Advanced Bayesian model updating method for aviation fuel gear pump flow model

Abstract

The invention discloses an advanced Bayesian model updating method for an aviation fuel gear pump flow model. The method comprises the following steps: 1, establishing an aviation fuel gear pump flow model, and defining parameters to be calibrated; 2, calling an advanced Bayesian model updating method to calibrate the rotating speed, the end surface clearance and the radial clearance; according to the method, the calculation cost is remarkably reduced by optimizing the algorithm efficiency and the calculation process, the improved method is successfully applied to parameter calibration of the gear pump flow model, and an efficient and reliable solution is provided for engineering practice.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of aviation fuel gear pumps, and particularly relates to an advanced Bayesian model updating method for a flow model of an aviation fuel gear pump. BACKGROUND

[0002] The main function of the fuel gear pump is to provide fuel that meets the flow demand for the combustion chamber of the aviation engine and the working environment with fuel as the medium, and to control the flow after the pump under different states to ensure that the engine obtains the best fuel supply under various flight states such as take-off, cruising, and acceleration, thereby meeting the performance requirements of the engine within the full flight envelope.

[0003] As a constant-displacement positive displacement pump, the flow after the pump is theoretically proportional to the speed, but as a mechanical component, the end faces and tooth tips of the driving and driven gears must have a tolerance gap to ensure smooth operation of the gear pump without abnormal phenomena such as bore sweeping. Due to the pressure difference between the inlet and outlet, high-pressure fuel leaks from the end face and tooth gap to the low-pressure area, resulting in a loss of displacement. Among them, the end face gap is the main leakage path, and by setting the floating side plate bearing, the end face gap can be effectively controlled to be in the micron level, thereby improving the volumetric efficiency.

[0004] Under the action of multiple non-uniform fluids, gear meshing, and transmission systems, the actual thickness of the end face gap and the radial gap often deviates from the design value, showing a fluctuation around the design value. Since there is no high-precision, high-reliability micron-level gap thickness measurement technology developed at present, there is significant uncertainty in the understanding of these two key gaps. At the same time, due to the inaccurate transmission of multiple sets of gears with side clearance in the accessory casing, the rotational speed from the engine rotor to the fuel pump input shaft also has uncertainty. These uncertainties will be propagated forward through the flow simulation model, thereby affecting the prediction accuracy of the model, and ultimately leading to a significant decline in the prediction effect of the flow.

[0005] Bayesian model updating is a key method for calibrating cognitively uncertain parameters in the model, which can effectively reduce the uncertainty of the output performance prediction. However, the existing Bayesian updating technology faces a significant bottleneck in practical application: since thousands of simulation models need to be called to achieve efficient exploration of the parameter space, for complex engineering problems based on numerical methods such as finite elements, finite differences, or finite volumes, the computational cost is too high, which seriously limits its engineering practicability. This problem has become a major obstacle to the development of current Bayesian updating technology. SUMMARY

[0006] To overcome the shortcomings of the existing technology, the present invention aims to provide an advanced Bayesian model update method for aviation fuel gear pump flow models. By optimizing algorithm efficiency and calculation process, the calculation cost is significantly reduced, and the improved method is successfully applied to the parameter calibration of gear pump flow models, providing an efficient and reliable solution for engineering practice.

[0007] The technical solution adopted in this invention is: An advanced Bayesian model update method for an aviation fuel gear pump flow model includes the following steps; Step 1: Establish a flow model for the aviation fuel gear pump and define the parameters to be calibrated; the parameters to be calibrated are rotational speed, end face clearance, and radial clearance. Step 2: Use advanced Bayesian model update methods to calibrate rotational speed, end face clearance, and radial clearance; thereby correcting the model, improving prediction accuracy, and reducing errors.

