A spectrum normalization gaussian process based active learning reliability analysis method

Abstract

The application belongs to the field of structural reliability analysis and uncertainty quantification, and specifically discloses an active learning reliability analysis method based on spectral normalized Gaussian process, which comprises the following steps: generating a Monte Carlo sample set according to a joint probability density function of random variables; then constructing an initial training data set and a spectral normalized Gaussian process (SNGP) surrogate model of a structural performance function; predicting the Monte Carlo sample set by using the surrogate model, estimating a system failure probability and screening a candidate sample pool; judging whether the model converges and whether the system failure probability meets the accuracy requirement according to the prediction result; if yes, outputting the surrogate model and the system failure probability estimation result, and completing the analysis. The application can significantly reduce the number of calls to the original calculation model in complex high-dimensional nonlinear reliability analysis, greatly improve the analysis efficiency while ensuring the accuracy, and is suitable for reliability analysis and design optimization of high-dimensional strong nonlinear systems in the fields of aerospace, vehicle engineering and the like.

Description

Technical Field

[0001] This invention relates to the field of structural reliability analysis and uncertainty quantification technology, specifically to an active learning reliability analysis method based on spectral normalized Gaussian processes. Background Technology

[0002] Structural reliability analysis is a crucial technique for ensuring the safety and service performance of engineering systems. Its core lies in assessing the probability of structural failure under conditions of multi-source uncertainty. Real-world engineering structures often exhibit characteristics such as high-dimensional input parameters, strong nonlinear response relationships, and low failure probabilities, posing significant computational challenges to reliability analysis.

[0003] Monte Carlo simulation is widely used in reliability analysis due to its applicability and theoretical completeness. However, this method requires a large number of samples and repeated calls to high-precision numerical models, resulting in extremely high computational costs in high-dimensional and strongly nonlinear structures, making it difficult to meet practical engineering needs. To reduce computational costs, reliability analysis methods based on surrogate models have been extensively studied, with response surface methodology, support vector regression, neural networks, and Kriging (Gaussian process regression) being used to replace real structural models. Kriging models can simultaneously provide the predicted mean and prediction uncertainty, and can be combined with active learning strategies to develop the AK-MCS method, which guides samples to gradually approach the limit state boundary through a learning function, achieving good results in low- to medium-dimensional problems. However, under high-dimensional and strongly nonlinear conditions, training traditional Kriging models requires repeated construction and solving of high-dimensional correlation matrices, with computational complexity increasing sharply with dimensionality and sample size, leading to a significant decrease in modeling efficiency and numerical stability. At the same time, the increased sparsity of samples in high-dimensional parameter spaces makes it difficult for kernel-based correlation modeling to accurately characterize complex nonlinear response features, thus limiting its applicability in high-dimensional reliability analysis.

[0004] To address high-dimensional problems, some studies have introduced dimensionality reduction techniques to alleviate the curse of dimensionality. However, the dimensionality reduction process may inevitably lose key information, thus affecting the accuracy of reliability assessment. Meanwhile, deep learning uncertainty quantification methods are gaining attention. Methods such as Bayesian neural networks, MC-dropout, and deep ensembles can characterize prediction uncertainty. However, their inference process typically relies on multiple forward computations or multi-model training, resulting in significant computational overhead in large-scale Monte Carlo analyses. Furthermore, they generally lack explicit characterization of the "sample distance-uncertainty growth" relationship, limiting their potential for exploration in active learning reliability analysis.

[0005] In recent years, the Spectral Normalized Gaussian Process (SNGP), with its distance-aware characteristics, has achieved stable, calibrated predictive uncertainty that does not vanish with dense training samples by introducing spectral normalization constraints during deep feature extraction and combining it with a Gaussian process output layer. This has enabled the model to have stronger generalization and exploration capabilities in high-dimensional nonlinear problems. However, existing research lacks methods to integrate SNGP with active learning reliability analysis frameworks, especially in complex engineering structures. How to efficiently guide sample selection, reduce redundant computation, and achieve rapid convergence of failure probabilities using its distance-aware uncertainty remains a challenge.

