray gun attitude angle, with a change period of milliseconds. Medium-changing layer parameters include spray gun flow rate, spraying pressure, and spray gun movement speed, with a change period of seconds.

[0068] The gradient correction of the rapidly varying layer parameters is extracted, and the cross-scale propagation coefficient matrix is ​​calculated. This matrix is ​​a 3x2 Jacobian block matrix, where each element represents the sensitivity of the rapid variation of the layer parameters to the gradient of the intermediate layer. The specific calculation method of the cross-scale propagation coefficient matrix is ​​as follows: a small perturbation increment is applied to the rapidly varying layer parameters in the three-locked coupling solver. The perturbation increment is: an increase of 0.5 mm in the spray gun distance and an increase of 0.01 radians in the spray gun attitude angle. The perturbation magnitude is determined by sensitivity testing to ensure the accuracy of numerical difference while avoiding nonlinear instability. The change in the gradient of the intermediate layer is observed, and the two are divided to obtain the numerical Jacobian matrix.

[0069] The excess compensation is equal to the transpose of the rapid change layer correction multiplied by the cross-scale propagation coefficient matrix and then multiplied by two modulation factors. The first modulation factor is the signal-to-noise ratio factor, which is the reciprocal of the posterior significance probability of each parameter multiplied by the effect size, where the effect size is the absolute value of the local average gradient difference index divided by the posterior standard deviation. The second modulation factor is the uncertainty attenuation factor, which is 1 divided by 1 plus the average value of the uncertainty field in the weld region, where the average value of the uncertainty field in the weld region is the arithmetic mean of the variances of all grid points in the gradient difference space uncertainty field.

[0070] The final gradient update of the intermediate-variable layer is equal to the original gradient of the intermediate-variable layer plus the overcompensation.

[0071] Module 3 outputs the following results: The corrected gradient correction vector for the weld area is transmitted to module four to update the rapid change layer process parameters. The predicted gradient change of the ordinary region after nonlinear propagation mapping is transmitted to module four to update the gradient components related to the ordinary region. The gradient update amount of the intermediate layer after cross-layer overcompensation is transmitted to module four to update the intermediate layer process parameters.

[0072] For parameters marked as insignificant in Module 2, their gradient update amounts are not corrected and are directly output as the original gradient update amounts. The output of this branch is marked as a pass-through gradient and transmitted separately to Module 4. The correction amount of the fast-varying layer parameters only goes through links 3.1 and 3.2, without going through the overcompensation calculation in link 3.3. In addition to going through links 3.1 and 3.2, the overcompensation amount of link 3.3 is also superimposed on the medium-varying layer parameters. Furthermore, when the average value of the uncertainty field in the weld area is >0.6, the uncertainty is judged to be too high. This threshold of 0.6 is an engineering safety threshold set to prevent overcompensation under ultra-high uncertainty conditions. At this time, the uncertainty attenuation factor in the overcompensation amount will make the compensation intensity approach 0. Overcompensation is not performed, and only the original medium-varying layer gradient is output with a small perturbation close to 0. This small perturbation is set to 1% of the original medium-varying layer gradient.

[0073] Furthermore, for Module 4: Risk Perception Dual Closed-Loop Coupling Feedback Optimization Module, its working process includes 3 stages.

[0074] Module 4 receives three types of gradient results from Module 3: the gradient after correction in the weld region, the gradient after nonlinear propagation in the ordinary region, and the gradient after overcompensation in the intermediate layer. It also receives the gradient difference space uncertainty field and the posterior significance probability sequence from Module 2.

[0075] Step 4.1: Uncertainty verification of regional consistency closed loop.

[0076] All gradient updates output from Module 3 are superimposed onto the current process parameters to obtain tentative new process parameters. The tentative new process parameters are then input into the three-locked coupled digital twin model of Module 1 to recalculate the predicted thickness field. Finally, the Bayesian deep ensemble network of Module 2 is called to re-execute the saliency inference to obtain new posterior saliency probabilities and new uncertainty fields.

