Adaptive pc-kriging reliability analysis method and system based on active learning

Abstract

This invention discloses an adaptive PC-Kriging reliability analysis method and system based on active learning. The method includes: using weighted clustering to obtain uniformly distributed first candidate samples from a pre-generated MC sample pool and constructing an initial PC-Kriging model; using interval reduction to select samples and construct a new sample pool, dividing the new sample pool into two sub-sample pools: a safe domain and a failure domain; for each sub-sample pool, weighted clustering is used again to select uniformly distributed second candidate samples; using the second candidate samples distributed in the failure domain and the safe domain, crossing points are constructed; the constructed crossing points are used as new experimental points to iteratively update the PC-Kriging model; convergence is determined and the results are output. The technical solution of this invention focuses on key regions through interval reduction and updates the model based on crossing points, which can more accurately approximate the limit state surface, reduce the number of experimental points, improve prediction accuracy, and is more efficient in utilizing computational resources.

Description

Technical Field

[0001] This invention relates to the field of reliability analysis technology for mechanical systems, and in particular to an adaptive PC-Kriging reliability analysis method and system based on active learning. Background Technology

[0002] As the mechanisms of engineering machinery become increasingly complex, the demands for diverse working environments and higher requirements for precision and stability rise, greater demands are placed on the reliability of engineering equipment. Because mechanical systems operate in complex and variable service environments, they involve numerous uncertainties such as random excitations, material properties, and geometric dimensions. These factors are among the main causes of performance fluctuations and failures in mechanical systems. Against this backdrop, using advanced theories and methods to assess and analyze the reliability of mechanical equipment has become a crucial approach to ensuring product quality and performance.

[0003] The applicant's earlier application, CN 118862647A, proposed an RBDO method based on a PC-Kriging model and quantile evaluation. This method reduces the number of calls to the original model by constructing a unified surrogate model and introducing quantile judgment constraints. However, in highly nonlinear problems, this method may lead to deviations in optimization results due to the accumulation of errors in quantile sign judgment. Furthermore, its "MP+EI" combination point-addition strategy is prone to ill-conditioned correlation matrices when dealing with too close sample points, reducing model prediction stability. In addition, its global surrogate model enrichment strategy relies on a large number of space-filling samples in the initial stage, resulting in wasted computational resources and difficulty in adapting to the rapid convergence requirements of high-dimensional complex problems.

[0004] The paper "Reliability Analysis of Complex Mechanical Structures Based on Active Learning (Journal of Northeastern University: Natural Science Edition, 2020, Vol.41, Cao Runan et al.)" proposes the AK-MCS-K method, which combines k-means clustering with parallel computing to reduce simulation time by selecting multiple sample points in each iteration. Although its computational efficiency is significantly improved, it still has the following limitations: First, the selection of initial sample points relies on Latin hypercube sampling, which does not fully incorporate probability density distribution characteristics, resulting in insufficient coverage of low-probability regions and affecting the model's generalization ability. Second, the sample selection mechanism near the limit state surface is relatively coarse, failing to accurately locate key regions and leading to redundant iterations. Finally, the lack of an adaptive interval reduction mechanism when dynamically adjusting the sample pool makes it impossible to efficiently focus on the boundary region between the failure domain and the safety domain, affecting convergence speed and accuracy.

[0005] The core problems faced by the aforementioned existing technologies are: uneven initial sample distribution leading to limited model generalization ability, inaccurate approximation of the limiting state surface resulting in low convergence efficiency, and inflexible dynamic adjustment strategies causing waste of computational resources. These problems are amplified, especially when dealing with the reliability analysis of mechanical structures with high-dimensional, nonlinear implicit function functions, making it difficult to meet the dual requirements of accuracy and efficiency in practical engineering. Summary of the Invention

[0006] Purpose of the invention: In order to overcome the shortcomings of the existing technology, the present invention provides an adaptive PC-Kriging reliability analysis method and system based on active learning. It aims to optimize the initial sample distribution through weighted clustering, dynamically focus on key areas through interval reduction, and construct an accurate approximation of the limit state surface by combining crossing points, so as to solve the shortcomings of the existing technology and achieve a balance between high efficiency and accuracy in the reliability analysis of complex mechanical structures.

