A subset simulation active learning reliability analysis method and system based on DGP

Abstract

This invention discloses a subset simulation-based active learning reliability analysis method and system based on Depth Gaussian Process (DGP). First, the initial design variables and response values ​​of the engineering structure are obtained to construct an initial training dataset, and a multi-layer deep Gaussian process surrogate model is established. Second, Bayesian inference is performed using Markov chain Monte Carlo simulation to obtain the posterior prediction distribution. Then, the target failure event is decomposed into multiple intermediate failure events using subset simulation, and the sample points with the optimal failure information gain are selected through a hierarchical learning function at each subset level. The training dataset is then updated using high-fidelity simulation. This Bayesian inference and active learning point addition process is repeated until the convergence criterion is met, resulting in a converged deep Gaussian process surrogate model. Finally, the failure probability of the engineering structure is calculated using Monte Carlo simulation, completing an efficient, accurate, and low-cost reliability analysis.

Description

Technical Field

[0001] This invention belongs to the field of structural reliability analysis technology, specifically relating to a subset simulation active learning reliability analysis method and system based on DGP. Background Technology

[0002] Structural reliability analysis aims to quantify the probability that an engineering project will meet performance requirements under the influence of uncertainties (such as material properties, loads, and geometric dimensions). For complex structures, their limit state functions are often implicit and require evaluation through expensive numerical simulations such as nonlinear finite element methods. While direct Monte Carlo simulation offers high accuracy, its computational cost is enormous, making it unsuitable for estimating the probability of rare failures.

[0003] To alleviate computational burden, surrogate modeling techniques are widely adopted. Gaussian processes (GPs), as a nonparametric Bayesian method, can provide prediction uncertainty, but their stationarity assumption limits their ability to represent complex nonstationary limit state surfaces. Deep Gaussian processes (DGPs), through hierarchical combinations of multiple Gaussian processes, can automatically learn nonlinear transformations of the input space, capture multi-scale and nonstationary behavior, and have stronger expressive power.

[0004] Subset simulation (SS) efficiently estimates the probability of small failures by decomposing rare failure events into a product of a series of more likely intermediate failure events and combining it with Markov chain Monte Carlo (MCMC) sampling. Combining active learning (AL) with subset simulation (such as AK-co-SS) can further reduce the number of high-fidelity simulations, but the expressive power of traditional Kriging models is limited, and existing learning functions do not fully utilize the hierarchical uncertainty information of DGP. Summary of the Invention

[0005] To address the aforementioned shortcomings in existing technologies, the subset simulation active learning reliability analysis method and system based on DGP provided by this invention are suitable for the efficient assessment of rare failure probabilities in engineering structures, especially for structural reliability problems involving complex, non-stationary, discontinuous limit state functions and high costs of high-fidelity simulation.

[0006] To achieve the above-mentioned objectives, the technical solution adopted by this invention is: a subset simulation active learning reliability analysis method based on DGP, comprising the following steps: S100. Obtain the initial design variables and corresponding response values ​​of the engineering structure, and construct the initial training dataset; S200. Based on the constructed initial training dataset, a multi-layer deep Gaussian process surrogate model is constructed. S300, for the deep Gaussian process surrogate model, a Markov chain Monte Carlo sampling strategy is used to perform Bayesian inference to obtain the posterior prediction distribution; S400: Based on the posterior prediction distribution, the target failure event is decomposed into the product of several intermediate failure events using subset simulation. Within each subset simulation level, the sample point with the optimal failure information gain is selected using a hierarchical learning function, and its response value is obtained through high-fidelity simulation, thereby updating the training dataset. S500, repeat S300~S400, recalculate Bayesian inference and active learning points using the updated training dataset until the preset convergence criterion is met, obtain the converged deep Gaussian process surrogate model, and then use Monte Carlo simulation to calculate the failure probability of the engineering structure to complete the reliability analysis.