[0077] The determination of a closed loop with regional consistency is as follows: The percentage of significant improvement is obtained by dividing the number of process parameters whose posterior significance probability decreases within the current iteration cycle by the total number of all parameters marked as strongly or weakly significant conflicting parameters. Based on the percentage of significant improvement and the trend of the spatial uncertainty field, a mutual exclusion decision is performed. Confirmation Update: When the significant improvement rate is ≥80%, the differentiated intervention strategy is deemed effective globally. It is confirmed that new process parameters are used on a trial basis, the current amplitude reduction rate is maintained, and the system is put into a time-scale consistent closed loop for stability verification. 80% is the minimum percentage that can achieve stable optimization effect in system simulation test. Extended observation: When the significant improvement rate is <80%, but the average value of the new uncertainty field in the weld area decreases by more than 20% compared to before adjustment, it is determined that although the apparent significance of some parameters has not improved immediately, the model's cognitive uncertainty in this area has been greatly reduced, indicating that the optimization direction is correct but the physical field has not yet fully responded; a 20% decrease rate can effectively distinguish between the situation where the optimization direction is correct and the situation where the model's cognitive uncertainty is not clear in the simulation; at this time, the system does not trigger rollback, marks the tentative new process parameters as under observation, extends the observation period by three iterations, and sets the length of this observation period to the typical time span for the system to produce a physical response to parameter changes, i.e., 150 milliseconds, and continues to execute subsequent steps; Automatic rollback: When the significant improvement rate is less than 80%, and the average value of the new uncertainty field in the weld area does not decrease or even increases, it is determined that the intervention direction is wrong or has caused local divergence. At this time, the system immediately abandons the current tentative new process parameters, rolls back to the process parameters before adjustment, and reduces the amplitude reduction factor in the nonlinear neural operator gradient propagation mapping module to 70% of the original value. The 70% is based on experience and ensures that the intervention intensity will not drop to 0 too quickly after multiple rollbacks.

[0078] Step 4.2: Uncertainty verification of time-scale consistent closed loop.

[0079] Based on the updated intermediate layer parameters, an intermediate layer coupling matrix is ​​constructed. This matrix is ​​a 3×3 matrix, and its elements represent the mutual coupling strength between the three parameters: spray gun flow rate, spraying pressure, and spray gun moving speed. The coupling matrix is ​​obtained by extracting the state transition matrix of model predictive control: First, a state space model of the intermediate layer parameters is established, and the state transition matrix is ​​obtained through system identification. Then, a 3×3 sub-block directly related to the intermediate layer parameters in the matrix is ​​taken and singular value decomposition is calculated to obtain three singular values. The relative percentage change of the current singular value compared with the singular value of the previous period is calculated. Uncertainty weighting is introduced, and the weighted change is the relative percentage change multiplied by the reciprocal of the average value of the spatial uncertainty field in the weld area.

[0080] If the weighted change exceeds the preset change threshold of 0.15, the coupling matrix is ​​determined to have changed significantly. This change is then backpropagated to the physical constraint Fourier neural operator in module three, triggering online fine-tuning of the operator. Simultaneously, the change information is returned to the Bayesian deep ensemble network in module two to update the prior distribution of the uncertainty field. The threshold of 0.15 is selected based on engineering experience: when the weighted change exceeds 0.15, the coupling matrix is ​​considered to have changed substantially. If the weighted change is ≤0.15, it is determined that the coupling matrix has not changed significantly, the system does not trigger feedback, and directly enters the next step of control boundary adaptive adjustment driven by uncertainty field.

[0081] Section 4.3: Adaptive adjustment of control boundary driven by uncertainty field.

[0082] Obtain the gradient difference space uncertainty field from Module 2, calculate the average uncertainty field of the weld area and the average uncertainty field of the ordinary area; based on these two statistics, perform three adaptive adjustments.