[0008] Step one specifically involves: Based on the principle of lumped parameters, a simulation model for transient flow rate of a gear pump is developed. The fluid domain in the gear pump is divided into discrete control volumes (CVs): statistical analysis of the low-pressure suction chamber. High-pressure transmission volume and a series of interdental volumes and The subscripts “1” and “2” represent the driving and driven gears, respectively, and the fluid characteristics of each CV are assumed to be uniform and depend only on time. The governing equations describing the pressure changes within a single control volume and the flow interactions between volumes are derived from the mass conservation equation and the fluid state equation, and are expressed as follows: (1) in and It is the first For laminar flow interaction, the inflow and outflow velocities of the first CV, starting from the first... CV to the first The flow rate of each CV is: (2) and located at the small hole interaction, from the first CV to the first The flow rate of each CV is: (3) in , and It refers to the gap width, length, and height. It is the cross-sectional area of ​​the orifice. is the flow coefficient; the above governing equations are solved by using the Runge-Kutta method, and the volume flow rates of mutual control, including the total delivery flow rate and the leakage flow rate, are calculated synchronously by using numerical integration.

[0009] The step two is specifically: Consider a simulation model , where is the output of interest, which is a vector of multi-dimensional outputs or a scalar of one-dimensional output; , is the rotational speed, the end face clearance and the radial clearance; In the modeling process, a wide prior distribution is assumed to ensure the inclusion of the true value, and the observed data is obtained by experiment, denoted as , and the mathematical relationship between the experimental value and the predicted value is expressed as: (4) where represents the measurement error; Based on Bayes' theorem, when a set of independent observation data and a reasonable assumed prior probability density function (PDF) are given, denoted as , the posterior PDF of the parameter is derived as: (5) where represents the likelihood function of the observed value, which is defined as a multi-dimensional Gaussian likelihood function, is called the evidence, which is defined as the integral of the product of the likelihood function and the prior in the probability space; The Transitional Markov Chain Monte Carlo (TMCMC) sampling algorithm with better global convergence is used, and the surrogate model of the simulation model is established by Gaussian Process Regression (GPR), which reduces the number of calls to the simulation model, thereby reducing the computational cost and improving the computational efficiency.

[0010] According to the TMCMC theory, the mean of the sample weight is positively correlated with the evidence, that is, , as the calculation accuracy of continuously improves, the weight with spatial correlation will also show global convergence, and the posterior mean and variance of the evidence are: (6) (7) From the above two formulas, the following learning function is proposed:​​ (8) The physical explanation of the above learning function is the quantization point of the posterior variance of , so this learning function is named as Posterior Variance Contribution (PVC) function, and then, the uncertainty of is reduced to the maximum.

[0011] (9) Because the integrand in the formula has a nonlinear relationship with the established GPR model, the PVC function cannot be solved analytically. The posterior samples of the GPR model are obtained by using the Karhunen-Loeve expansion (KL expansion), and the estimator of the PVC function is derived. The PVC function contains two independent and identically distributed parameters, and ; based on the prior distribution, samples and are generated respectively; therefore, the sample matrix of the Gaussian process at and is as follows: (10) (11) Therefore, the sample matrix of the likelihood function is: (12) (13) Based on the above two sample matrices, the estimator of the PVC function is derived as: (14) Once the sample weights meet the required accuracy, the adaptive training point design process must be terminated, and a proper stopping criterion must be defined. The relative error of the evidence is a reliable indicator to evaluate whether the sample weights have reached convergence, as shown in the following equation: (15) (16) Therefore, the stopping criterion is defined as: (17) where is a predefined threshold, within the range of .

[0012] TMCMC (Transitional Markov Chain Monte Carlo) is a sampling method used to solve the Bayes formula to obtain samples of the posterior distribution to approximate the posterior distribution. GPR (Gaussian Process Regression) is also known as Gaussian Process Regression, which is a surrogate model algorithm that mainly surrogates simulation models (that is, flow models). Because flow models need to be numerically solved, the calculation cost is high, so after establishing a surrogate model through GPR, the GPR model is used to replace the flow model, and the calculation cost of the GPR model is relatively low, thereby achieving the purpose of reducing the calculation cost.