[0006] Therefore, it is necessary to propose a reliability analysis method that integrates spectral normalized Gaussian process and active learning strategy to overcome the limitations of traditional Gaussian process and existing deep uncertainty method in the reliability analysis of high-dimensional strongly nonlinear structures, and to achieve a balance between high accuracy and high efficiency. Summary of the Invention

[0007] To address the problems existing in the prior art, this invention provides an active learning reliability analysis method based on spectral normalized Gaussian processes. This method integrates uncertainty quantification modeling, active learning, and high-dimensional surrogate model techniques to construct an efficient and robust structural reliability analysis framework. It aims to achieve accurate identification of failure boundaries in high-dimensional strongly nonlinear systems with low computational cost and obtain high-precision failure probability estimates. First, a spectral normalized Gaussian process (SNGP) surrogate model integrating a deep feature extractor and a Gaussian process output layer is constructed. Second, the calibrated uncertainty estimate output by this model is used to drive a two-stage adaptive active learning sampling strategy to efficiently explore the high-dimensional parameter space and locate information samples near the failure boundary. Then, a distance correction learning function is used in the active learning process to balance the prediction mean, uncertainty, and sample space distribution, thereby improving sampling efficiency and robustness. Finally, a large-scale Monte Carlo simulation is performed based on the surrogate model to estimate the system failure probability, and a stopping criterion based on error evaluation is introduced to directly judge the accuracy of the failure probability estimation. This ensures the reliability of the results while avoiding the premature termination or invalid iteration problems caused by the traditional convergence criterion based on uncertainty threshold, thus solving the problems mentioned in the background.

[0008] To achieve the above objectives, the present invention provides the following technical solution: an active learning reliability analysis method based on a spectral normalized Gaussian process, comprising the following steps: S1. Generate a large-scale Monte Carlo sample set based on the joint probability density function of the random variables; S2. Generate an initial training sample set in the input parameter space, call the real model to calculate the training sample set, and construct the initial training dataset; S3. Based on the initial training dataset, construct the SNGP surrogate model of the structure performance function spectrum normalized Gaussian process; S4. Using the surrogate model trained in step S3, predict the Monte Carlo sample set, estimate the system failure probability, and screen the candidate sample pool. S5. Based on the prediction results of step S4, determine whether the model has converged. If not, select new training sample points, calculate their true responses, and update the training dataset. If the model has converged, proceed to step S6. S6. Determine whether the calculated system failure probability meets the accuracy requirements. If not, expand the Monte Carlo sample set. If it does, output the surrogate model and the system failure probability estimation results to complete the analysis.

[0009] Preferably, step S1 specifically includes: based on the joint probability density function of the structural random variables. f X (x), within the domain of the random variable, the number of samples generated using Monte Carlo sampling is... N MC Input parameters Monte Carlo sample set S MC .

[0010] Preferably, step S2 specifically includes: in the joint probability density function of the random variables f X In the multidimensional parameter space defined by (x), a spatially uniform coverage sampling strategy is executed to generate a training sample set. S 0, and call the real model. G (x) is calculated to construct the initial training dataset. .

[0011] Preferably, step S3 specifically includes: based on the current... i The training dataset corresponding to the next active learning iteration ( i =0,1,2,…), construct an SNGP surrogate model for approximating the structure performance function. The SNGP proxy model By integrating a deep feature extractor f φ (x) and the Gaussian process GP output layer enable distance-aware uncertainty modeling, and can simultaneously output the predicted mean of the structural response. and uncertainty .

[0012] Preferably, step S4 specifically includes the following steps: S41. Utilizing the constructed proxy model For the Monte Carlo sample setS MC Perform batch response prediction; S42. Estimate the system failure probability based on the response results predicted in step S41. ; S43, Monte Carlo sample set S MC The prediction results are calculated based on the absolute value of the predicted mean. Sort in ascending order and filter by the smallest absolute value. p × N MC These sample points constitute the candidate sample pool for the current iteration. S candidate .

[0013] Preferably, in step S5, determining whether the model has converged, and if not, selecting new training sample points, calculating their true responses, and updating the training dataset, specifically includes: based on the prediction results of the surrogate model, determining the current... i Does the surrogate model trained during the next active learning iteration meet the preset convergence criteria? If not, the distance-aware adaptive sampling strategy DA-AS is used in the candidate sample pool. S candidate Select new training sample points x ∗ Call the real model G (x) is used to calculate the sample points x. ∗ and its response Add to the current training dataset To update the training dataset i+1 = ∪{(x ∗ , Continue training and updating the proxy model.

