Variable-fidelity Surrogate Models Research Articles

Efficient multi-objective CMA-ES algorithm assisted by knowledge-extraction-based variable-fidelity surrogate model

To accelerate the multi-objective optimization for expensive engineering cases, a Knowledge-Extraction-based Variable-Fidelity Surrogate-assisted Covariance Matrix Adaptation Evolution Strategy (KE-VFS-CMA-ES) is presented. In the first part, the KE-VFS model is established. Firstly, the optimization is performed using the low-fidelity surrogate model to obtain the Low-Fidelity Non-Dominated Solutions (LF-NDS). Secondly, aiming to obtain the High-Fidelity (HF) sample points located in promising areas, the K-means clustering algorithm and the space-filling strategy are used to extract knowledge from the LF-NDS to the HF space. Finally, the KE-VFS model is established by means of the obtained HF and LF sample points. In the second part, a novel model management based on the Modified Hypervolume Improvement (MHVI) criterion and pre-screening strategy is proposed. In each generation of KE-VFS-CMA-ES, excessive candidate points are firstly generated and then calculated by the MHVI criterion to find out a few potential points, which will be evaluated by the HF model. Through the above two parts, the promising areas can be detected and the potential points can be screened out, which contributes to speeding up the optimization process twofold. Three classic benchmark functions and a time-consuming engineering case of the aerospace integrally stiffened shell are studied, and results illustrate the excellent efficiency, robustness and applicability of KE-VFS-CMA-ES compared with other four known multi-objective optimization algorithms.

Optimization of EDM process parameters based on variable-fidelity surrogate model

Optimization of EDM process parameters based on variable-fidelity surrogate model

A variable-fidelity multi-objective optimization method for aerospace structural design optimization

Variable-fidelity (VF) surrogate models have been widespreadly applied to aerospace structural design and optimization problems with multiple objectives to alleviate the optimization cost. To enhance the performance of the existing multi-objective optimization algorithms based on VF surrogate model, a variable-fidelity hypervolume expected improvement (VF-HVEI) method is proposed. Co-Kriging model is utilized to replace computational expensive objective functions in the proposed method, and it is sequentially updated with the VF-HEVI method during the optimization process. The proposed infilling criterion effectively considers the prediction uncertainty of the VF surrogate model, the contribution of sample points of different fidelity on the improvement of the current Pareto front and the computation cost of different simulation models at the same time. The test results in analytical and engineering examples indicate that the proposed method obtains more accurate and robust Pareto front under the same simulation cost.

Optimization design of horizontal well fracture stage placement in shale gas reservoirs based on an efficient variable-fidelity surrogate model and intelligent algorithm

Hydraulic fracturing technique increases shale gas well productivity significantly. Horizontal well fracture optimization has been studied by many researchers worldwide in the past decade. However, most of these researches relied on computationally expensive numerical simulation models for objectives evaluation during the optimization process. This affects the optimization efficiency significantly. To address this issue, surrogate model methods which adapt a simple approximate model are employed to lessen the computational burden. In this study therefore, a novel intelligent variable-fidelity radial basis function (VFRBF) surrogate-assisted model for multi-objective fracture stage placement optimization, namely VFRBF-FSO, is proposed to reduce the computational burden of the numerical simulation-based production optimization. In the VFBRF-FSO method, low-fidelity (LF) and high-fidelity (HF) samples were adopted simultaneously to establish the variable-fidelity (VF) surrogate model. To the best of our knowledge, this is the first time that variable-fidelity model is used for shale gas horizontal well fracture stage placement optimization. The uniqueness of this proposed method is that a scaling factor and an augment matrix are used to integrate the LF and HF samples to increase the accuracy of the surrogate model. Moreover, two cases with different wells and well types were studied to illustrate the effectiveness and accuracy of the VFRBF-FSO method. The optimization results showed that the VFRBF-FSO method performed comparably with the HF model-based method in terms of convergence and diversity. However, the VFRBF-FSO reduced the simulation runs on the two cases with different wells and fracture types to about five times that of the HF model.