[0013] BUAT is an active learning algorithm that combines TMCMC and GPR algorithms. The GPR model is not directly trained, but is trained through active learning methods, which can reduce the calculation cost while ensuring calculation accuracy. BUAT (Beyasian Updating with Adaptive TMCMC) includes the following four steps: Step 1: Initialize functions and parameters. First, input the necessary functions and parameters, including simulation models , observation data , prior , measurement noise covariance matrix , working conditions , sample size and , initial training data set size , stop precision threshold and delay stop number threshold , then construct the likelihood function ; Step 2: Initialize the sample set. Select three random sample sets , and with size , is the initial sample in the TMCMC sampling process, and are the sample sets of parameters and , respectively, used for numerical estimation of the PVC function; then from randomly selected samples to form a subset , calculate the model response as , and define the initial training set as , and the initial set initialized as ; Step 3: Transition distribution sampling. This step is the most critical part of the algorithm, because it involves sampling from the -1 transition distribution between the prior and posterior distribution. In this process, resampling, MCMC sampling, training and sampling GPR model and active learning are comprehensively utilized; in general, this step can be summarized as two loops: first, the outer loop starts with the transition order , and then enters the inner loop, in which the GPR model of the response function is trained, and then the GPR model is updated through active learning and delayed stopping strategy; once the convergence criterion is met, the algorithm estimates the exponential and sample weight of the transition distribution, then obtains samples of the transition distribution through resampling and MCMC sampling, and finally updates the transition order and starts the next outer loop iteration; Step 4: Posterior distribution sampling and result output. When the exponential of the transition distribution reaches or exceeds 1, the algorithm transitions to the posterior distribution sampling stage; first, let , the transition order +1= ; similar to the inner loop in step three, the delayed stopping iteration strategy and the adaptive design of training points are applied simultaneously until the convergence threshold is met; then the sample weight is accurately estimated, and the samples of the posterior distribution are obtained using resampling and MCMC techniques; finally, these posterior samples are output for subsequent visualization and necessary analysis. Advantages of the present application:

[0014] Advantages of the present application: ​The present application integrates the global convergence of TMCMC challenge posterior sampling and the superior advantage of GPR quantitative prediction uncertainty over other agent modeling techniques, and provides an effective method for inferring the posterior distribution of the determined but unknown parameters by proposing an advanced Bayesian update framework BUAT. A new active learning strategy based on PVC function is developed, which uses the proportional relationship between the mean of sample weight and the evidence of transition distribution. An unbiased PVC function estimator is derived from the GPR posterior sample to solve the analytically challenging case involving nonlinear coupling between the integrand function and the GPR model. This adaptive strategy accurately identifies the points that contribute most to the evidence posterior variance and iteratively updates the GPR agent model to minimize the epistemic uncertainty in the sample weight. On the basis of significantly reducing the simulation evaluation, a strict continuous delay stop criterion is implemented to ensure the reliability of the posterior distribution samples obtained by the proposed BUAT method.

[0015] In the example of an aero-engine fuel pump, the BUAT accurately calibrates the posterior distribution of the lateral gap height, the radial gap height and the rotational speed, and reduces the computational cost by 63% compared with the traditional TMCMC method. This unique advantage makes the BUAT particularly suitable for high-cost engineering simulation models, providing a powerful tool for inverse uncertainty quantification in complex engineering systems. In addition, its accurate sampling capability can be applied to other Bayesian inference tasks, such as reliability estimation, sensitivity analysis and numerical research.

[0016] The present application improves the prediction accuracy of the flow model, which can more reliably carry out engineering practices such as structure design, parameter optimization and performance prediction based on the flow model. BRIEF DESCRIPTION OF DRAWINGS

[0017] Figure 1 The flowchart of the BUAT method of the present application.

[0018] Figure 2 The instantaneous flow model of the gear pump based on the lumped parameter principle of the present application.

[0019] Figure 3 The flow performance measurement test system of the gear pump of the present application. DETAILED DESCRIPTION

[0020] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the scope of protection of the present application.

[0021] An advanced Bayesian model updating method for an aviation fuel gear pump flow model, comprising: Step one, describe the proposed improved Bayesian model updating framework: Consider a simulation model Where represents the output of interest, which is a vector of multi-dimensional output or a scalar of one-dimensional output; The model input includes two different types: And ; Specifically, represents the deterministic and known operating condition parameters, which are the inlet and outlet pressures in the fuel gear pump flow model, while usually corresponds to the deterministic but unknown model parameters or other operating condition parameters, which are the rotational speed, end face clearance and radial clearance in the fuel gear pump flow model.

[0022] Due to insufficient information, in the modeling process, a broad prior distribution is usually assumed to ensure the inclusion of the true value. This often leads to excessive variability in model predictions, which exceeds the acceptable range of engineering applications. In order to achieve a realistic high-fidelity approximation, experimental methods are used to obtain observed data, denoted as , which refers to the measured inlet and outlet flow rates in the fuel gear pump flow model. The mathematical relationship between experimental values and predicted values can be expressed as: Where represents the measurement error, which refers to the measurement error of the flowmeter in the fuel gear pump flow model.