[0007] Further, step S200 includes the following sub-steps: S201. Set the number of layers in the deep Gaussian process and ensure that the dimensions of the latent variables in each layer are consistent with the input dimensions; S202. For each layer, the latent function is defined to follow a prior of a Gaussian process with zero mean and an isotropic squared exponential kernel function. S203. The covariance structure is simplified by adopting the conditional independence assumption, so that the correlation between different latent variable nodes in the same layer is zero. S204. Set the prior distribution of hyperparameters and complete the construction of a multi-layer deep Gaussian process surrogate model.

[0008] Furthermore, step S300 includes the following sub-steps: S301. Perform joint Bayesian inference on hyperparameters and latent variables in the deep Gaussian process surrogate model, and generate posterior samples using Markov chain Monte Carlo sampling. S302. Calculate the mean and variance of the posterior samples based on Markov chain Monte Carlo, and then obtain the posterior predicted distribution.

[0009] Furthermore, in S400, the hierarchical learning function is the sum of the output layer learning function and the hidden layer uncertainty aggregator.

[0010] Furthermore, the output layer learning function Represented as: In the formula, and The deep Gaussian process surrogate model at point The predicted mean and predicted standard deviation at the location, and These are the cumulative distribution function and probability density function of the standard normal distribution, respectively.

[0011] Furthermore, in S400, the method for determining the hidden layer uncertainty aggregator is as follows: For each hidden layer of the deep Gaussian process surrogate model, the variance of the activation values ​​of each node under different posterior samples is calculated using the posterior samples obtained by Markov chain Monte Carlo sampling. The maximum variance of all nodes in each hidden layer is taken as the layer uncertainty of that hidden layer. Aggregate the layer uncertainties of all hidden layers to obtain the overall hidden layer uncertainty as a hidden layer uncertainty aggregator.

[0012] Furthermore, in S400, subset simulation is used to decompose the target failure event into a product of several intermediate failure events. Within each subset simulation level, a hierarchical learning function is used to select the sample points with the optimal failure information gain, including: S401. Introduce several intermediate failure thresholds to decompose the target failure event into the product of several intermediate failure events; S402. For the first subset simulation layer, estimate the corresponding failure probability from the initial candidate sample pool through Monte Carlo simulation; S403, regarding the first The subset simulation layer uses Markov chain Monte Carlo sampling to generate samples that follow a conditional distribution and estimates the corresponding conditional probabilities. S404. In the candidate sample pool of each subset simulation layer, independently perform Bayesian inference and active learning to add points for the deep Gaussian process surrogate model to obtain the sample points with the optimal failure information gain.

[0013] Furthermore, in S500, the preset convergence criterion is: In the formula, For point The hierarchical learning function value at that location, For the preset threshold, This is the current candidate sample pool.

[0014] Furthermore, in S500, the failure probability for: In the formula, The total number of samples in the Monte Carlo simulation. For the converged deep Gaussian process surrogate model in the first... Predicted values ​​at each Monte Carlo sample point This is an indicator function that takes the value 1 when the condition is true and 0 otherwise.

[0015] A subset simulation active learning reliability analysis system includes: Data acquisition module: used to acquire the initial design variables and corresponding response values ​​of the engineering structure, and to construct the initial data training set; Model building module: used to build a multi-layer deep Gaussian process surrogate model based on the initial data training set; Bayesian inference module: used to perform Markov chain Monte Carlo sampling on the parameters of the deep Gaussian process surrogate model to obtain the posterior prediction distribution; Active learning and subset simulation-driven module: It is used to decompose failure events based on the posterior prediction distribution and the subset simulation strategy, use the hierarchical learning function to select the sample points with the optimal failure information gain, and obtain their response values ​​through high-fidelity simulation, thereby updating the training dataset. Convergence determination module: used to repeatedly execute the Bayesian inference, active learning and subset simulation processes in the model building module and the active learning and subset simulation driven module until the deep Gaussian multilayer surrogate model satisfies the preset convergence rules; Reliability assessment module: Used to calculate the failure probability of the converged deep Gaussian multilayer surrogate model through Monte Carlo simulation, and complete the reliability analysis.

[0016] The beneficial effects of this invention are as follows: (1) Significantly improves the proxy accuracy of complex limit states This invention constructs a multi-layer deep Gaussian process model through hierarchical nonlinear transformation of deep Gaussian processes, automatically capturing the non-stationary, multi-scale, and discontinuous characteristics of the limit state function. Compared with the traditional Kriging model, the surrogate model reduces the prediction error by more than 50% in complex boundary problems.