Transfer learning based variable-fidelity surrogate model for shell buckling prediction

This paper provides a novel establishment method of the variable-fidelity surrogate model (VFSM) driven by transfer learning for shell buckling prediction problems such as buckling prediction of variable-stiffness composite shells, aiming at achieving higher prediction accuracy than the traditional VFSM constructed by the bridge function. The transfer learning based variable-fidelity surrogate model (TL-VFSM) includes two steps. In the first step, low-fidelity model and high-fidelity model points are calculated and collected into the source dataset and target dataset for transfer learning, respectively. Based on the source dataset, a Deep Neural Network (DNN) model is initialized. Then, the hyperparameter tuning of the DNN model is carried out based on Bayesian Optimization (BO), which uses the target dataset as the validation set. After the hyperparameter pre-tuning, the pre-trained DNN model is obtained. In the second step, early layers of the pre-trained DNN model are remained, while the last layer is replaced. Then, another hyperparameter tuning is carried out to fine-tune the pre-trained DNN model using BO. After that, the transfer learning is finished and the re-trained DNN model is obtained. Through above two steps, the TL-VFSM is constructed. The effectiveness and efficiency of the proposed TL-VFSM are verified by three test functions and two engineering examples of variable-stiffness composite shells and hierarchical stiffened shells. Example results indicate that, using the same computational cost, the proposed TL-VFSM achieves higher prediction accuracy than traditional VFSMs constructed by the bridge function, and it can be a potential method for time-consuming buckling prediction problems with multi-source data or variable fidelities.

Variable-fidelity probability of improvement method for efficient global optimization of expensive black-box problems

Variable-fidelity (VF) surrogate models have attracted significant attention recently in simulation-based design because they can achieve a desirable accuracy at a reasonable cost by making use of the data from both low-fidelity (LF) and high-fidelity (HF) simulations. To facilitate the usage of VF surrogate models assisted efficient global optimization, there are still challenging issues on (1) how to construct the VF surrogate model for simulations with variable-fidelity levels under the non-nested sampling data, (2) how to determine the location and fidelity level of the samples simultaneously, and (3) how to handle constraints when VF surrogate models are also used for constraints. In this work, a variable-fidelity probability of improvement (VF-PI) method is proposed for computationally expensive black-box problems. First, a multi-level generalized Co-Kriging (GCK) model, which is extended from the two-level GCK model, is developed for VF surrogate modeling of simulations with three or more levels of fidelities under non-nested sampling data. Second, to determine the location and fidelity level of the sequential samples, an extended probability of improvement (EPI) function is developed. In EPI function, the model correlation and cost ratio between the LF and HF models, together with the sample density, are considered. Third, the probability of satisfying the constraints is introduced and combined with the EPI function, enabling the proposed approach to handle VF optimization problems with constraints. The comparison results illustrate that the proposed VF-PI method is more efficient and robust than the four compared methods on the illustrated cases.

Enhanced variable-fidelity surrogate-based optimization framework by Gaussian process regression and fuzzy clustering

In order to improve the global optimizing ability of surrogate-based optimizations, an enhanced variable-fidelity surrogate model (VFSM) optimization framework is developed based on the Gaussian process regression algorithm and the fuzzy clustering algorithm, which makes full use of the information fusion of high-fidelity model (HFM) and low-fidelity model (LFM). Firstly, a screening criterion is proposed for determining competitive sampling points based on low-fidelity surrogate model (LFSM). Then, the competitive sampling strategy is combined with the fuzzy clustering algorithm, which can adaptively obtain the reduced design space with more possibility to find the optimal result. In the reduced design space, the VFSM is constructed by means of the Gaussian process regression algorithm. Based on the VFSM, a gradient-based optimization can start from the competitive sampling point to search out the optimal solution quickly. In order to verify the effectiveness of the proposed optimization method, three test functions and an engineering example of hierarchical stiffened shells are carried out. Optimization results indicate that, under the similar computational cost, the proposed optimization method achieves significantly higher global optimizing ability than the high-fidelity surrogate model (HFSM) optimization method, the LFSM optimization method and the traditional VFSM optimization method by direct sampling.