[0023] Based on Bayes' theorem, when a set of independent observation data and a reasonably assumed prior probability density function (PDF) are given, denoted as the posterior PDF of the parameter can be derived as: Where represents the likelihood function of the observed value, defined as a multi-dimensional Gaussian likelihood function. The evidence, defined as the integral of the likelihood function and the prior over the probability space, is generally not directly solved for the posterior PDF due to the complexity of high dimensionality and nonlinearity, and the posterior distribution of parameters is obtained by sampling algorithms. The present application adopts the Transitional Markov Chain Monte Carlo (TMCMC) sampling algorithm with better global convergence, and establishes a surrogate model of the simulation model through Gaussian Process Regression (GPR), thereby reducing the number of calls to the simulation model and reducing the computational cost and improving the computational efficiency.

[0024] In order to further reduce the evaluation of the simulation model and ensure that the accuracy is not affected, a new active learning strategy is proposed, including adaptive design of training points and continuous delayed stopping criterion. According to the TMCMC theory, the mean of the sample weight is positively correlated with the evidence, i.e. , with the increasing calculation accuracy of , the weight with spatial correlation will also show global convergence, and the posterior mean and variance of the evidence are: (6) (7) From the above two formulas, the following learning function is proposed: (8) The above learning function has a physical interpretation that the quantization point contributes to the posterior variance of , so this learning function is named as Posterior Variance Contribution (PVC) function. Subsequently, the is added to the training set to maximize the reduction of the uncertainty of . (9) Since the integrand in the formula has a nonlinear relationship with the established GPR model, the PVC function cannot be solved analytically, and the present application uses the Karhunen-Loeve expansion to obtain the posterior samples of the GPR model, thereby deducing the estimator of the PVC function. The PVC function contains two independent and identically distributed parameters, and ; based on the prior distribution, samples and are generated respectively; therefore, the sample matrix of the Gaussian process at and is as follows: (10) (11) Thus, the sample matrix of the likelihood function is (12) (13) Based on the above two sample matrices, the estimator of the PVC function is derived as (14) Once the sample weights satisfy the required accuracy, the adaptive training point design process must be terminated. A proper stopping criterion must be defined. The relative error of the evidence is a reliable indicator to assess whether the sample weights have reached convergence, as shown in the following equation: (15) (16) Thus, the stopping criterion is defined as (17) where is a predefined threshold value within the range of

[0025] When the above condition is satisfied, it is reasonable to believe that the normalized sample weights effectively reflect the probability of sample duplication in the resampling process. It is worth noting that during the initial training period with small sample size, the randomness of the samples may cause the stopping criterion to be satisfied even if the weights have not converged. By introducing a delayed decision-making strategy, this problem can be effectively avoided, which means that the stopping criterion will only be triggered when it is satisfied for a number of consecutive times. This ensures that the sample design process is terminated only under reliable conditions. The proposed method is named BUAT (Beyasian Updating with Adaptive TMCMC), as shown in Figure 1 The complete procedure includes the following four steps: Step 1: Initialize functions and parameters. First, input the necessary functions and parameters, including the simulation model , the observation data , the prior , the measurement noise covariance matrix , the working condition , the sample size and , the initial training dataset size , the stopping precision threshold and the delayed stopping number threshold , then construct the likelihood function ; Step 2: Initialize the sample set. Select three random sample sets , and with a size of​ , These are the initial samples in the TMCMC sampling process. and These are parameters and The sample set is used for numerical estimation of the PVC function; then From randomly selected samples Composition of subsets The computational model response is The initial training set is defined as Initial set Size initialized to ; Step 3: Transitional Distribution Sampling. This step is the most critical part of the algorithm because it involves sampling from the prior and posterior distributions. Sampling is performed in the transition distribution -1. This process comprehensively utilizes resampling, MCMC sampling, training and sampling of the GPR model, and active learning; in general, this step can be summarized as two loops: first, the outer loop is formed by the transition sequence. The algorithm begins by entering the inner loop, where a GPR model of the response function is trained. This GPR model is then updated using active learning and a delayed stopping strategy. Once the convergence criterion is met, the algorithm estimates the exponent of the transition distribution. and sample weights Then, through resampling and MCMC sampling, we obtain... Transitional distribution samples Finally, the transition order is updated to And begin the next outer loop iteration; Step 4: Posterior distribution sampling and result output. When the exponent of the transition distribution... When the value reaches or exceeds 1, the algorithm transitions to the posterior distribution sampling phase; first, let... transition stage +1= Similar to the inner loop in step three, the delayed stopping iteration strategy and the adaptive design of training points are applied simultaneously until the convergence threshold is met; then the sample weights are accurately estimated. And use resampling and MCMC techniques to obtain the posterior distribution. samples Finally, these posterior samples are output for subsequent visualization and necessary analysis.