[0083] Gradient update step size adjustment: For the gradient component in the dominant direction of the weld region, the step size is multiplied by the step size decay factor one, which is equal to 1 divided by 1 plus the average value of the uncertainty field in the weld region; for the gradient component in the dominant direction of the ordinary region, the step size is multiplied by the step size decay factor two, which is equal to 1 divided by 1 plus the average value of the uncertainty field in the ordinary region; this design ensures that the step size decreases as uncertainty increases.

[0084] Propagation tolerance adjustment: When the average uncertainty field value in the weld area is >0.3, the random inactivation rate of the physical constraint Fourier neural operator in Module 3 is automatically increased to 0.3 to enhance robustness; when the average uncertainty field value in the weld area is <0.1, the random inactivation rate is reduced to 0.05; 0.3 and 0.1 are based on historical data of the statistical distribution of uncertainty predicted by the model, corresponding to typical working conditions of high uncertainty and low uncertainty, respectively.

[0085] Convergence condition adjustment: The thickness variance threshold in the final convergence condition is dynamically adjusted to the original threshold multiplied by 1 plus the average value of the uncertainty field in the weld area. That is, the higher the uncertainty, the more relaxed the convergence condition, to avoid the system over-optimizing in areas that cannot be accurately predicted. The original threshold is set to 0.01 square millimeters, which is calculated based on the upper limit of variance corresponding to the product coating uniformity requirement of ±0.1 millimeters.

[0086] Module 4 outputs the following results: The final set of process parameters after regional consistency verification is divided into three categories: parameters that have been updated are output during confirmation, parameters that have been rolled back to their original state during rollback, and parameters that are temporarily marked as being under observation and require an extended observation period during observation. This result is then transmitted to the final convergence determination stage. Time-scale feedback signal; if the coupling matrix changes significantly, the feedback signal is sent to Module 3 and Module 2; if the change is not significant, no feedback is sent, and the signal contains information on the direction and magnitude of the singular value changes; The updated control boundary parameters include step size decay factor one, step size decay factor two, dynamic random inactivation rate, and dynamic convergence threshold; these parameters are recorded internally by the system and used for subsequent calculations in the current iteration cycle and parameter settings in the next iteration cycle.

[0087] In the region-consistent closed loop, if the system enters the confirmation branch, it directly enters the time-scale consistent closed loop; if it enters the rollback branch, the system immediately terminates the current iteration cycle, discards all correction results from Module 3, restarts the next iteration with the parameters before adjustment, and feeds back the reduction factor decay information to Module 3; if it enters the observation branch, the system continues to execute the time-scale consistent closed loop, but does not immediately output the final parameters after the loop is completed, but waits for three additional iteration steps before making the final convergence judgment; in the time-scale consistent closed loop, if feedback is triggered, the physical constraint Fourier neural operator of Module 3 and the Bayesian deep ensemble network of Module 2 will be updated online, and the updated network will be used for the next iteration cycle; if feedback is not triggered, the system directly enters the convergence judgment stage.

[0088] In the straight seam welded pipe spraying scenario, this system executes the following process in a 50-millisecond cycle: Module 1 collects temperature field, flow field, thickness field, and process parameter data in real time, constructs a three-locked coupled partial differential equation digital twin basic model, and initializes the Bayesian deep ensemble network and physically constrained Fourier neural operators; Module 2 automatically differentiates and calculates the gradient contribution of each process parameter in the weld area and the ordinary area, and uses the Bayesian deep ensemble network to output the spatial uncertainty field and posterior significance probability of the gradient difference, thereby quantitatively determining whether the gradient difference between the two areas is a real physical conflict or random noise, solving the problem that traditional methods cannot distinguish the nature of the difference; Module 3 applies spatial differences to strongly significant conflict parameters based on the posterior significance probability. The system employs a multi-layered approach: First, it locks the gradient update direction to the weld area demand and reduces the amplitude. Second, it uses a physically constrained Fourier neural operator to replace the linear generalized inverse for nonlinear propagation mapping, based on an uncertainty field. This accurately predicts the propagation effect of weld corrections to ordinary areas, thus avoiding gradient distortion in linear approximations under abrupt property changes. Third, it constructs a dual closed loop for regional consistency and time-scale consistency, dynamically modulating the rollback threshold, step size factor, and convergence conditions with an uncertainty field. When the coupling matrix changes significantly, it triggers online fine-tuning and automatically relaxes the tolerance in low-confidence areas to prevent erroneous rollback and over-optimization. Finally, the system stably controls the coating thickness uniformity error in both the weld area and ordinary areas to within ±0.1 mm.

