Structural Reliability Analysis Via Combining Active Learning And Probability Density Evolution Method

How to perform the structural reliability analysis with good accuracy and high efficiency has always been a challenging issue in the field of science and engineering.The probability density evolution method(PDEM)has been gradually proved to be an effective tool to address this problem.However,in the existing numerical algorithm of the PDEM,the generation of representative points only takes advantage of the probabilistic information of basic random variables,and fails to account for the specific characteristic of the reliability problem,giving rise to the relatively-high computational burden.To this end,this study focuses on combining the active-learning process and the PDEM.Specifically,according to the demand of the PDEM on the surrogate model,the main modules in the active-learning process are built accordingly,including the learning function,the stopping condition and the surrogate model.In this regard,the computational cost of reliability analysis is expected to be determined according to the specific feature of the reliability problem.The main researches are stated as follows.The basic framework of combining the active-learning and the PDEM is proposed.Then,the learning function is defined from the perspective of ensuring the global Kriging accuracy at the while representative point set.Then,the prediction error of the Kriging at the current iteration serves as a substitute for the global prediction error,thereby specifying a pertinent stopping condition.In comparison with the standard PDEM,the proposed approach enables to determines the location and number of training samples according to the specific problems and reduces effectively the number of computational model evaluations.However,in the case of small samples,the global Kriging accuracy is relatively unsatisfactory in the dynamic nonlinear problems.According to the relative contribution of different representative points to the failure probability,the region of interest(ROI)is defined for the PDEM.In this regard,the precision of failure probability estimate can be effectively secured by only maintaining the Kriging accuracy in the ROI.To this end,the PDEM-oriented information entropy(PIE)function is proposed,and the stopping condition is then defined based on the PIE value.The results show that in comparison with the high computational cost incurred by maintaining the global Kriging accuracy,the treatment of only securing the Kriging accuracy in the ROI gains a good accordance with the standard PDEM,whilst effectively alleviating the computational burden.However,the stopping condition is prone to be conservative.To further reduce the computational cost,the stopping condition associated with the upper and lower bound of failure probability is proposed;meanwhile,the updating scheme of the boundary of the ROI is devised,which enables to reduce the computational time per iteration.Then,the Polynomial-Chaos Kriging(PCK)is employed to provide consistent predictions at the whole representative points,and the PDEM-oriented expected improvement function(PEIF)is formulated.Due to the direct link to the failure probability,the stopping condition associated with the bound of failure probability estimate shows more superiority over the one based on the learning function value.In high-dimensional regime,the partial least-squares(PLS)-based dimensionality reduction is combined with the PCK model,giving rise to the PLS-PCK model.The nonparametric approach is employed to build the orthogonal polynomial basis of the PCK in the low-dimensional latent subspace,and three different schemes are devised to determine the optimal dimension of PLS-based subspace.Then,the PLS-PCK model is combined with the PDEM,whilst maintaining other modules unchanged.In comparison with training directly the PCK model in the high-dimensional regime,the PLS-PCK model reduces substantially the training time,thereby the total computational time consumed by the active-learning process is alleviated accordingly.To extend the combination of active learning and the PDEM to other types of metamodels,the empirical probability distribution of metamodel is defined according to those sub-models in the leave-one-out cross-validation strategy.On this basis,two kinds of general learning functions are proposed according to the definition of the PEIF.Then,the pseudo-interpolation replacement strategy is devised for those regressiontype metamodels,so as to avoid the prediction error at the training samples.The support vector regression(SVR)shows favorable superiority over the PCK,and the learning function based on the empirical probability distribution itself is more advantageous than the one based on both empirical mean and variance.Finally,the ensemble of metamodels(EM)is established based on the PCK,the low-rank approximation(LRA)and the SVR,which provides both the predicted value and the prediction variance.Then,the multi-point enrichment strategy is devised,where at most 4 new training samples are selected per iteration.The results show that the accuracy of the EM is close to the component metamodel with the largest weight,but the EM effectively avoids the improper selection of single metamodel.In comparison with the single-point enrichment strategy,the multi-point enrichment strategy reduces substantially the number of iterations in the active-learning process,thereby alleviating the total computational time.