[0026] Step two: Based on the principle of lumped parameters, a simulation model for the transient flow rate of a gear pump is developed, the implementation details of which are as follows: Figure 2 As shown.

[0027] The fluid domain in a gear pump is divided into discrete control volumes (CVs): a low-pressure suction volume , a high-pressure delivery volume , and a series of inter-tooth volumes and , where the subscripts “1” and “2” denote the driving and driven gears, respectively. The fluid properties in each CV are assumed to be homogeneous and depend only on time.

[0028] The governing equations describing the pressure variation within a single control volume and the flow interaction between volumes are derived from the mass conservation equation and the fluid state equation, expressed as: (1) where and are the inflow and outflow flow rates of the th CV. For laminar interactions, the flow rate from the th CV to the th CV is: (2) and for orifice interactions, the flow rate from the th CV to the th CV is: (3) where , and are the gap width, length, height, is the orifice cross-sectional area, is the flow coefficient.

[0029] The above governing equations are solved using the Runge-Kutta method, and the inter-control volume flow rates, including the total delivery flow rate and leakage flow rate, are calculated simultaneously using numerical integration.

[0030] An experimental setup for obtaining gear pump flow observation data is shown in the schematic diagram, including physical diagrams of key components, as shown in Figure 3 . The rotational speed and torque signals are converted into current through a human-machine interaction window to drive the motor and gear box system, which operates the test gear pump synchronously. The fuel in the tank is initially pressurized by a centrifugal pump to ensure sufficient suction pressure and to mitigate cavitation effects. Then, the gear pump delivers fuel at the required pressure and flow rate, completing a closed-loop cycle by returning the fuel to the tank. Two pressure sensors are installed at key locations: the gear pump inlet and outlet. The system also integrates a safety valve, two filters, and additional components (e.g., a temperature control system, an on-off valve, and a throttle valve) to ensure operational safety and reliability. The suction and delivery flow rates of the gear pump are measured using two mass flow meters installed at the inlet and outlet, respectively, with the delivery flow rate representing the total flow rate of the gear pump, and the total leakage flow rate equaling the difference between the suction and delivery flow rates.

Claims

1. An advanced Bayesian model updating method for an aviation fuel gear pump flow model, characterized in that, The method comprises the following steps; Step one: establishing an aviation fuel gear pump flow model and defining parameters to be calibrated; the parameters to be calibrated are rotating speed, end face gap and radial gap; Step two: calling an advanced Bayesian model updating method to calibrate the rotating speed, end face gap and radial gap; thereby correcting the model, improving the prediction accuracy and reducing the error.

2. The advanced Bayesian model updating method of an aviation fuel gear pump flow model according to claim 1, characterized in that, The step one is specifically: Based on the lumped parameter principle, a transient flow simulation model of gear pump is developed. The fluid domain in the gear pump is divided into discrete control volumes: a low pressure suction volume , a high pressure delivery volume , and a series of inter-tooth volumes and , where the subscripts "1" and "2" represent the driving and driven gears, respectively. The fluid properties of each CV are assumed to be uniform, depending only on time; The control equations describing the pressure variation within a single control volume and the flow interaction between volumes are derived from the mass conservation equation and the fluid state equation, expressed as: (1) wherein and is the flow rate into and out of the th CV, and for laminar flow interaction, the flow rate from the th CV to the th CV is: (2) and the flow rate from the first CV to the second CV is: and the flow rate from the first CV to the second CV is: and the flow rate from the first CV to the second CV is: (3) where , and are the gap width, length, height, is the orifice cross-sectional area, is the flow coefficient; the above control equations are solved by using the Runge-Kutta method, and the volume flow rates controlled synchronously, including the total delivery flow rate and the leakage flow rate, are calculated by using numerical integration.