[0007] Technical solution: To achieve the above objectives, the present invention provides an adaptive PC-Kriging reliability analysis method based on active learning, the method comprising:

[0008] The weighted clustering method is used to obtain the first candidate samples with a uniform distribution from the pre-generated MC sample pool, and the initial PC-Kriging model is constructed accordingly.

[0009] Using the interval reduction method, samples whose predicted values ​​of the limit state function fall within a specific interval are selected as the new sample pool; based on the sign of the predicted value, the new sample pool is divided into two sub-sample pools: the safe region and the failure region.

[0010] For the two sub-sample pools, a weighted clustering method is used to select a uniformly distributed second candidate sample from each pool.

[0011] Crossing points are constructed using second candidate samples distributed in the failure domain and the safety domain;

[0012] The constructed crossing points are used as new experimental points to update the PC-Kriging model;

[0013] Based on the predicted values ​​of the current PC-Kriging model, the upper and lower thresholds of the sample pool are dynamically adjusted, and the results are output when the termination iteration condition is met.

[0014] Furthermore, the step of obtaining uniformly distributed first candidate samples from a pre-generated MC sample pool using weighted clustering, and directly using the first candidate sample points as experimental points to construct the initial PC-Kriging model includes:

[0015] Weight coefficients are constructed based on sample probabilities, and a uniformly distributed weighted sample set is obtained using a weighted sampling method.

[0016] K-means is used to further select representative uniform samples based on the weighted sample set.

[0017] Furthermore, the interval reduction method is based on an interval reduction function, which is:

[0018]

[0019] in, and These are the upper and lower thresholds of the predicted value of the limit state function in the j-th iteration, respectively. s represents the surrogate model's prediction of sample x; mc Let be the sample set; β represents the reduction rate, which is a function of τ, specifically:

[0020]

[0021] In the formula, β min β represents the minimum reduction rate. max κ represents the maximum reduction rate; κ determines the rate of change of the function; τ represents the sample elimination rate, which is the ratio of the number of samples eliminated after the previous iteration to the total number of samples in the pool; τ c It is the position parameter of the function's center.

[0022] Furthermore, the construction of crossing points using second candidate samples distributed in the failure domain and the safety domain includes:

[0023] Based on the source of the second candidate sample, the second candidate sample is divided into a safe domain sample subset and a failure domain sample subset. Any sample point in the safe domain sample subset and any sample point in the failure domain sample subset are connected to form a crossing line.

[0024] Multiple points are discretized on each crossing line, and the crossing points near the limit state surface are selected using the KO learning function and the WKO learning function.

[0025] The KO learning function is represented as:

[0026]

[0027] Among them, P KO (·) represents the probability that the sample point is located near the limiting state surface. Let ε be the desired response function, and ε be the error in the desired region. For a pair of sample points The k-th discrete point on the crossing line; Φ(·) represents the cumulative distribution function of the standard normal distribution; The standard deviation; The mathematical expectation of a normal distribution;

[0028] The WKO learning function is represented as:

[0029]

[0030] Where WKO(·) represents the weighted probability of Kriging occurrence; W(·) represents the probability weight of discrete points; P ij It is a discrete set of points; For a pair of sample points The crossing points on the line between them.

[0031] Furthermore, the improved WKO learning function is defined as follows:

[0032]

[0033] The distance condition from the initial sample points is:

[0034]

[0035] The final crossing point x new for:

[0036]

[0037] Furthermore, the condition for terminating the iteration is expressed as:

[0038]

[0039] in, Let ε be the predicted failure probability value for the i-th iteration. cv This is the critical value.