[0017] (2) Significantly reduce the cost of high-fidelity simulation The hierarchical learning function HLF constructed in this invention utilizes the dual uncertainties of the output layer and hidden layer to significantly improve active learning efficiency. Experiments show that the number of calls to 2D examples is reduced by 47%, the number of calls to 4-bar truss examples is reduced by 87%, and the number of calls to 12-dimensional truck crane boom examples is reduced by 76%, saving an average of more than 70% in high-fidelity simulation and greatly reducing engineering analysis costs.

[0018] (3) Improve the robustness and accuracy of rare failure probability estimation In this invention, the DGP is updated independently for each subset of the simulation hierarchy, effectively suppressing the progressive amplification of errors. In all examples, the relative error of the failure probability estimation is <5% (0.20% for 2D, 0.18% for truss, and 3.66% for crane), and the standard deviation is significantly lower than that of the comparative methods.

[0019] (4) Provides reliable full Bayesian uncertainty quantification This invention employs MCMC joint sampling to avoid the underestimation of uncertainty in variational inference. The accurate posterior prediction covariance provides a solid statistical basis for HLF and convergence criteria, making the termination of active learning more reliable.

[0020] (5) Modular system design with good engineering practicality The system of this invention is divided into independent modules such as data acquisition, model building, Bayesian inference, active learning and subset simulation, convergence determination, and reliability assessment. It supports seamless integration with finite element software and is suitable for efficient reliability analysis in fields such as aerospace, vehicle engineering, civil engineering, and mechanical equipment. Attached Figure Description

[0021] Figure 1 This is a flowchart of the subset simulation active learning reliability analysis method based on DGP in an embodiment of the present invention.

[0022] Figure 2 This is a schematic diagram of the structure of the deep Gaussian process proxy model in an embodiment of the present invention.

[0023] Figure 3 This is a schematic diagram of the iterative process of the two-dimensional dual failure mode system in an embodiment of the present invention.

[0024] Figure 4 This is a schematic diagram of a four-bar truss structure in an embodiment of the present invention.

[0025] Figure 5 This is an overall structural diagram of the truck crane boom system in an embodiment of the present invention.

[0026] Figure 6 The diagram shows the finite element model of the boom of a truck crane in an embodiment of the present invention, where (a) represents the load and boundary conditions, and (b) represents the mesh generation.

[0027] Figure 7 This is a diagram showing the size configuration and variable parameters of the crane boom in an embodiment of the present invention.

[0028] Figure 8 The figure shows the finite element analysis results of the boom of the truck crane in the embodiment of the present invention, where (a) is the stress distribution, (b) is the deformation of the variable amplitude plane, and (c) is the deformation of the slewing plane.

[0029] Figure 9 This is a convergence diagram of the baseline failure probability of the crane boom in an embodiment of the present invention. Detailed Implementation

[0030] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0031] Example 1:

[0032] This invention provides a subset simulation active learning reliability analysis method based on DGP, which can make full use of DGP hierarchical uncertainty and deeply integrate with subset simulation to achieve accurate estimation of rare failure probability of complex structures with a small number of high-fidelity simulations.

[0033] refer to Figure 1 A subset simulation active learning reliability analysis method based on DGP includes the following steps: S100. Obtain the initial design variables and corresponding response values ​​of the engineering structure, and construct the initial training dataset; S200. Based on the constructed initial training dataset, a multi-layer deep Gaussian process surrogate model is constructed. S300, for the deep Gaussian process surrogate model, a Markov chain Monte Carlo sampling strategy is used to perform Bayesian inference to obtain the posterior prediction distribution; S400: Based on the posterior prediction distribution, the target failure event is decomposed into the product of several intermediate failure events using subset simulation. Within each subset simulation level, the sample point with the optimal failure information gain is selected using a hierarchical learning function, and its response value is obtained through high-fidelity simulation, thereby updating the training dataset. S500, repeat S300~S400, recalculate Bayesian inference and active learning points using the updated training dataset until the preset convergence criterion is met, obtain the converged deep Gaussian process surrogate model, and then use Monte Carlo simulation to calculate the failure probability of the engineering structure to complete the reliability analysis.