Efficient Aerodynamic Optimization of Propeller using Hierarchical Kriging Models

The number of aerodynamic analysis in the design optimization of propeller has been reduced significantly by using the surrogate model, such as kriging. In this paper, a more efficient aerodynamic design optimization method of propeller is proposed by using the HK (hierarchical kriging) model. The high-fidelity model is defined as a RANS (Reynolds- Averaged Navier-Stokes) simulation for propeller and the low-fidelity model is defined as the results by blade-element/vortex theory. Initial samples are selected for two levels of fidelity via LHS (Latin Hypercube Sampling). The aerodynamic performance of varying fidelity is used to build variable-fidelity surrogate models for functions of objective (e.g. thrust) and constraint (e.g. shaft power). Infill sampling criteria, including MSP (minimizing of surrogate prediction), EI (expected improvement), PI (probability of improvement), LCB (lower-confidence bounding) and MSE (mean-squared error) are used to obtain new samples, and the surrogate models are repetitively updated until a global optimum is found. High-altitude propellers are optimized by the kriging model and hierarchical kriging model, respectively. Compared to kriging model, the number of RANS solving by the hierarchical kriging model is reduced 37.5%, and the optimization time is reduced 24.3%. The results have shown that the proposed design method for propeller can significantly improve the optimization efficiency.

Toward the robust establishment of variable-fidelity surrogate models for hierarchical stiffened shells by two-step adaptive updating approach

Since the high-fidelity model (HFM) of hierarchical stiffened shells is time-consuming, the sampling points based on HFM are generally few, which would result in a certain randomness of the sampling process. In some cases, the prediction accuracy of the variable-fidelity surrogate model (VFSM) is prone to be not robust and reliable. In order to improve the robustness of the prediction accuracy of VFSM, a two-step adaptive updating approach is proposed for the robust establishment of VFSM. In the first step, the leave-one-out (LOO) cross validation is carried out for sampling points of the low-fidelity model (LFM), aiming at finding out those with large prediction error. Then, these points are evaluated by HFM and then added into the original HFM set. In the second step, another LOO cross validation is performed on sampling points of the hybrid bridge function linking HFM and LFM. Based on the Voronoi diagram method, new updating points are chosen from where the largest prediction error of the bridge function lies, and then the VFSM is updated. After above two-step updating process, the VFSM is established. Three simple examples of test functions are firstly presented to verify the effectiveness and efficiency of the proposed method. Further, the proposed method is applied to an engineering example of hierarchical stiffened shells. In order to provide evaluation indexes for prediction accuracy and robustness of VFSM, the VFSM is established by multiple times, and the mean value and the standard deviation of the relative root mean square error (RRMSE) values of the multiple sets of VFSM are calculated. Results indicate that, under the similar computational cost, the mean value and the standard deviation of the RRMSE values of the proposed method decrease by 24.1% and 82.0% than those of the traditional VFSM based on the direct sampling method, respectively. Therefore, the high prediction accuracy and robustness of the proposed method is verified. Additionally, the total computational time of the proposed VFSM decreases by 70% than that of the surrogate model based on HFM when achieving the similar prediction accuracy, indicating the high prediction efficiency of the proposed VFSM.