[0040] An adaptive PC-Kriging reliability analysis system based on active learning, the system comprising:

[0041] The model building module is used to obtain uniformly distributed first candidate samples from the pre-generated MC sample pool using weighted clustering, and directly use the first candidate sample points as experimental points to build the initial PC-Kriging model.

[0042] The first selection module is used to select samples whose predicted values ​​of the limit state function are in a specific interval as a new sample pool using the interval reduction method; and divides the new sample pool into two sub-sample pools, a safe region and a failure region, based on the positive or negative of the predicted value.

[0043] The second selection module is used to select uniformly distributed second candidate samples from the two sub-sample pools by applying weighted clustering again.

[0044] The crossing point construction module is used to construct crossing points using second candidate samples distributed in the failure domain and the safety domain;

[0045] The iterative update module is used to iteratively update the PC-Kriging model by taking the constructed crossing points as new experimental points;

[0046] The discrimination output module is used to dynamically adjust the upper and lower thresholds of the sample pool based on the predicted values ​​of the current PC-Kriging model, gradually focusing on the sample interval near the limit state surface. When the relative error of the predicted failure probability values ​​of two adjacent iterations is less than a preset critical value, the iteration is terminated and the result is output.

[0047] Beneficial Effects: The adaptive PC-Kriging reliability analysis method and system based on active learning of the present invention have the following beneficial effects:

[0048] (1) Compared with the prior art mentioned in the background section, the technical solution of the present invention has obvious advantages. By focusing on key areas through interval reduction and updating the model based on the crossing point, it can more accurately approximate the limit state surface, reduce the number of experimental points, and improve the prediction accuracy. Moreover, it is more efficient in utilizing computing resources, and the intermediate process does not require calling the real performance function. In terms of dynamic adjustment, it can flexibly optimize sample selection and model update based on the model prediction value. When facing the reliability analysis of complex and ever-changing mechanical structures, it can better balance computational efficiency and accuracy, better adapt to actual engineering needs, and ensure that the analysis results are accurate and reliable.

[0049] (2) When selecting samples, weight coefficients are constructed based on sample probabilities, avoiding the shortcomings of using traditional joint probability distribution as weights. This allows for the allocation of smaller weights to samples with high probability density, resulting in a uniform distribution of the obtained weighted sample set. Normalization and random number feature value filtering ensure that the weighted samples are also distributed in low-probability regions. Furthermore, K-means is used to select uniform samples from the weighted sample set, resulting in a better distribution of experimental points for the initial PC-Kriging model, effectively improving the model's generalization ability and reducing bias.

[0050] (3) Based on the interval shrinking function, the upper and lower thresholds of the limit state function in the MC sample pool can be predicted by the PC-Kriging model, and these thresholds can be used to select sample points close to LSS from the sample pool as the sample pool for the next iteration update of the PC-Kriging model. The above algorithm can accelerate the convergence speed of the function in the early stage of iteration, and as the number of iterations increases, the number of samples in the sample pool decreases at an exponential rate without causing the model update to terminate prematurely.

[0051] (4) By dividing the sample subset to form the crossing line, the possible intersection region with the limit state surface can be accurately located. Points are discretized on the crossing line, and the KO and WKO learning functions are used to filter the crossing points. The point with the highest probability of the response value being on the limit state surface can be found from the set of discrete points. The WKO function, which considers the probability weight of the sample points, takes into account the probability of the discrete point position and the spatial distribution of the sample. The selected crossing points are more representative and can effectively improve the approximation accuracy of the PC-Kriging model near the limit state surface. Attached Figure Description

[0052] Figure 1 This is a flowchart illustrating the adaptive PC-Kriging reliability analysis method based on active learning.

[0053] Figure 2 This is a distribution diagram of the initial uniform sampling points;

[0054] Figure 3 This refers to the reduced interval after the first iteration, along with the corresponding uniform sampling points, crossing points, and newly added experimental points.

[0055] Figure 4 This refers to the reduced interval after the second iteration, along with the corresponding uniform sampling points, crossing points, and newly added experimental points.