[0034] S200 of this embodiment of the invention includes the following steps: S201, Set the number of layers in the depth Gaussian process. And ensure that the dimensions of latent variables in each layer are consistent with the input dimensions; S202, For each layer The latent function is defined to follow a priori of a Gaussian process with zero mean and an isotropic quadratic exponential kernel function. Among them, the isotropic squared exponential kernel function for: In the formula, For length scale parameters, To output scale parameters, and They represent two different 3D input vector (such as structural design variables, material parameters, loads, etc.); S203. The covariance structure is simplified by adopting the conditional independence assumption, so that the correlation between different latent variable nodes within the same layer is zero. This is expressed as: In the formula, For the first The latent variable output matrix of all nodes in the layer. For the first Layer The output of implicit variables for each node. This represents the number of nodes in this layer. Indicates that in a given number Under the condition of layer output, the first The joint conditional probability distribution of all latent variables in the layer. For the first The number of nodes in the layer; S204. Set the prior distribution of hyperparameters and complete the construction of a multi-layer deep Gaussian process surrogate model.

[0035] S300 in this embodiment of the invention includes the following sub-steps: S301. Perform joint Bayesian inference on hyperparameters and latent variables in the deep Gaussian process surrogate model, and generate posterior samples using Markov chain Monte Carlo sampling. S302. Calculate the mean and variance of the posterior samples based on Markov chain Monte Carlo, and then obtain the posterior predicted distribution. Among them, the mean of the posterior sample and variance They are respectively: In the formula, Indicates the test point to be predicted. Indicates test point The corresponding unknown output (actual response value). This represents the training input matrix (i.e., the set of input vectors for all evaluated sample points). This represents the training output vector (i.e., the high-fidelity simulation response values ​​corresponding to all evaluated sample points). The number of samples reserved for Markov chains and Monte Carlo chains. and The first Hyperparameters and latent variables of each sample.

[0036] In S400 of this embodiment of the invention, the hierarchical learning function is the output layer learning function. With hidden layer uncertainty aggregator The sum, expressed as: In this embodiment, the output layer learning function Using the expected improvement function, it is expressed as: In the formula, and The deep Gaussian process surrogate model at point The predicted mean and predicted standard deviation at the location, and These are the cumulative distribution function and probability density function of the standard normal distribution, respectively.

[0037] In this embodiment, the method for determining the hidden layer uncertainty aggregator is as follows: For each hidden layer of the deep Gaussian process surrogate model, the variance of the activation values ​​of each node under different posterior samples is calculated using the posterior samples obtained by Markov chain Monte Carlo sampling. Among them, for the first The first hidden layer The variance of the activation values ​​of each node under different posterior samples is: In the formula, Indicates the first Layer The activation value function for each node. Indicates the first Activation values ​​under MCMC samples This represents the sample mean of the activation values; The maximum variance of all nodes in each hidden layer is taken as the layer uncertainty of that hidden layer. Among them, the Uncertainty in the hidden layer : Aggregating the uncertainties of all hidden layers yields the overall hidden layer uncertainty, which serves as a hidden layer uncertainty aggregator. .

[0038] The hierarchical learning function HLF provided by this invention utilizes both the output layer prediction uncertainty (improved through expectation) and the hidden layer cognitive uncertainty (through the hidden layer uncertainty aggregator) to achieve a balance between active learning near the decision boundary and in areas where model knowledge is lacking, thereby significantly improving sampling efficiency.