[0014] Preferably, the preset convergence criterion is an error-based stopping criterion, i.e., an error-type stopping criterion (ESC), which uses an upper limit of error. With misclassification probability This approach combines various methods to ensure accuracy requirements are met while avoiding oversampling; specifically, it includes: For each sample point x j Its misclassification probability for: ; in, and These are proxy models For sample point x j The predicted mean and uncertainty, The cumulative distribution function of the standard normal distribution is calculated by ESC for each sample point. Quantification model prediction error ϵ The number of misclassified samples is modeled using a Poisson binomial distribution; when the model's maximum prediction error... ϵ max Meet the preset information level α Corresponding error threshold ϵ thr When the model converges, it is considered to have converged.

[0015] Preferably, the distance-aware adaptive sampling strategy DA-AS specifically includes: Define the distance metric d(x) in the space of standardized random variables. k , ), which represents the candidate point x k up to the current training set The closest front at medium distance m The mean of the Euclidean distances of the training samples, i.e.: ; Where, x i,j It is the training set Mid-distance candidate sample point x k The most recent j The sample points; ||⋅|| represents the Euclidean distance; this metric is essentially the reciprocal of the local sample density, and the larger the value, the sparser the training samples in the neighborhood of the candidate point, and the higher the potential for spatial exploration; Using d(x) k , ) respectively for traditional learning functions U / EFF Corrections are made to control the spatial distribution of samples while uncovering model uncertainties: a) Targeting U The function penalizes by distance to encourage exploration of regions far from existing training samples, and the optimal sample is selected by minimizing the correction function; b) For the EFF function, a reward is given based on distance to encourage the spatial dispersion of samples, and the optimal sample is selected by maximizing the correction function.

[0016] Preferably, the step S6 of determining whether the calculated system failure probability meets the accuracy requirements specifically includes: determining the failure probability estimated by the current surrogate model. Coefficient of variation COV Does it meet the preset accuracy requirements? When the convergence value is less than the preset convergence threshold of 0.05, the surrogate model is considered to have reached sufficient accuracy, the training process ends, and the final trained surrogate model is output. and system failure probability Complete the analysis.

[0017] The beneficial effects of this invention are: 1) The Spectrum Normalized Gaussian Process (SNGP) surrogate model used in this invention incorporates a distance-aware deep feature extractor. f φ (x) and the Gaussian process output layer can provide well-calibrated mean and variance estimates, breaking through the limitations of conventional Gaussian process regression in predicting uncertainty and inaccuracy in high-dimensional and strongly nonlinear problems, and providing key technical support for the reliability analysis of high-dimensional and strongly nonlinear problems.

[0018] 2) This invention integrates the SNGP agent with an uncertainty-guided active learning strategy. By utilizing its distance awareness and well-calibrated uncertainty, it can systematically identify unreliable prediction regions and guide adaptive sampling to key and underexplored regions in the sample space, thereby achieving efficient improvement of the agent model and accurate estimation of failure probability.

[0019] 3) The distance-aware adaptive sampling strategy proposed in this invention, through coupling U / EFF Learning functions and distance metrics effectively balance "effective information utilization" and "exploration of unknown regions" in the model update process, significantly improving sampling efficiency and accelerating the convergence of surrogate models and failure probability estimates, providing an efficient solution to the problem of high computational costs in high-dimensional reliability analysis. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of the reliability analysis method based on spectral normalized Gaussian process and active learning in an embodiment of the present invention; Figure 2 This is a schematic diagram of the geometric model of a high-dimensional nonlinear cantilever beam structure in an embodiment of the present invention; Figure 3 This is a schematic diagram of the SNGP proxy model structure in an embodiment of the present invention; Figure 4 This is a comparison chart of the probability density function of the high-dimensional nonlinear cantilever beam structure response predicted by the spectral normalized Gaussian process surrogate model in an embodiment of the present invention and the Monte Carlo simulation results. Figure 5 This is a comparison chart of the cumulative probability distribution function of the high-dimensional nonlinear cantilever beam structure response predicted by the spectral normalized Gaussian process surrogate model in this embodiment of the invention and the Monte Carlo simulation results. Detailed Implementation

[0021] 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 skilled in the art without creative effort are within the scope of protection of the present invention.

[0022] Example 1 This invention provides a technical solution: an active learning reliability analysis method based on a spectral normalized Gaussian process, the process of which is as follows: Figure 1 As shown, it includes the following steps: S1. Generate a large-scale Monte Carlo sample set based on the joint probability density function of the random variables, specifically including: based on the joint probability density function of the structural random variables. f X (x), within the domain of the random variable, the number of samples generated using Monte Carlo sampling is... N MC Input parameters Monte Carlo sample set S MC .

