Sparse Optimization Of PCE Models Based On Information Entropy

There are many uncertainties in the modeling and Simulation of real physical systems,such as engineering design and product manufacturing,which will lead to product performance fluctuation and risk.Uncertainty quantification(UQ)provides a complete research framework for modeling and optimization of complex systems,Establishing the polynomial chaos expansions(PCE)model is one of the common methods to solve the problem of UQ.However,the PCE model has some problems such as dimension disaster.The existing sparse methods mainly start from the aspect of numerical accuracy,and focus on selecting the features that have the greatest impact on the coefficients,which may ignore the information carried by those high-order features.This paper mainly studies the problems faced by the existing sparse methods of the PCE model,such as ignoring the overall distribution,the sparsity does not meet the expectation,the loss of important features,and so on.The idea of selecting polynomial chaos basis based on information entropy and KL divergence is proposed.Two sparse PCE optimization methods based on information entropy and a sparse PCE model method based on KL divergence are established,which are verified by numerical experiments and applied to physical and engineering experiments.This paper mainly completes the following aspects:1)A shallow optimization method based on information entropy is established,aiming at the problem that the sparsity of the model does not meet the expectation due to paying too much attention to the numerical difference and ignoring the overall distribution in the classical sparse method,which is a plug-in sparse model secondary optimization method with strong applicability.Based on the definition of information entropy and the principle of maximum entropy,the entropy of the active basis function is calculated on the first sparse result,the entropy is used to reorder,and the expected sparsity is given at the same time so that the model can achieve secondary sparsity.Shallow optimization is applicable to almost all classical sparse methods and can be regarded as an external plug-in.It discards those basis functions with less uncertainty to achieve the ideal sparsity and reduces the difference between the response distribution and the real distribution while maintaining the model accuracy.Experiments verify the effectiveness of the method in the low-dimensional case,In the case of lognormal radio function,the logarithm of KL divergence is reduced by 50 % compared with the classical method,and the sparsity is reduced by 50 %.2)A more targeted deep optimization method based on the overall entropy increase of the model is established to retain the important high-order features in the model,which can greatly improve the calculation accuracy and efficiency of the model in high-dimensional cases.Considering the time-consuming of quadratic optimization and its limitation that it can only ’discard’ but not ’retain’ under high-dimensional conditions,to retain the high-order features that have a great impact on the model response,the information entropy is used to modify the regular term of the classical sparse optimization method.In the step of model selection,deep optimization fully considers the information entropy of features,embeds the regular term based on information entropy in the known sparse method,to retain those features with a large amount of information,and designs a parameter selection algorithm,which can select ideal parameters under different design accuracy and different dimensions.The experimental results show that deep optimization can achieve a better optimization effect in high-dimensional complex cases,and the model response distribution is closer to the real value.In the case of OAKLEY&O’HAGAN,the KL divergence after LARS deep optimization is reduced by 85 %.3)a new sparse method based on KL divergence is constructed using the distribution difference between the front and back of the model response corresponding coefficient during feature selection,which can greatly reduce the KL divergence between the corresponding distribution and the real distribution on the premise of ensuring the model accuracy and sparsity.The role of KL divergence in uncertainty is analyzed,and its feasibility for model selection is explained.Using the principle of linear regression for analysis and theoretical derivation,the sparse objective function of the model is given.It is found that it not only reduces the residual to control the numerical accuracy,but also adds the consideration of the active basis function matrix.The algorithm is formed and simulated.The results show that compared with the classical sparse method,On the premise of maintaining sparsity and relative MSE,the PCE sparsity method based on KL divergence has a great improvement in response distribution and KL divergence.In the case of Ishigami,only 100 sample points are needed to reach the KL divergence(log)of-3,while the classical sparse algorithm still stays at about-2,down by an order of magnitude.