This study proposes a selection and stacking ensemble-based method to facilitate the determination of the covariance function type for Gaussian process regression. The proposed method operates at the model ensemble level and involves the use of multiple Gaussian process regression models with different types of covariance functions as base learners in a stacking ensemble with a final Gaussian process regression model as the meta-learner. First, the Pearson correlation coefficients between the leave-one-out predicted responses from each candidate base learner and the actual responses are computed and sorted in ascending order, after which the interquartile range (IQR) is calculated, and the candidate base learners that fall below the lower 1.5 & times; IQR are removed. Afterward, the adjacent gaps between the Pearson correlation coefficients that correspond to the remaining candidate base learners are calculated and sorted in ascending order, and some of the remaining candidate base learners are further removed according to the gaps that lie above the upper 1.5 & times; IQR. Finally, a newly constructed Gaussian process regression model with a linear covariance function is used as the meta-learner for final predictions. To validate the effectiveness of the proposed method, six analytical test functions, three engineering datasets, and one simulation case are used for a performance study along with three representative approaches. The results demonstrate that the proposed method achieves competitive accuracy and generalization ability. Furthermore, its performance is evaluated across four widely used open-source toolkits for Gaussian process regression, and the results confirm the robustness of the method.

This paper proposes a new infill criterion for the optimization of expensive black-box design problems. The method complements the classical Efficient Global Optimization algorithm by considering the distribution of improvement instead of merely the expectation. During the optimization process, we maximize a penalized expected improvement acquisition function from a specially collected infill candidate set. Specifically, the acquisition function is formulated by penalizing the expected improvement with the variation of improvement, and the infill candidate set is composed of some global and local maxima of the expected improvement function which are identified to be “mutually non-dominated”. Some conditions necessary for setting the penalty coefficient of the acquisition function are investigated, and the definition of “mutually non-dominated infill candidates” is presented. The proposed method is demonstrated with a 1-D analytical function and benchmarked using six 10-D analytical functions and an underwater vehicle structural optimization problem. The results show that the proposed method is efficient for the optimization of expensive black-box design problems.

This paper proposes a new latent variable Gaussian process modeling method for problems involving both qualitative and quantitative input variables. By exploring a full-dimensional latent space where the values of qualitative variables are appropriately represented by certain normalized points therein, we generalize the concept of distance to be applicable to qualitative variables and thereby adapt the mixed qualitative-quantitative inputs to any conventional quantitative-only correlation structures. Specifically, we present distinct treatments for ordinal and nominal qualitative variables. For each ordinal qualitative variable, the corresponding values are represented by intrinsically ordered points of a one-dimensional unit latent space. For each nominal qualitative variable, the corresponding values are represented by points residing in an axis-aligned latent space, with the dimensionality equal to the number of values, and each point anchored to a separate unit axis. The coordinates of the latent points, along with other hyper-parameters of the Gaussian process model, are jointly estimated via maximum likelihood estimation. An experimental study was conducted to compare four representative Gaussian process modeling methods for mixed qualitative and quantitative factor problems and the proposed method, using twelve analytical test functions from relevant literature and a dataset of cooling system noise. The effectiveness of the proposed method was validated by the results. We further demonstrated how to gain insights into the qualitative factors via the proposed LVGP-Full method.

开展车下安装的轨道交通变流器结构分析通常涉及耗时的仿真计算,以评估结构强度是否符合安全要求.对于典型的轨道交通变流器产品,进行1次静强度分析(4个工况)需耗时1h以上,进行1次单方向的随机振动分析需要8 h以上.当采用传统的工程优化方法对变流器进行结构优化时,由于需要大量调用仿真进行评估,优化效率受到严重限制.因此采用机器学习方法,通过拉丁超立方采样试验设计确定仿真方案(采样位置),根据仿真样本构建Kriging代理模型(高斯过程),以近似反映设计变量与响应之间的函数关系.在Kriging模型提供信息的基础上,实现高效的灵敏度分析、设计空间探索和全局优化,为设计人员提供优化设计的参考依据与推荐方案,避免出现安全裕量过低或设计冗余过大等极端情况,在提高产品性能的同时缩短设计周期,提升产品的竞争力.基于Kriging代理模型,文章给出了进行分析与优化的完整技术路线,并对其中关键步骤(试验设计、代理模型、自适应采样、灵敏度分析、设计空间探索、全局优化等)的具体实现方法进行了详细说明.通过对某牵引变流器吊耳结构进行分析与优化,验证了所提技术路线的有效性.

Kriging surrogate model is a powerful tool to facilitate engineering system analysis and mechanical design by emulating the time-consuming simulations. However, estimating the hyper parameters of Kriging for high-dimensional problems can itself be computationally-expensive, because the large correlation matrix needs to be inverted a lot of times. This paper investigates a method for accelerating the training process of Kriging with a relatively trivial loss of model accuracy. The main idea is to reduce the number of hyperparameters by projecting the original high-dimensional hyperparameter space onto a low-dimensional subspace spanned by a small set of orthogonal directions. Specifically, the identification of the subspace whose bases are linear combinations of the original hyperparameters is accomplished by employing the Active Subspace Method (ASM). Provided that the primary variability of the likelihood function is well described in the active subspace of hyperparameters, the maximum likelihood estimation process is expected to be facilitated. The main steps of the whole modeling procedure are demonstrated, and practical implementation details are given. An experimental study including a set of analytical functions from 20-D to 80-D and a rear subframe modal analysis problem parameterized with 35 design variables is conducted for performance analysis and comparison. Results indicate that for high-dimensional problems a reasonable trade-off between the modeling efficiency and model accuracy of Kriging can be achieved by the proposed method.