[0089] Example 2: Please see Figure 4 Based on Example 1, this example also provides a digital twin modeling and parameter self-optimization method for steel pipe spraying process, including steps 1 to 9, which are executed sequentially according to the order of data processing and model calculation.

[0090] Step 1: Multi-source data acquisition and preprocessing: Real-time acquisition of temperature field, flow field, thickness field data and process parameter data from sensors deployed on the production line.

[0091] An infrared thermal imaging sensor collects the surface temperature distribution of the steel pipe at a frequency of 50 Hz, a laser Doppler velocimeter collects the velocity field of the sprayed droplets at a frequency of 100 Hz, and a laser thickness gauge collects the coating thickness at a frequency of 10 Hz; process parameters are directly read by a programmable logic controller.

[0092] All data are timestamped and unified into the structured grid coordinate system of the digital twin using cubic spline interpolation. This grid coordinate system is divided into 200 nodes in the circumferential direction and 500 nodes in the axial direction. The temperature data is filtered by median filtering to remove salt-and-pepper noise, with a filtering window size of 3×3. The thickness data is subjected to outlier detection. The mean and standard deviation of each grid point over the past 20 sampling periods are calculated. Data points exceeding 3 times the standard deviation are marked as outliers and replaced with the mean of the nearest neighbor points.

[0093] After processing, a time-synchronized and spatially registered multiphysics dataset is obtained, which includes a temperature field matrix, a flow field matrix, a thickness field matrix, and a current process parameter vector. The current process parameter vector includes five dimensions: spray gun distance, spray gun attitude angle, spray gun flow rate, spraying pressure, and spray gun movement speed.

[0094] Step 2: Digital Twin Initialization and Network Pre-training: Construct a three-locked coupled digital twin basic model and complete the pre-training of the Bayesian deep ensemble network and the physically constrained Fourier neural operator.

[0095] A set of coupled partial differential equations was established based on the heat conduction equation, the Navier-Stokes equation, and the mass conservation equation. The finite volume method was used to discretize the equations on a 200×500 structured grid. The mesh resolution in the weld area was increased to 0.5 mm, and the mesh resolution in the ordinary area was 1 mm.

[0096] Initial material properties are set as follows: the thermal conductivity of the weld area is 35 W / m Kelvin, and the thermal conductivity of the ordinary area is 58 W / m Kelvin; the Bayesian deep ensemble network uses five replicas, each replica is a three-layer fully connected network with 128 hidden layer neurons, and uses the hyperbolic tangent activation function; the physical constraint Fourier neural operator uses a four-layer Fourier neural operator, with 32 Fourier modes retained in each layer.

[0097] 100 pre-training rounds were performed using 1000 samples from the historical dataset. The optimizer was the Adam optimizer with a learning rate of 0.001. The loss function was the mean squared error plus the physical constraint residual, where the physical constraint residual was the sum of squared residuals of the three-locked coupled partial differential equation system on the predicted thickness field. At the same time, a regularization term was added, using the solution of the traditional linear Moore-Penrose generalized inverse as the constraint. The regularization weight was initially 0.5, decayed by 5% every 20 rounds, and finally decayed to 0.05.

[0098] After processing, we obtain a digital twin model capable of performing forward computation and automatic differentiation, initial weight parameters of a Bayesian deep ensemble network, and initial weight parameters of a physically constrained Fourier neural operator.

[0099] Step 3: Calculate the gradient of process parameters: Calculate the average gradient contribution of each process parameter in the weld area and the normal area, and calculate the local average gradient difference index.