[0039] In S400 of this embodiment of the invention, subset simulation is used to decompose the target failure event into a product of several intermediate failure events. Within each subset simulation level, a hierarchical learning function is used to select the sample points with the optimal failure information gain, including: S401. Introduce several intermediate failure thresholds to decompose the target failure event into the product of several intermediate failure events; Among them, for a series of intermediate failure thresholds Decomposition failure events Represented as: In the formula, ; S402. For the first subset simulation layer, estimate the corresponding failure probability from the initial candidate sample pool using Monte Carlo simulation. for: In the formula, The total number of samples in the initial candidate sample pool. For indicator functions; S403, regarding the first Subset simulation layers are generated using Markov chain Monte Carlo sampling to follow a conditional distribution. The sample is used to estimate the corresponding conditional probability. ; S404. In the candidate sample pool of each subset simulation layer, independently perform Bayesian inference and active learning to add points for the deep Gaussian process surrogate model to obtain the sample points with the optimal failure information gain.

[0040] In S500 of this embodiment, the preset convergence criterion is: In the formula, For point The hierarchical learning function value at that location, For the preset threshold, This is the current candidate sample pool; where the preset threshold is set to... or .

[0041] In S500 of this embodiment, the failure probability for: In the formula, The total number of samples in the Monte Carlo simulation. For the converged deep Gaussian process surrogate model in the first... Predicted values ​​at each Monte Carlo sample point This is an indicator function that takes the value 1 when the condition is true and 0 otherwise.

[0042] Example 2:

[0043] This embodiment further defines the subset simulation active learning reliability analysis method based on DGP in Embodiment 1, and provides a subset simulation active learning reliability analysis system. For other contents not mentioned, please refer to the method description in Embodiment 1 and the description of the prior art.

[0044] The subset simulation active learning reliability analysis system provided in this embodiment includes: Data acquisition module: used to acquire the initial design variables and corresponding response values ​​of the engineering structure, and to construct the initial data training set; Model building module: used to build a multi-layer deep Gaussian process surrogate model based on the initial data training set; Bayesian inference module: used to perform Markov chain Monte Carlo sampling on the parameters of the deep Gaussian process surrogate model to obtain the posterior prediction distribution; Active learning and subset simulation-driven module: It is used to decompose failure events based on the posterior prediction distribution and the subset simulation strategy, use the hierarchical learning function to select the sample points with the optimal failure information gain, and obtain their response values ​​through high-fidelity simulation, thereby updating the training dataset. Convergence determination module: used to repeatedly execute the Bayesian inference, active learning and subset simulation processes in the model building module and the active learning and subset simulation driven module until the deep Gaussian multilayer surrogate model satisfies the preset convergence rules; Reliability assessment module: Used to calculate the failure probability of the converged deep Gaussian multilayer surrogate model through Monte Carlo simulation, and complete the reliability analysis.

[0045] In the model building module of this invention embodiment, during the construction of a multi-layer deep Gaussian process surrogate model, the traditional Gaussian process is extended to a multi-layer deep Gaussian process (DGP), and implicit transformation of the input space is achieved through layer-by-layer nonlinear mapping; specifically, the model building process is as follows: Set the number of layers in the deep Gaussian process and ensure that the dimensions of the latent variables in each layer are consistent with the input dimensions; For each layer, the latent function is defined to follow a prior of zero mean and an isotropic quadratic exponential Gaussian process. The covariance structure is simplified by assuming conditional independence, so that the correlation between different latent variable nodes in the same layer is zero; the prior distribution of hyperparameters is set to complete the construction of a multi-layer deep Gaussian process surrogate model.

[0046] In the Bayesian inference module of this embodiment, Markov chain Monte Carlo is used to jointly sample the hyperparameters and latent variables of the deep Gaussian process surrogate model to obtain the posterior prediction distribution.

[0047] In the active learning and subset simulation-driven module of this embodiment, the hierarchical learning function is the sum of the output layer learning function and the hidden layer uncertainty aggregator. The output layer learning function... The desired improvement function is adopted.

[0048] The method for determining the hidden layer uncertainty aggregator is as follows: For each hidden layer of the deep Gaussian process surrogate model, the variance of the activation values ​​of each node under different posterior samples is calculated using the posterior samples obtained by Markov chain Monte Carlo sampling. The maximum variance of all nodes in each hidden layer is taken as the layer uncertainty of that hidden layer. Aggregate the layer uncertainties of all hidden layers to obtain the overall hidden layer uncertainty as a hidden layer uncertainty aggregator.