[0023] random variable X = (X F , X M , X G () is a vector of random input variables, including external loads in the model. F Material properties M and model geometry G The uncertainty of the sample set. S MC Sample set used for subsequent failure probability assessment based on surrogate model. S MC size N MC It is linked to the model's predicted failure probability.

[0024] S2. Generate an initial training sample set in the input parameter space, call the real model to calculate the training sample set, and construct the initial training dataset; specifically, this includes: in the joint probability density function of random variables... f X In the multidimensional parameter space defined by (x), a spatially uniform coverage sampling strategy is executed to generate a training sample set. S 0, and call the real model. G (x) is calculated to construct the initial training dataset. .

[0025] Based on the joint probability distribution of random variables f X(x), implement a hierarchical uniform sampling scheme in its corresponding high-dimensional parameter space, and use Latin Hypercube Sampling (LHS) to generate a preset number of N The initial experimental design (DoE) of 0 forms the input parameter sample set. S 0; subsequently, for the sample set S Each sample point x in 0 k ( k = 1, ..., N 0 Call the actual structure performance function one by one G (x) Perform numerical calculations to obtain the corresponding structural response values. G (x k This allows us to construct the initial training dataset. LHS provides a uniform and representative initial sample distribution in a high-dimensional parameter space, which is beneficial for the SNGP model to establish a stable distance-aware relationship, enabling the prediction uncertainty to reasonably reflect the sample sparsity, thereby improving the robustness and practicality of the surrogate model in the reliability analysis of complex engineering projects.

[0026] S3. Based on the initial training dataset, construct a surrogate model of the Spectral Normalized Gaussian Process (SNGP) for the structure-performance function, achieving efficient approximation of high-dimensional, strongly nonlinear limit state functions; specifically including: based on the current... i The training dataset corresponding to the next active learning iteration ( i =0,1,2,…), construct an SNGP surrogate model for approximating the structure performance function. The SNGP proxy model By integrating a deep feature extractor f φ (x) and the Gaussian process GP output layer enable distance-aware uncertainty modeling, and can simultaneously output the predicted mean of the structural response. and uncertainty This provides consistent and informative guidance for active learning adaptive sampling, enabling efficient approximation of performance functions for high-dimensional, strongly nonlinear structures. Specifically, the deep feature extractor employs spectral normalization constraints to limit the network's Lipschitz constant, thereby maintaining the geometric stability and distance consistency of the latent feature space.

[0027] S4. Using the surrogate model trained in step S3, predict the Monte Carlo sample set, estimate the system failure probability, and filter the candidate sample pool. Specifically, this includes the following steps: S41. Utilizing the constructed proxy model For the Monte Carlo sample set S MC Perform batch response prediction; obtain each sample point x j ( j = 1, 2, …, N MC The predicted mean of the corresponding structural performance function and standard deviation ; S42. Estimate the system failure probability based on the response results predicted in step S41. ; Definition of the first j Reliability indicators for each sample point β j for:

[0028] In the formula, e To ensure numerical stability, the preferred value for the regularization constant is [value to be filled in]. e =10 -6 Based on this, the system failure state is assessed, and the first... i The system failure probability estimate in the next iteration Defined as:

[0029] In the formula, Φ(⋅) is the cumulative distribution function of the standard normal distribution. This method, by combining the predicted mean and the predicted uncertainty, achieves a smooth and stable estimate of the sample failure probability, effectively avoiding reliance on thresholds (such as...). G (x)≤0) to determine the possible numerical oscillation problem.

[0030] S43, Monte Carlo sample set S MC The prediction results are calculated based on the absolute value of the predicted mean. Sort in ascending order and filter by the smallest absolute value. p × N MC These sample points constitute the candidate sample pool for the current iteration. S candidate This ensures that the candidate sample set is located at the predicted failure boundary. High-value samples near ≈0 are prioritized for inclusion in the subsequent active learning selection process.

[0031] S5. Based on the prediction results of step S4, determine whether the model has converged. If not, select new training sample points, calculate their true responses, and update the training dataset. If the model has converged, proceed to step S6.

[0032] Based on the prediction results of the surrogate model, determine the current number of... i Does the surrogate model trained during the next active learning iteration meet the preset convergence criteria? If not, the distance-aware adaptive sampling strategy DA-AS is used in the candidate sample pool. S candidate Select new training sample points x ∗ Call the real model G (x) is used to calculate the sample points x. ∗ and its response Add to the current training dataset To update the training dataset i+1 = ∪{(x ∗ , Continue training and updating the proxy model.