[0100] The current process parameter vector is input into the digital twin model for forward computation to obtain the predicted thickness field; the loss function is the mean square error between the predicted thickness field and the measured thickness field; the partial derivative of the loss function with respect to each process parameter is calculated using automatic differentiation technology; for each parameter, the arithmetic mean of the partial derivatives at all grid points in the weld region is calculated to obtain the average gradient contribution of the weld region; and the arithmetic mean of the partial derivatives at all grid points in the ordinary region is calculated to obtain the average gradient contribution of the ordinary region; the local average gradient difference index is calculated to be equal to the average gradient contribution of the weld region minus the average gradient contribution of the ordinary region.

[0101] After processing, the average gradient contribution of the weld area, the average gradient contribution of the ordinary area, and the local average gradient difference index corresponding to each parameter are obtained.

[0102] Step 4: Gradient difference uncertainty field quantification and significance inference: The spatial uncertainty field of gradient difference is output using a Bayesian deep ensemble network, and the posterior significance probability of each parameter is calculated.

[0103] The current process parameter vector and sensor data are input into five replicas of the Bayesian deep ensemble network, with each replica independently outputting a gradient contribution prediction. For each grid point, the arithmetic mean of the five predicted values ​​is calculated as the gradient difference mean, and the sample variance of the five predicted values ​​is calculated as the gradient difference variance. This yields the gradient difference mean matrix and the gradient difference variance matrix, which together constitute the gradient difference spatial uncertainty field. Based on the mean matrix and variance matrix, the Z-score for each parameter is calculated as the local average gradient difference index divided by the posterior standard deviation. The posterior significance probability is then calculated as 1 minus twice the standard normal cumulative distribution function at the absolute value of the Z-score. Based on the comparison of the posterior significance probability with thresholds of 0.05 and 0.30, each parameter is labeled as a strongly significant conflict parameter, a weakly significant conflict parameter, or an insignificant parameter.

[0104] After processing, the posterior significance probability and label classification, gradient difference space uncertainty field containing mean and variance are obtained for five parameters.

[0105] Step 5: Differentiated gradient correction and nonlinear propagation mapping: Perform differentiated gradient correction on the significantly conflicting parameters and calculate the nonlinear propagation effect through the physically constrained Fourier neural operator.

[0106] For parameters labeled as strongly or weakly significant conflict parameters, the signs of the average gradient contribution in the weld region and the average gradient contribution in the ordinary region are compared. If the signs are the same, the original gradient is retained; if the signs are opposite, the gradient direction is locked to the direction of the weld region, and the amplitude is reduced by a factor of β. The reduction factor β is 0.3 for strongly significant conflict parameters and 0.6 for weakly significant conflict parameters. The gradient difference space uncertainty field is encoded into a 128-dimensional conditional vector, concatenated with the modified gradient vector, and then input into the physical constraint Fourier neural operator. The operator performs a four-layer frequency domain transformation and cross-attention fusion, and outputs a prediction of the nonlinear gradient change in the ordinary region.

[0107] After processing, the corrected gradient vector of the weld area and the gradient vector of the ordinary area after nonlinear propagation are obtained.

[0108] Step 6: Cross-layer overcompensation calculation: Calculate the overcompensation amount of the gradient correction of the fast-changing layer parameters on the medium-changing layer parameters.

[0109] The gradient correction of the rapidly varying layer parameters is extracted, and a cross-scale propagation coefficient matrix (3×2 matrix) is constructed. Small perturbations are applied to the rapidly varying layer parameters in a three-locked coupling solver, with increments of 0.5 mm for the spray gun distance and 0.01 radians for the spray gun attitude angle. The gradient changes in the intermediate-variable layer are observed for numerical estimation. Each element is calculated by dividing the gradient change by the perturbation increment. The signal-to-noise ratio (SNR) factor is calculated as the reciprocal of the posterior significance probability multiplied by the absolute value of the local average gradient difference index divided by the posterior standard deviation. The uncertainty attenuation factor is calculated as 1 divided by 1 plus the average uncertainty field value in the weld region. The excess compensation is calculated as the transpose of the gradient correction multiplied by the cross-scale propagation coefficient matrix, multiplied by the SNR factor, and then multiplied by the uncertainty attenuation factor. The excess compensation is then superimposed on the original gradient of the intermediate-variable layer.