[0056] Figure 5 This refers to the reduced interval after the third iteration, along with the corresponding uniform sampling points, crossing points, and newly added experimental points.

[0057] Figure 6 This refers to the reduced interval after the fourth iteration, along with the corresponding uniform sampling points, crossing points, and newly added experimental points.

[0058] Figure 7 This refers to the reduced interval after the fifth iteration, along with the corresponding uniform sampling points, crossing points, and newly added experimental points.

[0059] Figure 8 A schematic diagram of the KO function;

[0060] Figure 9 This is a schematic diagram of the system configuration of an adaptive PC-Kriging reliability analysis system based on active learning. Detailed Implementation

[0061] The invention will now be further described with reference to the accompanying drawings.

[0062] like Figure 1 The adaptive PC-Kriging reliability analysis method based on active learning shown includes the following steps S101-S106:

[0063] Step S101: Use weighted clustering to obtain uniformly distributed first candidate samples from the pre-generated MC sample pool, and directly use the first candidate sample points as experimental points to construct the initial PC-Kriging model.

[0064] Step S102: Using the interval reduction method, select samples whose predicted values ​​of the limit state function are in a specific interval as a new sample pool; based on the sign of the predicted value, divide the new sample pool into two sub-sample pools: the safe region and the failure region.

[0065] Step S103: For the two sub-sample pools, the weighted clustering method is used again to select the second candidate samples that are uniformly distributed from them.

[0066] Step S104: Construct crossing points using second candidate samples distributed in the failure domain and the safe domain;

[0067] Step S105: The constructed crossing points are used as new experimental points to iteratively update the PC-Kriging model;

[0068] Step S106: Based on the predicted values ​​of the current PC-Kriging model, dynamically adjust the upper and lower thresholds of the sample pool, gradually focusing on the sample interval near the limit state surface. When the relative error of the predicted failure probability values ​​of two adjacent iterations is less than the preset critical value, terminate the iteration and output the result.

[0069] Compared to the existing technologies mentioned in the background section, the technical solution of this invention has significant advantages. By focusing on key areas through interval reduction and updating the model based on crossing points, it can more accurately approximate the limit state surface, reduce the number of experimental points, and improve prediction accuracy. Furthermore, it is more efficient in utilizing computational resources, as the intermediate process does not require calling the actual performance function. In terms of dynamic adjustment, it can flexibly optimize sample selection and model updates based on model predictions. When facing the reliability analysis of complex and ever-changing mechanical structures, it can better balance computational efficiency and accuracy, better adapt to actual engineering needs, and ensure accurate and reliable analysis results.

[0070] To achieve better generalization ability and lower bias in the constructed PC-Kriging model, it is desirable that the acquired crossing points be widely distributed along the limiting state surface. Traditional MC samples are not uniform in the variable space. If candidate points are drawn from the sample pool with equal probability, the selected candidate points will inevitably cluster in regions with high probability density. This leads to the clustering of crossing points in local areas, while regions with low probability density remain unacquired. To address this issue, this invention proposes a K-Weight Clustering method, which includes (i.e., step S101 above includes) the following steps S201-S202:

[0071] Step S201: Construct weight coefficients based on sample probabilities, and obtain a uniformly distributed weight sample set using a weighted sampling method;

[0072] To obtain uniformly distributed candidate points, samples with higher probability densities need to be assigned smaller weighting coefficients. Since the evaluation of the joint probability distribution at a single point may not represent the probability of points within the specified neighborhood, and ignores the influence of the shape of the joint probability distribution around the study point, the joint probability distribution cannot be directly used as weights. Therefore, this step specifically includes:

[0073] - Define the probability of a sample point as the probability of a sample point occurring within a neighborhood of 10% of the standard deviation of each input random variable.