[0033] Furthermore, the preset convergence criterion is an error-based stopping criterion, i.e., an error-based stopping criterion (ESC). Traditional convergence criteria such as max( EFF ≤10 -3 or min( U The condition ≥ 2 applies to Gaussian process (GP) models, but it cannot accurately reflect the variation of uncertainty in the SNGP model as a function of the training samples. The error-based stopping criterion (ESC) uses an upper limit of error. With misclassification probability This approach combines various methods to ensure accuracy requirements are met while avoiding oversampling; specifically, it includes: For each sample point x j Its misclassification probability for:

[0034] in, and These are proxy models For sample point x j The prediction results. The cumulative distribution function is the standard normal distribution. ESC calculates the cumulative distribution function for each sample point. Quantification model prediction error ϵ The number of misclassified samples is modeled using a Poisson binomial distribution; for rare events (low failure probability) problems, it can be further approximated as a Poisson distribution. The active learning process terminates when the following condition is met:

[0035] When the model's maximum prediction error ϵ max Meet the preset information level α Corresponding error threshold ϵ thr When the model converges, it is considered to have converged.

[0036] Furthermore, the DA-AS is a candidate sample set-based algorithm. S candidate The proposed distance-aware active learning sample selection method employs a distance-aware adaptive sampling (DA-AS) strategy combined with a spectral normalized Gaussian process (SNGP) model, improving the sampling efficiency and accuracy of the surrogate model in high-dimensional complex systems. It avoids the sample clustering and redundant sampling problems that traditional learning functions easily encounter when dealing with high-dimensional nonlinear problems, enhancing the spatial coverage and information utilization efficiency of active learning sampling, thereby achieving efficient and stable exploration of key regions in the reliability analysis of high-dimensional nonlinear structures. The DA-AS strategy specifically includes: Define the distance metric d(x) in the space of standardized random variables. k , ), which represents the candidate point x k up to the current training set The closest front at medium distance m The mean of the Euclidean distances of the training samples, i.e.:

[0037] The input variables have been normalized according to their statistical scales before calculation, x i,j It is the training set Mid-distance candidate sample point x k The most recent j _x_{i=1}^2 sample points; ||⋅|| represents the Euclidean distance; this metric is essentially the reciprocal of the local sample density, and a larger value indicates that the training samples in the neighborhood of the candidate point are sparser, and the higher the potential for spatial exploration; using d(x_{i=1}^2 sample points; ||⋅|| represents the Euclidean distance; this metric is essentially the reciprocal of the local sample density, and a larger value k , ) respectively for traditional learning functions U / EFF Corrections are made to control the spatial distribution of samples while uncovering model uncertainties: a) Targeting U The function penalizes by distance to encourage exploration of regions far from existing training samples, and the optimal sample is selected by minimizing the correction function;

[0038] b) For the EFF function, a reward is given based on distance to encourage the spatial dispersion of samples, and the optimal sample is selected by maximizing the correction function; .

[0039] S6. Determine whether the calculated system failure probability meets the accuracy requirements. If not, expand the Monte Carlo sample set. If it does, output the surrogate model and the system failure probability estimation results to complete the analysis.

[0040] Furthermore, determining whether the calculated system failure probability meets the accuracy requirements specifically includes: the failure probability predicted based on the current surrogate model. Calculate the coefficient of variation of the predicted value:

[0041] This reflects the relative uncertainty in the failure probability estimation and helps determine the failure probability estimated by the current surrogate model. Coefficient of variation COV Does it meet the preset accuracy requirements? When the convergence threshold is less than the preset threshold (usually set to 0.05), the surrogate model is considered to have reached sufficient accuracy, the training process ends, and the final trained surrogate model is output. and system failure probability Complete the analysis. If COV If the accuracy requirement is not met, return to step S4 to expand the sample set. S MC Retrain the surrogate model and predict its failure probability until... Continue until the requirements are met.

[0042] Example 2: High-dimensional nonlinear cantilever beam structure To verify the applicability and effectiveness of the reliability analysis method based on spectral normalized Gaussian process and active learning proposed in this invention in high-dimensional, strongly nonlinear structural problems, this embodiment selects a high-dimensional nonlinear cantilever beam structure as the analysis object for illustration. Its structural geometric model is shown below. Figure 2 As shown, the cantilever beam is fixed at one end and free at the other, which is a typical nonlinear structural mechanics problem under high-dimensional random loads.