[0110] After processing, the gradient update amount after overcompensation of the intermediate layer is obtained.

[0111] Step 7: Risk perception dual closed-loop verification: Verify whether the gradient correction direction is correct through a region-consistent closed loop, verify the stability of the coupling matrix through a time-scale consistent closed loop, and adaptively adjust the control boundary according to the uncertainty field.

[0112] Apply all gradient updates from steps 5 and 6 to the current process parameters to obtain exploratory parameters; re-invoke the Bayesian deep ensemble network to perform significance inference, obtaining new posterior significance probabilities and new uncertainty fields; compare the old and new significance probabilities: if the new posterior significance probabilities generally decrease, confirm the update; if the new posterior significance probabilities increase but the average value of the new uncertainty field decreases by more than 20%, proceed to the observation period of three iterations; if both increase, roll back the gradient and reduce β to 70%.

[0113] Based on the updated intermediate layer parameters, an intermediate layer coupling matrix is ​​constructed, the percentage change of singular values ​​is calculated, and the weighted change is obtained by weighting with the reciprocal of the average value of the uncertainty field in the weld region. This weighted change is compared with a threshold of 0.15 to determine whether to trigger online fine-tuning of the physical constraint Fourier neural operator. Finally, the step size decay factor 1 and step size decay factor 2, the random inactivation rate of the physical constraint Fourier neural operator, and the convergence threshold are adjusted according to the average value of the uncertainty field in the weld region and the average value in the ordinary region.

[0114] After processing, the final set of process parameters verified by the double closed loop and the updated control boundary parameters are obtained. The result has three branches: the parameters of the confirmation branch enter step 8; the parameters of the rollback branch are rolled back to the state before step 3 and restarted; the parameters of the observation branch enter step 8 but the marking requires three additional observation steps; at the same time, if the time scale feedback is triggered, the physical constraint Fourier neural operator and the Bayesian deep ensemble network receive the online fine-tuning signal.

[0115] Step 8: Convergence condition judgment: Determine whether the current optimization meets the termination conditions, including the weld area thickness variance threshold and the significance probability threshold.

[0116] Calculate the thickness variance of the coating in the weld area. The dynamic thickness variance threshold is equal to the preset target value of 0.01 square millimeters multiplied by 1 plus the average value of the uncertainty field in the weld area. Check whether the posterior significance probability of all process parameters is greater than 0.30, that is, no parameter is marked as a strongly significant conflict parameter or a weakly significant conflict parameter. At the same time, check whether the current iteration number has reached the maximum iteration number of 500 times. The maximum iteration number is determined based on the production line cycle time and response time requirements. If the thickness variance is less than the dynamic threshold and all parameters are insignificant, or the iteration number reaches 500 times, then convergence is determined; otherwise, non-convergence is determined.

[0117] If convergence is determined, proceed to step 9 to output the optimal parameters; if convergence is determined, use the current process parameters as the new starting point and return to step 3 to start the next round of iteration; the parameters of the observation branch carry the observation counter when returning, and if no deterioration occurs after three consecutive iteration steps, it is converted to the confirmation branch.

[0118] Step 9: Output of optimal process parameters and update of knowledge base: Output the final determined combination of optimal process parameters and save the uncertainty evolution trajectory and neural operator weights in the optimization process to the knowledge base.

[0119] The final process parameter vector confirmed in step 8 is written into the programmable logic controller to guide actual spraying production. At the same time, the following data is written into the offline knowledge base: the values ​​of the five dimensions of the final process parameters, the posterior significance probability sequence of each iteration, the evolution record of the uncertainty field, and the final weight parameters of the physical constraint Fourier neural operator. For the next batch of steel pipes of the same specification, the weights of the physical constraint Fourier neural operator saved this time can be directly loaded as the initial values, and the historical uncertainty field can be used as the prior distribution, thereby significantly reducing the number of preheating iterations required by the Bayesian deep integration network.