[0074] -The weight coefficient W of the i-th sample (i) Represented as:

[0075]

[0076] in,

[0077] P(x) is a sample x = [x1, x2, ..., xn] n The probability of occurrence, x (i) This represents the i-th sample; f(x) is the standard deviation of x; j ) represents a random variable x j The probability distribution function, Represents a random variable x j The cumulative distribution function; x1, x2, ..., x n They are independent of each other; α represents the weight smoothing adjustment parameter. The main purpose of this parameter is to prevent extremely high or low weights due to differences in probability distribution functions, and to increase the probability that each region is sampled equally. The smaller the value of α, the more important the low-weight samples will be; conversely, the larger the value, the greater the probability of sampling high-weight samples. In the method proposed in this invention, its value is set to 0.5-1.

[0078] - The weighting coefficients are normalized and expressed as follows:

[0079]

[0080] - Assign a random number u to each sample (i) =rand(0,1) and calculate the feature value for each sample. Finally, the M samples with the largest feature values ​​are selected as the result of weighted sampling. It should be noted that due to the subsequent reduction of the interval, the sample interval may be multi-regional. Therefore, in order to ensure that the weighted samples are also distributed in low-probability areas, the value of M cannot be too small. In this invention, the value of M is 1% of the MC sample size.

[0081] Step S202: Using K-means, further select representative uniform samples based on the weighted sample set.

[0082] In steps S201-S202 above, by constructing weight coefficients based on sample probabilities, the shortcomings of using traditional joint probability distribution as weights are avoided. This allows for the allocation of smaller weights to samples with high probability density, resulting in a uniform distribution of the obtained weight sample set. Normalization and random number feature value filtering ensure that the weight samples are also distributed in low-probability regions. Furthermore, K-means is used to select uniform samples from the weight sample set, resulting in a better distribution of experimental points for the initial PC-Kriging model, effectively improving the model's generalization ability and reducing bias.

[0083] The reliability prediction accuracy based on the surrogate model mainly depends on the prediction performance in important regions near the LSS (Limit State Surface where the limit state function equals zero). Therefore, the key to active learning is to accurately locate the sampling points on or near the LSS.

[0084] The interval reduction method in step S102 above is based on an interval reduction function, which is:

[0085]

[0086] in, and These are the upper and lower thresholds of the predicted value of the limit state function in the j-th iteration, respectively.

[0087]

[0088] In the formula, β min This represents the minimum reduction rate, β in this embodiment. min The value is 0.4; β max This represents the maximum reduction rate; in this embodiment, β... max The value is 0.7; κ determines the rate of change of the function, and in this embodiment, κ is set to 20; τ represents the sample rejection rate, which is the ratio of the number of samples rejected to the total number of samples in the sample pool after the last iteration; τ c It is the function center position parameter, τ in this embodiment c The value is 0.5.

[0089] Based on this function, the upper and lower thresholds of the limiting state function in the MC sample pool can be predicted using the PC-Kriging model. These thresholds are then used to select sample points close to the LSS from the sample pool, which will serve as the sample pool for the next iteration of the PC-Kriging model. This algorithm accelerates the convergence speed of the function in the early stages of iteration, and the exponential decrease in the number of samples in the sample pool as the number of iterations increases does not cause premature termination of the model update. Figures 2 to 7 This is a schematic diagram of the reduction of the example, where Figure 2 This is a distribution map of the initial uniform sampling points. Figures 3 to 7 The graph shows the distribution of the reduced interval after multiple iterations, along with the corresponding uniform sampling points, crossing points, and newly added experimental points. In each graph, purple dots represent sampling points, black dashed lines represent crossing lines, and black wavy lines represent limit state curves.

[0090] The method described above uses an interval reduction strategy to gradually filter out samples far from the limiting state surface (LSS) and focus on the sample interval near the LSS. This not only avoids computational overhead in non-sensitive regions, but also maintains the surrogate model's high performance in predicting the LSS even without other optimization measures. During iteration, the continuous change in the sample interval increases the diversity of candidate points and improves the quality of crossing points. Furthermore, with increasing iterations, candidate points and crossing points approach the true LSS more quickly, thus improving the model's convergence speed.