Efficient aerodynamic shape optimization using variable-fidelity surrogate models and multilevel computational grids

A variable-fidelity method can remarkably improve the efficiency of a design optimization based on a high-fidelity and expensive numerical simulation, with assistance of lower-fidelity and cheaper simulation(s). However, most existing works only incorporate “two” levels of fidelity, and thus efficiency improvement is very limited. In order to reduce the number of high-fidelity simulations as many as possible, there is a strong need to extend it to three or more fidelities. This article proposes a novel variable-fidelity optimization approach with application to aerodynamic design. Its key ingredient is the theory and algorithm of a Multi-level Hierarchical Kriging (MHK), which is referred to as a surrogate model that can incorporate simulation data with arbitrary levels of fidelity. The high-fidelity model is defined as a CFD simulation using a fine grid and the lower-fidelity models are defined as the same CFD model but with coarser grids, which are determined through a grid convergence study. First, sampling shapes are selected for each level of fidelity via technique of Design of Experiments (DoE). Then, CFD simulations are conducted and the output data of varying fidelity is used to build initial MHK models for objective (e.g. CD) and constraint (e.g. CL, Cm) functions. Next, new samples are selected through infill-sampling criteria and the surrogate models are repetitively updated until a global optimum is found. The proposed method is validated by analytical test cases and applied to aerodynamic shape optimization of a NACA0012 airfoil and an ONERA M6 wing in transonic flows. The results confirm that the proposed method can significantly improve the optimization efficiency and apparently outperforms the existing single-fidelity or two-level-fidelity method.

Variable-fidelity expected improvement method for efficient global optimization of expensive functions

The efficient global optimization method (EGO) based on kriging surrogate model and expected improvement (EI) has received much attention for optimization of high-fidelity, expensive functions. However, when the standard EI method is directly applied to a variable-fidelity optimization (VFO) introducing assistance from cheap, low-fidelity functions via hierarchical kriging (HK) or cokriging, only high-fidelity samples can be chosen to update the variable-fidelity surrogate model. The theory of infilling low-fidelity samples towards the improvement of high-fidelity function is still a blank area. This article proposes a variable-fidelity EI (VF-EI) method that can adaptively select new samples of both low and high fidelity. Based on the theory of HK model, the EI of the high-fidelity function associated with adding low- and high-fidelity sample points are analytically derived, and the resulting VF-EI is a function of both the design variables x and the fidelity level l. Through maximizing the VF-EI, both the sample location and fidelity level of next numerical evaluation are determined, which in turn drives the optimization converging to the global optimum of high-fidelity function. The proposed VF-EI is verified by six analytical test cases and demonstrated by two engineering problems, including aerodynamic shape optimizations of RAE 2822 airfoil and ONERA M6 wing. The results show that it can remarkably improve the optimization efficiency and compares favorably to the existing methods.

Theoretical investigations of the new Cokriging method for variable-fidelity surrogate modeling

Cokriging is a variable-fidelity surrogate modeling technique which emulates a target process based on the spatial correlation of sampled data of different levels of fidelity. In this work, we address two theoretical questions associated with the so-called new Cokriging method for variable-fidelity modeling:

A cooperative radial basis function method for variable-fidelity surrogate modeling

By coupling the low-fidelity (LF) model with the high-fidelity (HF) samples, the variable-fidelity model (VFM) offers an efficient way to overcome the expensive computing challenge in multidisciplinary design optimization (MDO). In this paper, a cooperative radial basis function (Co-RBF) method for the VFM is proposed by modifying the basis function of RBF. The RBF method is constructed on the HF samples, while the Co-RBF method incorporates the entire information of the LF model with the HF samples. In Co-RBF, the LF model is regard as a basis function of Co-RBF and the HF samples are utilized to compute the Co-RBF model coefficients. Two numerical functions and three engineering problems are adopted to verify the proposed Co-RBF method. The predictive results of Co-RBF are compared with those of RBF and Co-Kriging, which show that the Co-RBF method improves the efficiency, accuracy and robustness of the existing VFMs.

An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging

ABSTRACTVariable-fidelity (VF) modelling methods have been widely used in complex engineering system design to mitigate the computational burden. Building a VF model generally includes two parts: design of experiments and metamodel construction. In this article, an adaptive sampling method based on improved hierarchical kriging (ASM-IHK) is proposed to refine the improved VF model. First, an improved hierarchical kriging model is developed as the metamodel, in which the low-fidelity model is varied through a polynomial response surface function to capture the characteristics of a high-fidelity model. Secondly, to reduce local approximation errors, an active learning strategy based on a sequential sampling method is introduced to make full use of the already required information on the current sampling points and to guide the sampling process of the high-fidelity model. Finally, two numerical examples and the modelling of the aerodynamic coefficient for an aircraft are provided to demonstrate the approximation capability of the proposed approach, as well as three other metamodelling methods and two sequential sampling methods. The results show that ASM-IHK provides a more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.