[0120] After processing, the optimal combination of process parameters is output to complete the self-optimization closed loop; if there are observation branches that have not completed three iteration steps, the observation will continue to be completed in subsequent batches, and the observation results will be incorporated into the knowledge base for continuous model improvement.

[0121] The above embodiments can be implemented, in whole or in part, by software, hardware, firmware, or any other combination thereof. When implemented in software, the above embodiments can be implemented, in whole or in part, as a computer program product. Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution.

[0122] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment, depending on actual needs.

[0123] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application.

Claims

1. A digital twin modeling and parameter self-optimization system for steel pipe spraying process, characterized in that, include: The data acquisition and digital twin initialization module collects multi-physics field data and process parameters during the steel pipe spraying process, constructs a basic digital twin model based on the three-locked coupled partial differential equation, initializes the Bayesian deep ensemble network and the physical constraint Fourier neural operator, and outputs the spatiotemporally aligned multi-physics field dataset, the initialized model and network weights, and the current process parameter vector. The gradient difference Bayesian significance inference module obtains the average gradient contribution of each process parameter in the weld area and the ordinary area through forward computation and automatic differentiation of the digital twin basic model. It uses a Bayesian deep ensemble network to calculate the spatial uncertainty field of gradient difference and the posterior significance probability, generates an intervention intensity ranking sequence, and outputs the inference results. The nonlinear neural operator gradient propagation mapping module formulates a differentiated gradient correction strategy based on the posterior significance probability, obtains the gradient correction vector of the weld area, performs nonlinear gradient propagation mapping through the physical constraint Fourier neural operator, calculates the cross-layer overcompensation in the process parameters, and outputs the corrected gradient update vector. The risk perception dual-closed-loop coupling feedback optimization module applies the corrected gradient update vector to the digital twin base model to update the process parameters, re-triggers the gradient difference Bayesian significance inference module to form a regional consistency closed loop, and forms a time-scale consistency closed loop based on the stability monitoring of the coupling matrix, outputting the optimal combination of process parameters verified by the dual loop.

2. The digital twin modeling and parameter self-optimization system for steel pipe spraying process according to claim 1, characterized in that, The multiphysics data includes temperature field, flow field, and thickness field data; the process parameters include millisecond-level fast-changing layer parameters and second-level medium-changing layer parameters; the three-locked coupled partial differential equations include the heat conduction equation for describing temperature diffusion, the Navier-Stokes equation for describing the motion of sprayed droplets, and the mass conservation equation for describing the accumulation of coating thickness, and are coupled through constitutive parameters.

3. The digital twin modeling and parameter self-optimization system for steel pipe spraying process according to claim 1, characterized in that, The Bayesian deep ensemble network consists of multiple copies of deep neural networks with identical structures but randomly initialized weights; the physical constraint Fourier neural operator is based on the Fourier neural operator architecture, and the residuals of the three-locked coupled partial differential equations are embedded at the network output as physical constraints.

4. The digital twin modeling and parameter self-optimization system for steel pipe spraying process according to claim 1, characterized in that, In the gradient difference Bayesian significance inference module, the process of obtaining the average gradient contribution includes: inputting the current process parameter vector into the digital twin basic model for forward calculation to obtain the predicted thickness field; constructing the mean square error loss function with the predicted thickness field and the measured thickness field and obtaining the partial derivative through automatic differentiation; and calculating the average gradient contribution and local average gradient difference index of the weld area and the ordinary area respectively. The calculation process of the posterior significance probability includes: calculating the mean and variance of the gradient difference based on the forward inference results of each replica of the Bayesian deep ensemble network to form the gradient difference space uncertainty field; for each process parameter, calculating the Z score based on the difference in the average gradient contribution between the weld area and the ordinary area and the posterior standard deviation, and obtaining the corresponding posterior significance probability according to the standard normal distribution.