[0091] In the aforementioned method, a small number of initial training samples are first selected from the MC sample pool using a uniform sampling strategy to construct the initial PC-Kriging predictor. After j iterations, assuming the sample set after (j-1) region reductions is Based on the current PC-Kriging predictor It can be further Divided into and These correspond to two sub-sample pools, one with positive and one with negative predicted values. Then, using a uniform sampling strategy, n is selected from each of the two sub-sample pools. S and n F There are 1 sample points, denoted as _ . and

[0092] The step S104 above, which involves constructing crossing points using second candidate samples distributed in the failure domain and the safety domain, includes the following steps S301-S304:

[0093] Step S301: Based on the source of the second candidate sample, the second candidate sample is divided into a safe domain sample subset and a failure domain sample subset. A crossing line is formed by connecting any sample point within the safe domain sample subset and any sample point within the failure domain sample subset; that is, for any pair of sample points... The line connecting two points is called the crossing line. The crossing line intersects the limit state surface (or limit state curve) at least once, and this intersection point is the ideal crossing point.

[0094] Step S302: Discretize and generate multiple points on each crossing line, and use the KO (Kriging Occurrence) learning function and the WKO (Weighted Kriging Occurrence) learning function to select crossing points near the limit state surface;

[0095] In this step, a discrete point set P between two points is generated using linear interpolation. ij Suppose we discretize the crossing line into m points, then the k-th discrete point can be represented as:

[0096]

[0097] After obtaining the discrete points on the crossing line, the most crucial step is to select the crossing points from these discrete points. It's important to note that during the algorithm iteration process, because the PC-Kriging model cannot perfectly fit the limit state surface, the searched crossing points are not necessarily the intersections of the limit state surface, but rather the points in the discrete point set whose corresponding response values ​​have the highest probability of occurring on the limit state surface. The KO function introduced in this invention is defined as the response of point X (i.e., The probability of point X appearing in the desired region is the area of ​​the probability distribution function (PDF) of the response of point X within the range corresponding to the desired region. For example... Figure 8 As shown, the KO function represents the shaded region under the distribution of points from point A to point E, from which a candidate point should be selected to train the current Kriging predictor. For example, point A has a mean outside the desired region and a KO value of zero, meaning it will not be considered as a candidate point. Points B and D have means close to the desired region and positive KO values, but only a small fraction of their distributions are considered, so their chances of becoming candidate points are small. Points C and E have means within the desired region and a high chance of becoming candidate points. The error of the desired region (i.e., ε) is equal to the standard deviation (i.e., ... The function is usually chosen as ) The latter results in a larger expected area for point E compared to point C, which has a smaller standard deviation and a sharper distribution.

[0098] The mathematical definition of the KO function at any point X is represented as a Kriging predictor. The corresponding response appears in the expected region. The probability of is given by the following formula:

[0099]

[0100] Where f(·) represents the probability distribution function of the Kriging predictor;

[0101] Therefore, the KO learning function is expressed as:

[0102]

[0103] Among them, P KO (·) represents the probability that the sample point is located near the limiting state surface. Let ε be the desired response function, and ε be the error in the desired region. For a pair of sample points The k-th discrete point on the crossing line; Φ(·) represents the cumulative distribution function of the standard normal distribution; The standard deviation; The mathematical expectation of a normal distribution;

[0104] In the first embodiment, the WKO learning function is represented as:

[0105]

[0106] Where WKO(·) represents the weighted probability of Kriging occurrence, a comprehensive measure of the likelihood of a discrete point being near the limit state surface, considering both the probability weight of the sample point itself and its Kriging occurrence probability; when selecting crossing points, the larger the value of WKO(·), the higher the probability that the discrete point is near the limit state surface, and the more appropriate its probability weight is in the entire sample space, thus the higher the probability and importance of the discrete point as a crossing point; W(·) represents the probability weight of the discrete point; P ij It is a discrete set of points; For a pair of sample points The crossing points on the line between them.

[0107] By applying the WKO learning function to the discrete points of each crossing line, we can easily obtain n. S ×n F There are several crossing points, denoted as...