[0049] Furthermore, a deep coupling strategy of active learning and subset simulation is adopted. Within each subset simulation level, Bayesian inference and active learning addition of DGP are performed independently, and conditional samples generated by MCMC are used to approximate the real failure boundary layer by layer.

[0050] The system provided by this invention supports flexible configuration of multi-layer deep Gaussian processes and can interface with finite element software for data exchange. Furthermore, the system adopts a modular architecture, with each module operating independently and transmitting data through standardized interfaces.

[0051] In the system provided by this invention, active learning-driven DGP is embedded in each layer of subset simulation, enabling targeted updates of the surrogate model in each level of conditional failure region, avoiding the accumulation of errors between subset simulation layers, and ensuring the robustness of rare failure probability estimation.

[0052] Example 3:

[0053] This embodiment is a specific experimental example of using the method in Embodiment 1 to perform reliability analysis on a two-dimensional series system.

[0054] This embodiment uses a two-dimensional dual-failure-mode cascade system as a numerical example, and its limit state function is: System failure caused by Definition. Wherein, and Let be two independent random variables, both of which follow a standard normal distribution (mean 0, standard deviation 1). and Let each represent a limit state function for a failure mode; when Structural failure is determined at that time.

[0055] Perform the following steps according to the method of the present invention: S1: Sampling from a standard normal distribution Using an initial set of sample points, the response values ​​are calculated through the true limit state function to construct the initial training dataset.

[0056] S2: Construct a two-layer deep Gaussian process surrogate model, with the number of nodes in each layer being the same as the input dimension (2D). Use an isotropic squared exponential kernel function and set the prior distribution of hyperparameters.

[0057] S3: Bayesian inference is performed using Markov chain Monte Carlo (MCMC), and the chain length is set. Discard the first 100 preheating samples to obtain the posterior predicted distribution.

[0058] S4: First, in the initial candidate pool ( In this process, the hierarchical learning function for each candidate point is calculated. Select the point corresponding to the maximum value for true evaluation, add it to the training set, and repeat until... Then determine the first intermediate threshold. Current failure probability The corresponding quantiles. (The quantiles fall within...) The samples are used as seeds, and the next-level candidate pool is generated through MCMC, and so on. Figure 3 It demonstrates the gradual approximation process from k=1 to k=4.

[0059] S5: After convergence, the final DGP proxy model is obtained.

[0060] S6: Use A Monte Carlo sample, through Determine if a failure has occurred and calculate the probability of failure.

[0061] The results of 20 independent repeated experiments are shown in Table 1. The average number of calls to the real function by the method of this invention (ALDGPcoSS+HLF) is shown in Table 1. Estimate the probability of failure The relative error is only 0.20%, which is significantly better than the traditional Kriging method.

[0062] Table 1: Test Results of Two-Dimensional Series System Example 4:

[0063] This embodiment is a specific experimental example of using the method in Embodiment 1 to perform reliability analysis on a four-bar truss structure.

[0064] like Figure 4 As shown, this is a statically determinate four-bar truss with a cross-sectional area of... mm 2 elastic modulus N / mm 2 Bearing random lateral forces and axial force There are two other node coordinates These are random variables. Each random variable follows a normal distribution, and the specific parameters are shown in Table 2. The limit state function is defined as follows: the maximum stress does not exceed the yield strength, and the maximum displacement does not exceed the allowable value.

[0065] Table 2: Truss Parameters The reliability analysis was performed according to the method of this invention, following the same steps as in Example 1. The results are shown in Table 3. The method of this invention has an average of only 40.8 calls, and the failure probability is estimated... The relative error is 0.18%, while the traditional AKcoSS+U requires 488.9 calls with an error of 1.14%. The sample efficiency of this invention is improved by about 87%, and the accuracy is higher.

[0066] Table 3: Truss Results Example 5:

[0067] This embodiment is a specific experimental example of using the method in Embodiment 1 to conduct a reliability analysis of the boom structure of a truck crane.