[0043] Establish a structural rectangular coordinate system: the direction along the cantilever beam axis is defined as... x The direction, perpendicular to the width of the beam section, is defined as... y Direction, the vertical upward direction is defined as z Direction. The cantilever beam is 5 m long, and its upper edge is subjected to 106 random concentrated loads along the beam length, forming a high-dimensional random input space.

[0044] Among them, the first 6 random loads F 1~ F 6 follows a log-normal distribution, the rest are random loads. F 7~ F 106 Following a normal distribution; the location of each load. l F,1 ~ l F,106 , beam width w Liang Gao h and the yield strength of the material S y All of them are modeled as normally distributed random variables. All of the above random variables are statistically independent, and their probability distribution types and statistical parameters are listed in Table 1.

[0045] Table 1 Distribution parameters of random variables ; Based on classical beam bending theory and equivalent mechanical relationships, and considering the spatial distribution of random loads and the uncertainty of cross-sectional geometric parameters, the maximum equivalent normal stress generated in a cantilever beam under random loads is expressed as a function of the resultant load force and its moment on the cross-sectional bending capacity, with material yielding as the structural failure criterion. Therefore, the performance function of the cantilever beam structure is defined as follows:

[0046] Where x is the input vector consisting of all random variables; when G When (x) < 0, it indicates that the maximum equivalent stress in the cantilever beam exceeds the material's yield strength, and the structure fails; when G The structure is safe when (x)≥0.

[0047] Based on the established high-dimensional nonlinear cantilever beam model, this embodiment further employs an active learning reliability analysis method based on spectral normalized Gaussian process proposed in this invention to efficiently evaluate the failure probability of the cantilever beam structure under high-dimensional random loads. The specific implementation steps are as follows: Step 1, Define the random input variable vector X={ F 1,…, F 106, l F,1 ,…, l F,106 , w , h , S y}, Establish its joint probability distribution functionf X (x). Based on the joint probability distribution, a Monte Carlo method is used to generate a distribution containing... N MC Monte Carlo sample set of 1 sample S MC This is used for subsequent failure probability assessment based on the surrogate model, where the sample size is... N MC This is to ensure that the failure probability estimation results have sufficient statistical accuracy.

[0048] Step 2, based on the joint probability distribution f X In the 215-dimensional parameter space defined by (x), Latin hypercube sampling (LHS) is used to generate a parameter space containing... N Initial experimental design sample set with 0 samples S 0, to ensure uniform coverage and representativeness of the sample in high-dimensional space.

[0049] right S Each sample point x in 0 k ( k =1,2,…, N 0), call the actual performance function G (x) is used to calculate the corresponding true response value. G (x k This allows us to construct the initial training dataset. .

[0050] Step 3, based on the current number i The training dataset corresponding to the next active learning iteration (Initial time) Construct the SNGP proxy model A function used to approximate the true performance of a cantilever beam structure. G (x), whose model structure is shown in the diagram. Figure 3 As shown.

[0051] Step 4: Utilize the constructed SNGP proxy model For the Monte Carlo sample set S MC Perform batch prediction to obtain each sample point x j ( j =1, 2, …, N MC The performance function predicts the mean. and the predicted standard deviation .

[0052] Based on the prediction results, for the firsti System failure probability in the next iteration An estimation is performed. Then, the absolute value of the predicted mean for all Monte Carlo samples is calculated. Sort in ascending order and select the group with the smallest absolute value. M = 0.1× N MC These samples constitute the candidate sample pool for the current iteration. The focus is on key regions close to the limiting state surface.

[0053] Step 5: Based on the current proxy model, process the Monte Carlo sample set. S MC The prediction results are used to calculate the probability of misclassification. And based on this, quantify the prediction error of the model. ϵ i Determine whether it meets the error control-based stopping criterion ESC.

[0054] when ϵ i < ϵ max If the accuracy is ≤0.01, the current proxy model is considered to meet the requirements, and proceed to step 6; Otherwise, in the candidate sample pool The method employs a distance-aware adaptive sampling strategy to select the optimal new training sample point x. ∗ And calculate its true performance function value. G (x ∗ ). The newly acquired data pairs (x ∗ , G (x ∗ Add to the training dataset and update to +1 = ∪{(x ∗ , )},make i=i+ 1. Return to step 3 and continue iterating.