5. The digital twin modeling and parameter self-optimization system for steel pipe spraying process according to claim 1, characterized in that, The process of generating the intervention intensity ranking sequence includes: comparing the posterior significance probability with a preset confidence threshold, classifying and labeling the process parameters as strongly significant conflict parameters, weakly significant conflict parameters, or insignificant parameters; and arranging all labeled parameters in ascending order of posterior significance probability values ​​to generate the intervention intensity ranking sequence.

6. The digital twin modeling and parameter self-optimization system for steel pipe spraying process according to claim 5, characterized in that, The process of formulating a differentiated gradient correction strategy based on posterior significance probability includes: screening out process parameters that are marked as strongly significant conflict parameters or weakly significant conflict parameters; for each screened parameter, comparing the sign of the average gradient contribution of the weld region with that of the ordinary region; if the signs are opposite, locking the gradient update direction of the parameter to the same direction as the sign of the average gradient contribution of the weld region, and reducing the gradient update magnitude by a corresponding factor according to the marking type to obtain the gradient correction vector of the weld region.

7. The digital twin modeling and parameter self-optimization system for steel pipe spraying process according to claim 2, characterized in that, The process of performing nonlinear gradient propagation mapping includes: encoding the gradient difference space uncertainty field into a condition vector; concatenating the gradient correction vector of the weld region with the condition vector and inputting it into the physical constraint Fourier neural operator; performing fast Fourier transform, frequency domain linear transform and inverse Fourier transform in sequence; and then fusing it with the condition vector through a cross-attention mechanism to output the predicted value of the gradient change in the ordinary region. The process of calculating the cross-layer excess compensation amount in the process parameters includes: extracting the gradient correction amount of the rapidly changing layer parameters, constructing the cross-scale propagation coefficient matrix by applying small perturbations; calculating the signal-to-noise ratio factor and uncertainty attenuation factor based on the posterior significance probability and the uncertainty field of the gradient difference space; and multiplying the gradient correction amount of the rapidly changing layer parameters, the cross-scale propagation coefficient matrix, the signal-to-noise ratio factor and the uncertainty attenuation factor to obtain the cross-layer excess compensation amount. The cross-layer excess compensation is superimposed on the gradient of the intermediate layer to form a corrected gradient update vector.

8. The digital twin modeling and parameter self-optimization system for steel pipe spraying process according to claim 1, characterized in that, The process of re-triggering the gradient difference Bayesian significance inference module to form a regionally consistent closed loop includes: inputting the updated process parameters into the digital twin base model, and re-executing the gradient difference Bayesian significance inference module to obtain a new posterior significance probability and a new uncertainty field; and selecting one of the three mutually exclusive decision branches—confirmation update, extended observation, and automatic rollback—based on the improvement ratio of the posterior significance probability and the changing trend of the new uncertainty field.

9. The digital twin modeling and parameter self-optimization system for steel pipe spraying process according to claim 1, characterized in that, The risk-aware dual-closed-coupled feedback optimization module uses the gradient difference spatial uncertainty field as the global modulation signal to dynamically adjust at least one of the following control boundaries: the differential step size attenuation factor for the gradient components of the weld area and the ordinary area, the random inactivation rate of the physical constraint Fourier neural operator, and the thickness variance threshold in the system convergence determination.

10. The digital twin modeling and parameter self-optimization system for steel pipe spraying process according to claim 2, characterized in that, The process of forming a time-scale consistent closed loop based on the stability monitoring of the coupling matrix includes: performing singular value decomposition on the coupling matrix constructed from the intermediate variation layer parameters, calculating the singular value change weighted by the uncertainty field of the gradient difference space; if the weighted change exceeds the preset change threshold, it is determined that the coupling matrix has changed significantly, and a feedback signal is sent to the nonlinear neural operator gradient propagation mapping module and the gradient difference Bayesian significance inference module.

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