[0068] like Figure 5 The image shows the boom system of a 55-ton truck crane, consisting of five boom sections made of Q960 high-strength steel. The finite element model is shown below. Figure 6 As shown, the load includes gravity, lifting load, wind load, etc., with the boundary condition being a fixed pin hole. Key uncertainty parameters include... Figure 7 As shown, this includes the thickness of each arm section. Length of each arm section Lifting weight Wind pressure All of them follow a normal distribution, as shown in Table 4.

[0069] Table 4: Parameters of Truck Cranes The limit state function is for a series system, requiring the maximum von Mises stress to be less than the allowable stress of the material (960 / 1.5MPa), the amplitude plane deformation to be less than 40000×5%mm, and the rotation plane deformation to be less than 300×5%mm. Figure 8 The finite element analysis results of the boom structure are presented, where (a) is the stress distribution, (b) is the variable amplitude plane deformation, and (c) is the rotation plane deformation.

[0070] Table 5: Results of the Truck Crane Because finite element simulation is costly (approximately 30 minutes per simulation), the reference failure probability was obtained by gradually increasing the number of training samples until convergence, and was ultimately determined as follows: ( Figure 9 ).

[0071] The results of 20 independent experiments conducted using the method of this invention are shown in Table 5. The method of this invention (ALDGPcoSS+HLF) requires an average of only 154.3 calls to the real finite element method, and the failure probability estimation... The relative error is 3.66%. In contrast, the traditional Kriging method requires at least 656.6 calls with an error of 11.12%. The method of this invention saves approximately 76% of the computational cost in 12-dimensional problems while maintaining good accuracy.

[0072] The above embodiments demonstrate that this invention employs a deep Gaussian process as a surrogate model, whose hierarchical structure can automatically learn the nonlinear transformations of the input space, effectively capturing the non-stationary, multi-scale features of complex limit state surfaces. Through full Bayesian MCMC inference, not only is the output prediction distribution obtained, but the cognitive uncertainty of each hidden layer can also be quantified.

[0073] In the description of this invention, it should be understood that the terms "center," "thickness," "upper," "lower," "horizontal," "top," "bottom," "inner," "outer," and "radial," etc., indicating orientation or positional relationships based on the orientation or positional relationships shown in the accompanying drawings, are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying the relative importance or the number of technical features implicitly specified. Therefore, a feature defined by "first," "second," and "third" may explicitly or implicitly include one or more of that feature.

[0074] Specific embodiments have been used to illustrate the principles and implementation methods of this invention. The descriptions of the embodiments above are only for the purpose of helping to understand the method and core ideas of this invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.

[0075] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A subset simulation active learning reliability analysis method based on DGP, characterized in that, Includes the following steps: S100. Obtain the initial design variables and corresponding response values ​​of the engineering structure, and construct the initial training dataset; S200. Based on the constructed initial training dataset, a multi-layer deep Gaussian process surrogate model is constructed. S300, for the deep Gaussian process surrogate model, a Markov chain Monte Carlo sampling strategy is used to perform Bayesian inference to obtain the posterior prediction distribution; S400: Based on the posterior prediction distribution, the target failure event is decomposed into the product of several intermediate failure events using subset simulation. Within each subset simulation level, the sample point with the optimal failure information gain is selected using a hierarchical learning function, and its response value is obtained through high-fidelity simulation, thereby updating the training dataset. S500, repeat S300~S400, recalculate Bayesian inference and active learning points using the updated training dataset until the preset convergence criterion is met, obtain the converged deep Gaussian process surrogate model, and then use Monte Carlo simulation to calculate the failure probability of the engineering structure to complete the reliability analysis.

2. The subset simulation active learning reliability analysis method based on DGP according to claim 1, characterized in that, S200 includes the following steps: S201. Set the number of layers in the deep Gaussian process and ensure that the dimensions of the latent variables in each layer are consistent with the input dimensions; S202. For each layer, the latent function is defined to follow a prior of a Gaussian process with zero mean and an isotropic squared exponential kernel function. S203. The covariance structure is simplified by adopting the conditional independence assumption, so that the correlation between different latent variable nodes in the same layer is zero. S204. Set the prior distribution of hyperparameters and complete the construction of a multi-layer deep Gaussian process surrogate model.