[0055] Step 6: After satisfying the error stopping criterion, further calculate the surrogate model. Predicting failure probability The coefficient of variation is used to assess the statistical stability of the failure probability estimation results. When the coefficient of variation is less than 0.05, the failure probability is considered to be low. The preset accuracy requirement has been met. The active learning process is terminated, and the final trained SNGP surrogate model is output. and system failure probability .

[0056] If the coefficient of variation does not meet the requirements, then expand the Monte Carlo sample set. S MC Then return to step 4 to reassess the failure probability until the accuracy requirements are met.

[0057] Figure 4 and Figure 5 The diagrams show the probability density function and cumulative probability distribution function of the high-dimensional nonlinear cantilever beam structure response predicted by the spectral normalized Gaussian process surrogate model in the embodiments, and compare them with the Monte Carlo simulation results. The results show that although there are some differences in the predictions of the two methods far from the limit state region, in the neighborhood of the critical limit state that determines the failure probability, the response distribution of the method of this invention is highly consistent with that of the MCS, and can accurately characterize the structural failure features.

[0058] In addition, the method of the present invention was compared and verified with a variety of typical reliability analysis methods, and the results are summarized in Table 2.

[0059] Table 2 Comparison of Reliability Analysis Results for Different Method Combinations ; As shown in Table 2, while the SIR-GP-IS and GSIR-GP-IS methods have high computational accuracy, they require constructing an initial training set containing thousands of samples, resulting in high computational costs. The first-order reliability method (FORM), while computationally efficient, suffers from insufficient accuracy in strongly nonlinear high-dimensional problems. In contrast, the framework proposed in this invention relies on distance-aware uncertainty modeling and combines it with a distance-aware adaptive sampling strategy, achieving high-precision failure probability estimation with only a small number of active learning iterations.

[0060] Specifically, AL-SNGP+ U cor The (ESC) method converged with only 31 additional active learning samples (531 model calls in total), resulting in a failure probability of 1.95 × 10⁻⁶. -6 The coefficient of variation was 1.62%; AL-SNGP+ EFF cor The (ESC) method also converged within 32 iterations, and the results were highly consistent with the former. Therefore, this invention demonstrates significant advantages in reliability analysis of high-dimensional, strongly nonlinear structures, including high computational efficiency, small sample requirements, and high accuracy in failure probability estimation, making it valuable for engineering applications.

[0061] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0062] The terminology used in the embodiments of this invention is for the purpose of describing particular embodiments only and is not intended to limit the invention. The singular forms “a,” “the,” and “the” as used in the embodiments of this invention and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise.

[0063] It should be understood that the term "and / or" used in this article is merely a description of the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A existing alone, A and B existing simultaneously, and B existing alone. Additionally, the character " / " in this article generally indicates that the preceding and following related objects have an "or" relationship.

[0064] Depending on the context, the word "if" as used here can be interpreted as "when," "when," "in response to determination," or "in response to detection." Similarly, depending on the context, the phrase "if determination" or "if detection (of the stated condition or event)" can be interpreted as "when determination," "in response to determination," "when detection (of the stated condition or event)," or "in response to detection (of the stated condition or event)."

[0065] Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A reliable analysis method for active learning based on spectral normalized Gaussian processes, characterized in that, Includes the following steps: S1. Generate a large-scale Monte Carlo sample set based on the joint probability density function of the random variables; S2. Generate an initial training sample set in the input parameter space, call the real model to calculate the training sample set, and construct the initial training dataset; S3. Based on the initial training dataset, construct the SNGP surrogate model of the structure performance function spectrum normalized Gaussian process; S4. Using the surrogate model trained in step S3, predict the Monte Carlo sample set, estimate the system failure probability, and screen the candidate sample pool. S5. Based on the prediction results of step S4, determine whether the model has converged. If not, select new training sample points, calculate their true responses, and update the training dataset. If the model has converged, proceed to step S6. S6. Determine whether the calculated system failure probability meets the accuracy requirements. If not, expand the Monte Carlo sample set. If it does, output the surrogate model and the system failure probability estimation results to complete the analysis.

2. The active learning reliability analysis method based on spectral normalized Gaussian process according to claim 1, characterized in that: Step S1 specifically includes: based on the joint probability density function of the structural random variables. f X (x), within the domain of the random variable, the number of samples generated using Monte Carlo sampling is... N MC Input parameters Monte Carlo sample set S MC .