[0108] In the above method, by dividing the sample subset to form the crossing line, the possible intersection region with the limit state surface can be accurately located. Points are discretized on the crossing line, and the Kjörk (KO) and WKO (Warning Kjörk) learning functions are used to filter the crossing points. This allows the identification of the points with the highest probability of their response values ​​falling on the limit state surface from the set of discrete points. The WKO function, which considers the probability weights of the sample points, comprehensively considers both the probability of the discrete point locations and the spatial distribution of the samples, resulting in more representative crossing points that effectively improve the approximation accuracy of the PC-Kriging model near the limit state surface.

[0109] In a preferred embodiment, the improved WKO learning function is defined as:

[0110]

[0111] The distance condition from the initial sample points is:

[0112]

[0113] The final crossing point x new for:

[0114]

[0115] Compared to the previous embodiment, this embodiment improves the WKO learning function to avoid sample points being too close together.

[0116] The condition for terminating the iteration in step S106 is expressed as follows:

[0117]

[0118] in, Let ε be the predicted failure probability value for the i-th iteration. cv This is the critical value.

[0119] In the above embodiment, i, j, k, etc. are all local variables.

[0120] Generally, when the error fluctuation of the failure probability is less than 0.005, the calculation of the failure probability is considered relatively stable. However, depending on the varying accuracy and efficiency requirements of the reliability problem being studied, setting strict convergence criteria can lead to model non-convergence or excessive computational cost. To balance computational efficiency and accuracy, the critical value of the relative error is typically around 10. -5 ≤ε cv ≤10 -2 Within the range.

[0121] This invention also provides an adaptive PC-Kriging reliability analysis system based on active learning, the system comprising:

[0122] The model building module 401 is used to obtain uniformly distributed first candidate samples from the pre-generated MC sample pool using the weighted clustering method, and directly use the first candidate sample points as experimental points to build the initial PC-Kriging model.

[0123] The first selection module 402 is used to select samples whose predicted values ​​of the limit state function are in a specific interval as a new sample pool using the interval reduction method; and divides the new sample pool into two sub-sample pools, a safe region and a failure region, according to the positive or negative of the predicted value.

[0124] The second selection module 403 is used to select uniformly distributed second candidate samples from the two sub-sample pools by applying weighted clustering again.

[0125] Crossing point construction module 404 is used to construct crossing points using second candidate samples distributed in the failure domain and the safety domain;

[0126] The iterative update module 405 is used to iteratively update the PC-Kriging model by taking the constructed crossing points as new experimental points;

[0127] The discrimination output module 406 is used to dynamically adjust the upper and lower thresholds of the sample pool based on the predicted value of the current PC-Kriging model, gradually focusing on the sample interval near the limit state surface. When the relative error of the predicted failure probability values ​​of two adjacent iterations is less than a preset critical value, the iteration is terminated and the result is output.

[0128] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. An adaptive PC-Kriging reliability analysis method based on active learning, characterized in that, The method includes: The weighted clustering method is used to obtain the first candidate samples with a uniform distribution from the pre-generated MC sample pool, and the initial PC-Kriging model is constructed accordingly. Using the interval reduction method, samples whose predicted values ​​of the limit state function fall within a specific interval are selected as the new sample pool; based on the sign of the predicted value, the new sample pool is divided into two sub-sample pools: the safe region and the failure region. For the two sub-sample pools, a weighted clustering method is used to select a uniformly distributed second candidate sample from each pool. Crossing points are constructed using second candidate samples distributed in the failure domain and the safety domain; The constructed crossing points are used as new experimental points to update the PC-Kriging model; Based on the predicted values ​​of the current PC-Kriging model, the upper and lower thresholds of the sample pool are dynamically adjusted, and the results are output when the termination iteration condition is met. The interval reduction method is based on an interval reduction function, which is: ； in, and They are the first Upper and lower thresholds of the predicted value of the limit state function in the next iteration; For the surrogate model to sample The predicted value; For the sample set; Indicates the reduction rate, which is about The function is as follows: ； In the formula, This represents the minimum value of the reduction rate; This represents the maximum value of the reduction rate; Determines the rate of change of the function; The sample rejection rate is the ratio of the number of samples rejected after the last iteration to the total number of samples in the sample pool. It is the position parameter of the function's center.