3. The subset simulation active learning reliability analysis method based on DGP according to claim 1, characterized in that, S300 includes the following steps: S301. Perform joint Bayesian inference on the hyperparameters and latent variables in the deep Gaussian process surrogate model, and use Markov chain Monte Carlo sampling to generate posterior samples. S302. Calculate the mean and variance of the posterior samples based on Markov chain Monte Carlo, and then obtain the posterior predicted distribution.

4. The subset simulation active learning reliability analysis method based on DGP according to claim 1, characterized in that, In step S400, the hierarchical learning function is the sum of the output layer learning function and the hidden layer uncertainty aggregator.

5. The GDP-based subset simulation active learning reliability analysis method according to claim 4, characterized in that, The output layer learning function Represented as: In the formula, and The deep Gaussian process surrogate model at point The predicted mean and predicted standard deviation at the location, and These are the cumulative distribution function and probability density function of the standard normal distribution, respectively.

6. The subset simulation active learning reliability analysis method based on DGP according to claim 4, characterized in that, In step S400, the method for determining the hidden layer uncertainty aggregator is as follows: For each hidden layer of the deep Gaussian process surrogate model, the variance of the activation values ​​of each node under different posterior samples is calculated using the posterior samples obtained by Markov chain Monte Carlo sampling. The maximum variance of all nodes in each hidden layer is taken as the layer uncertainty of that hidden layer. Aggregate the layer uncertainties of all hidden layers to obtain the overall hidden layer uncertainty as a hidden layer uncertainty aggregator.

7. The subset simulation active learning reliability analysis method based on DGP according to claim 4, characterized in that, In S400, subset simulation is used to decompose the target failure event into a product of several intermediate failure events. Within each subset simulation level, a hierarchical learning function is used to select the sample points with the optimal failure information gain, including: S401. Introduce several intermediate failure thresholds to decompose the target failure event into the product of several intermediate failure events; S402. For the first subset simulation layer, estimate the corresponding failure probability from the initial candidate sample pool through Monte Carlo simulation; S403, regarding the first The subset simulation layer uses Markov chain Monte Carlo sampling to generate samples that follow a conditional distribution and estimates the corresponding conditional probabilities. S404. In the candidate sample pool of each subset simulation layer, independently perform Bayesian inference and active learning to add points for the deep Gaussian process surrogate model to obtain the sample points with the optimal failure information gain.

8. The DPG-based subset simulation active learning reliability analysis method according to claim 4, characterized in that, In S500, the preset convergence criterion is: In the formula, For point The hierarchical learning function value at that location, For the preset threshold, This is the current candidate sample pool.

9. The subset simulation active learning reliability analysis method based on DGP according to claim 1, characterized in that, In S500, the failure probability for: In the formula, The total number of samples in the Monte Carlo simulation. For the converged deep Gaussian process surrogate model in the th... Predicted values ​​at each Monte Carlo sample point This is an indicator function that takes the value 1 when the condition is true and 0 otherwise.

10. A subset simulation active learning reliability analysis system, implemented based on the DGP-based subset simulation active learning reliability analysis method according to any one of claims 1 to 9, characterized in that, include: Data acquisition module: used to acquire the initial design variables and corresponding response values ​​of the engineering structure, and to construct the initial data training set; Model building module: used to build a multi-layer deep Gaussian process surrogate model based on the initial data training set; Bayesian inference module: used to perform Markov chain Monte Carlo sampling on the parameters of the deep Gaussian process surrogate model to obtain the posterior prediction distribution; Active learning and subset simulation-driven module: It is used to decompose failure events based on the posterior prediction distribution and the subset simulation strategy, use the hierarchical learning function to select the sample points with the optimal failure information gain, and obtain their response values ​​through high-fidelity simulation, thereby updating the training dataset. Convergence determination module: used to repeatedly execute the Bayesian inference, active learning and subset simulation processes in the model building module and the active learning and subset simulation driven module until the deep Gaussian multilayer surrogate model satisfies the preset convergence rules; Reliability assessment module: Used to calculate the failure probability of the converged deep Gaussian multilayer surrogate model through Monte Carlo simulation, and complete the reliability analysis.

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