ObjectivesThe structural optimization of ships usually involves the use of high-fidelity numerical simulations which are time-consuming and thus difficult to evaluated frequently, and this intrinsic property hinders the optimization process. To promote efficient design optimization, this paper explores the use of Gradient-enhanced Kriging (GEK) surrogate mode in order to shorten the design loop and save design cost. A reduced GEK-based infill criterion is proposed to decrease the number of simulations by calculating the gradients only for sample locations where improvement occurs.MethodsA multi-start local optimization algorithm is employed to search the local optima of the "expected improvement" function and locate candidate infill points. The associated "approximate probability of stationary point (APSP)" values are also evaluated, and infill decisions are made according to the extent of consistency between these two quantities, thereby improving optimization efficiency. The proposed method is then applied to the structural optimization of an underwater vehicle to increase the seventh-order natural frequency under unconstrained free vibration in an underwater environment, and the validity is verifed.ResultsThe result shows that，compared with the baseline，the optimized design achieves a 14.6% improvement.ConclusionsThe proposed GEK-based optimization method can be generalized to cases when gradients can only be evaluated by finite difference.

This paper explores the use of Gradient-enhanced Kriging for optimization of expensive black-box design problems, which is not completely limited by the conventional Efficient Global Optimization algorithm framework. Specifically, we give the best linear unbiased predictor and mean squared prediction error of the partial derivatives of Gradient-enhanced Kriging and then propose a measure named “Approximate Probability of Stationary Point” to estimate the approximate probability of a candidate infill point be a stationary point of the underlying function. When it comes to the selection of infill point, we not only maximize the well-known Expected Improvement but also evaluate the Approximate Probability of Stationary Point as a “double-check” step. Then the infill decision is made according to the extent of consistency between these two quantities. Furthermore, to examine whether the optimization process will gain from sparing more costs for response evaluation, we investigate also the cases that the gradient evaluation step is conditionally skipped in some iterations. Three new infill criteria are proposed and experimented with three analytical test functions and an airfoil optimal shape design. Results show that the optimization performance can be improved by exploiting the auxiliary gradient information in the proposed way.

By exploring the auxiliary information from gradients, the accuracy of Kriging model can be improved. However, the dramatically increased time for model training tends to be unaffordable. Therefore, a novel gradient-enhanced Kriging modeling method which utilizes only a partial set of gradients, is developed in this article. Within the framework of this method, a balance between model accuracy and modeling efficiency can be achieved. More specifically, the influence of each input variable on output is estimated and ranked by feature selection technique. Then an empirical evaluation rule is proposed to facilitate the selection of gradients. Five representative numerical benchmarks from 10-D to 30-D and an airfoil optimal shape design with 18 variables are used for validation. Results show that when compared with the conventional Gradient-enhanced Kriging, the modeling time of the proposed method is significantly reduced, while the loss of accuracy is negligible. As a consequence, the proposed surrogate modeling method can provide an alternative way for approximating high-dimensional problems. (C) 2018 Elsevier Inc. All rights reserved.

Surrogate models are usually used as a time-saving approach to reduce the computational burden of expensive computer simulations for engineering design. However, it is difficult to choose an appropriate model for an unknown design space. To tackle this problem, an effective method is forming an ensemble model that combines several surrogate models. Many efforts were made to determine the weight factors of ensemble, which include global and local measures. This article investigates the characteristics of global and local measures, and presents a new ensemble model which combines the advantages of these two measures. In the proposed method, the design space is divided into two parts, and different strategies are introduced to evaluate the weight factors in these two parts respectively. The results from numerical and engineering design cases show that the proposed ensemble model has satisfactory robustness and accuracy (it performs best for most cases tested in this article), while spending almost the equivalent modeling time (the additional cost is not more than 6.7% for any case tested in this article) compared with the combined global and local ensemble models.

Sample data may be corrupted by noise in engineering problems. In order to make satisfactory approximations for the data with noise, some regression metamodels are adopted in current researches. The commonly used nugget-effect Kriging regards the variance of noise as a constant and ignores the difference of the noise influence, thus may not be effective enough in some cases. Therefore, a Kriging-based metamodel which combines the merits of Kriging and Support Vector Regression (SVR) is put forward for improving the performance in metamodeling with noisy data. The developed method, termed as SVEK, can capture the underlying trend of an unknown function efficiently by classifying the sample points and then regressing these classified points with different extents. Besides, a criterion for selecting the error margin ε in SVR training is proposed to facilitate the parameter setting process. Moreover, a one-variable test example is used to illustrate the modeling theory and construction procedures of SVEK. Eight numerical benchmark problems with different important characteristics are used to validate the proposed method. Then an overall comparison between the nugget-effect Kriging and the proposed method has been made. Results show that SVEK is promising in metamodeling with noisy data.