3. The advanced Bayesian model updating method of an aviation fuel gear pump flow model according to claim 1, wherein, The step two is specifically: Constructing simulation models wherein y represents the output of interest, being a vector of multi-dimensional outputs or a scalar of one-dimensional output; The model input is , is the rotational speed, the end face clearance and the radial clearance; In the modeling process, the Assuming a broad prior distribution to ensure the inclusion of true values, the experimental data is obtained by an experimental method, denoted as The mathematical relationship between the experimental value and the predicted value is expressed as: (4) wherein represents the measurement error; Based on Bayes' theorem, given a set of independent observations and a reasonable assumed prior probability density function, denoted as the posterior PDF of the parameter is derived as: (5) wherein denotes the likelihood function of the observations, defined as the multi-dimensional Gaussian likelihood function, called the evidence, is defined as the integral over the probability space of the product of the likelihood function and the prior; A transition Markov chain Monte Carlo sampling algorithm with better global convergence is adopted, and a simulation model is established by means of Gaussian process regression; According to the TMCMC theory, the mean of the sample weights is positively correlated with the evidence, that is , with the increasing of the calculation accuracy of , the weight with spatial correlation will also show global convergence, and the posterior mean and variance of the evidence are: (6) （7） From the above two equations, the following learning function is presented: (8) The physical interpretation of the above learning function is the quantization point To the posterior variance of the contribution, add Maximum reduction The uncertainty of (9) The posterior samples of the GPR model are obtained by using KL expansion, and an estimate of the PVC function is obtained, the PVC function containing two independent and identically distributed parameters, and ; based on the prior distribution, samples and are generated respectively; at and The sample matrix of the Gaussian process is as follows: (10) （11） Thus, the sample matrix of the likelihood function is: (12) （13） Based on the above two sample matrices, the estimator of the PVC function is derived as: (14) The relative error of the evidence is a reliable indicator to assess whether the sample weights have reached convergence, as shown in the following equation: (15) （16） Thus, the stop criterion is defined as: (17) wherein is a predefined threshold value, within the range of Bayes' theorem is the fundamental theorem of Bayesian updating, which is to solve the posteriori by priori and likelihood, the formula is as follows: (18) TMCMC is used to solve the Bayesian formula to obtain samples of the posterior distribution to approximate the posterior distribution.

4. The advanced Bayesian model updating method for the aviation fuel gear pump flow model according to claim 1, characterized in that, BUAT is a combination of TMCMC and GPR algorithm, wherein the GPR model is not directly trained, but is trained by means of active learning; BUAT comprises the following four steps: Step 1: Initialize functions and parameters. First, input the necessary functions and parameters, including the simulation model , the observation data , the prior , the measurement noise covariance matrix , the working condition , the sample size , and , the initial training data set size , the stop precision threshold , and the delay stop number threshold . Then, construct the likelihood function ; Step 2: initialization of sample set; Three random sample sets , and of size , are selected. The initial sample is the set of samples in the TMCMC sampling process, and are the sample sets of parameters and , respectively, used for the numerical estimation of the PVC function. Then a subset is formed by randomly selecting samples from the initial sample set , the model response is calculated as , the initial training set is defined as , and the size of the initial set is initialized. Step 3: Transition distribution sampling; first, the outer loop is sampled by the transition order Start, then enter the inner loop, train the GPR model of the response function in the inner loop, then update the GPR model through active learning and delayed stopping strategy; once the convergence criteria are met, the algorithm estimates the index of the transition distribution And sample weights Then, by resampling and MCMC sampling, get Samples of the transition distribution Finally, update the transition order to And start the next outer loop iteration;​ Step 4: posterior distribution sampling and result output; When the index of the transition distribution reaches or exceeds 1, the algorithm transitions to the posterior distribution sampling phase; first, let , the transition phase +1= ; Similar to the inner loop in step three, the delayed stopping iteration strategy and the adaptive design of training points are applied simultaneously until the convergence threshold is met; Then the sample weights are estimated accurately and the posterior distribution is obtained using resampling and MCMC techniques samples ; finally, these posterior samples are output for subsequent visualization and necessary analysis.

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