2. The adaptive PC-Kriging reliability analysis method based on active learning according to claim 1, characterized in that, The step involves using weighted clustering to obtain uniformly distributed first candidate samples from a pre-generated MC sample pool, and directly using these first candidate sample points as experimental points to construct an initial PC-Kriging model, including: Weight coefficients are constructed based on sample probabilities, and a uniformly distributed weighted sample set is obtained using a weighted sampling method. K-means is used to further select representative uniform samples based on the weighted sample set.

3. The adaptive PC-Kriging reliability analysis method based on active learning according to claim 1, characterized in that, The process of constructing crossing points using second candidate samples distributed in the failure domain and the safe domain includes: Based on the source of the second candidate sample, the second candidate sample is divided into a safe domain sample subset and a failure domain sample subset. Any sample point in the safe domain sample subset and any sample point in the failure domain sample subset are connected to form a crossing line. Multiple points are discretized on each crossing line, and the crossing points near the limit state surface are selected using the KO learning function and the WKO learning function. The KO learning function is represented as: ； in, This represents the probability that a sample point lies near the limiting state surface. Let be the desired response function. The error is for the desired region; For a pair of sample points The kth discrete point on the line crossing between them; The cumulative distribution function representing the standard normal distribution; The standard deviation; The mathematical expectation of a normal distribution; The WKO learning function is represented as: ； ； in, This represents the weighted probability of Kriging occurring; Represents the probability weights of discrete points; It is a discrete set of points; For a pair of sample points The crossing points on the line between them.

4. The adaptive PC-Kriging reliability analysis method based on active learning according to claim 3, characterized in that, The improved WKO learning function is defined as follows: ； The distance condition from the initial sample points is: ； ； Final crossing point for: 。 5. The adaptive PC-Kriging reliability analysis method based on active learning according to claim 1, characterized in that, The condition for terminating the iteration is expressed as: ； in, Let be the predicted failure probability value for the i-th iteration. This is the critical value.

6. An adaptive PC-Kriging reliability analysis system based on active learning, characterized in that, The system includes: The model building module is used to obtain uniformly distributed first candidate samples from the pre-generated MC sample pool using weighted clustering, and directly use the first candidate sample points as experimental points to build the initial PC-Kriging model. The first selection module is used to select samples whose predicted values ​​of the limit state function are in a specific interval as a new sample pool using the interval reduction method; and divides the new sample pool into two sub-sample pools, a safe region and a failure region, based on the positive or negative of the predicted value. The second selection module is used to select uniformly distributed second candidate samples from the two sub-sample pools by applying weighted clustering again. The crossing point construction module is used to construct crossing points using second candidate samples distributed in the failure domain and the safety domain; The iterative update module is used to iteratively update the PC-Kriging model by taking the constructed crossing points as new experimental points; The discrimination output module is used to dynamically adjust the upper and lower thresholds of the sample pool based on the predicted values ​​of the current PC-Kriging model, gradually focusing on the sample interval near the limit state surface. When the relative error of the predicted failure probability values ​​of two adjacent iterations is less than a preset critical value, the iteration is terminated and the result is output. The interval reduction method is based on an interval reduction function, which is: ； in, and They are the first Upper and lower thresholds of the predicted value of the limit state function in the next iteration; For the surrogate model to sample The predicted value; For the sample set; Indicates the reduction rate, which is about The function is as follows: ； In the formula, This represents the minimum value of the reduction rate; This represents the maximum value of the reduction rate; Determines the rate of change of the function; The sample rejection rate is the ratio of the number of samples rejected after the last iteration to the total number of samples in the sample pool. It is the position parameter of the function's center.

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