3. The active learning reliability analysis method based on spectral normalized Gaussian process according to claim 1, characterized in that: Step S2 specifically includes: in the joint probability density function of the random variables f X In the multidimensional parameter space defined by (x), a spatially uniform coverage sampling strategy is executed to generate a training sample set. S 0, and call the real model. G (x) is calculated to construct the initial training dataset. .

4. The active learning reliability analysis method based on spectral normalized Gaussian process according to claim 1, characterized in that: Step S3 specifically includes: based on the current... i The training dataset corresponding to the next active learning iteration ( i =0,1,2,…), construct an SNGP surrogate model for approximating the structure performance function. The SNGP proxy model By integrating a deep feature extractor f φ (x) and the Gaussian process GP output layer enable distance-aware uncertainty modeling, and can simultaneously output the predicted mean of the structural response. and uncertainty .

5. The active learning reliability analysis method based on spectral normalized Gaussian process according to claim 1, characterized in that: Step S4 specifically includes the following steps: S41. Utilizing the constructed proxy model For the Monte Carlo sample set S MC Perform batch response prediction; S42. Estimate the system failure probability based on the response results predicted in step S41. ; S43, Monte Carlo sample set S MC The prediction results are calculated based on the absolute value of the predicted mean. Sort in ascending order and filter by the smallest absolute value. p × N MC These sample points constitute the candidate sample pool for the current iteration. S candidate .

6. The active learning reliability analysis method based on spectral normalized Gaussian process according to claim 1, characterized in that: In step S5, it is determined whether the model has converged. If not, a new training sample point is selected, its true response is calculated, and the training dataset is updated. Specifically, this includes: based on the prediction results of the surrogate model, determining whether the current... i Does the surrogate model trained during the next active learning iteration meet the preset convergence criteria? If not, the distance-aware adaptive sampling strategy DA-AS is used in the candidate sample pool. S candidate Select new training sample points x ∗ Call the real model G (x) is used to calculate the sample points x. ∗ and its response Add to the current training dataset To update the training dataset i+1 = i ∪{(x ∗ , Continue training and updating the proxy model.

7. The active learning reliability analysis method based on spectral normalized Gaussian process according to claim 6, characterized in that: The preset convergence criterion is an error-based stopping criterion, namely the error-type stopping criterion ESC. The error-type stopping criterion ESC is based on an upper limit of error. With misclassification probability This approach ensures that accuracy requirements are met while avoiding oversampling. Specifically, it includes: For each sample point x j Its misclassification probability for: ； in, and These are proxy models For sample point x j The predicted mean and uncertainty, The cumulative distribution function of the standard normal distribution is calculated by ESC for each sample point. Quantization model prediction error ϵ The number of misclassified samples is modeled using a Poisson binomial distribution; when the model's maximum prediction error... ϵ max Meets the preset information level α Corresponding error threshold ϵ thr When the model converges, it is considered to have converged.

8. The reliability analysis method based on spectral normalized Gaussian process and active learning according to claim 6, characterized in that: The distance-aware adaptive sampling strategy DA-AS specifically includes: Define the distance metric d(x) in the space of standardized random variables. k , ), which represents the candidate point x k up to the current training set The closest front at medium distance m The mean of the Euclidean distances of the training samples, i.e.: ； Where, x i,j It is the training set Mid-distance candidate sample point x k The most recent j The sample points; ||⋅|| represents the Euclidean distance; this metric is essentially the reciprocal of the local sample density, and the larger the value, the sparser the training samples in the neighborhood of the candidate point, and the higher the potential for spatial exploration; Using d(x) k , ) respectively for traditional learning functions U / EFF Corrections are made to control the spatial distribution of samples while uncovering model uncertainties: a) Targeting U The function penalizes by distance to encourage exploration of regions far from existing training samples, and the optimal sample is selected by minimizing the correction function; b) For the EFF function, a reward is given based on distance to encourage the spatial dispersion of samples, and the optimal sample is selected by maximizing the correction function.

9. The reliability analysis method based on spectral normalized Gaussian process and active learning according to claim 1, characterized in that: The step S6, determining whether the calculated system failure probability meets the accuracy requirements, specifically includes: determining the failure probability estimated by the current surrogate model. Coefficient of variation COV Does it meet the preset accuracy requirements? When the convergence value is less than the preset convergence threshold of 0.05, the surrogate model is considered to have reached sufficient accuracy, the training process ends, and the final trained surrogate model is output. and system failure probability Complete the analysis.

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