계층적 크리깅 모델을 이용한 설계 최적화 기법의 유용성 검증

근사모델을 이용한 최적설계 문제에서는 설계변수의 수가 증가함에 따라 근사모델의 정확도를 확보하기 위한 계산 횟수가 급격히 증가한다. 이를 해결하기 위해 저정확도 모델을 바탕으로 고정확도 모델로 보정하는 Variable-Fidelity Modeling을 이용하였다. 본 논문에서 Variable-Fidelity Model로는 계층적 크리깅 모델을 이용하였으며, 다목적 유전자 알고리즘과 결합하여 최적화 프레임워크를 제안하였다. 이 방법의 유용성을 검증하기 위하여 천음속 영역에 대한 익형 최적 설계를 하였다. 설계변수로는 PARSEC의 파라메터를 이용하였으며, 서로 다른 격자수를 가지는 경우 그리고 서로 다른 정확도를 가지는 해석자를 이용한 경우에 관하여 해석을 수행하였다. 검증을 위해 단일 정확도 모델에 대한 최적화 결과와 비교하였다. 모든 경우에 관하여 파레토 라인이 유사하게 나오는 것을 확인 할 수 있었으며, 계산시간은 계층적 크리깅 모델을 이용한 Variable-Fidelity Model이 단일 정확도 모델에 비하여 훨씬 줄어들었다. 이를 바탕으로 본 논문의 방법이 단일 정확도를 가지는 모델에 대한 최적화 방법과 유사한 정확도를 가지며 더욱 효율적임을 확인 할 수 있다. On the optimization design problem using surrogate model, it requires considerable number of sampling points to construct a surrogate model which retains the accuracy. As an alternative to reduce construction cost of the surrogate model, Variable-Fidelity Modeling(VFM) technique, where correct high fidelity model based on the low fidelity surrogate model is introduced. In this study, hierarchical kriging model for variable-fidelity surrogate modeling is used and an optimization framework with multi-objective genetic algorithm(MOGA) is presented. To prove the feasibility of this framework, airfoil design optimization process is performed for the transonic region. The parameters of PARSEC are used to design variables and the optimization process is performed in case of varying number of grid and varying fidelity. The results showed that pareto front of all variable-fidelity models are similar with its single-level of fidelity model and calculation time is considerably reduced. Based on computational results, it is shown that VFM is a more efficient way and has an accuracy as high as that single-level of fidelity model optimization.

Component sizing of a plug-in hybrid electric vehicle powertrain, Part A: coupling bio-inspired techniques to meshless variable-fidelity surrogate models

In the present investigation, the authors propose a variable fidelity optimisation framework for component sizing of a plug-in hybrid electric vehicle PHEV powertrain. The proposed computational framework can be divided into two different stages. At the first stage, finite element grids of different resolutions are used to capture initial information regarding the behaviour of physical system. To generate those grids, maximum power of electric motor P

Derivative-Enhanced Variable Fidelity Surrogate Modeling for Aerodynamic Functions

In this paper a derivative-enhanced variable-fidelity surrogate model approach is developed based on a cokriging formulation. In this approach the absolute values of a high-fidelity function as well as the trends obtained by low-fidelity function values are utilized to develop an accurate surrogate model. Derivative information of arbitrary fidelity levels can be also utilized to develop a more accurate surrogate model. The efficiencies of the developed approaches are investigated by analytic function-fitting, aerodynamic data modeling and two-dimensional airfoil-drag-minimization problems. In the aerodynamic problems low-fidelity levels are defined by a different physical model or coarser computational mesh. The numerical examples show that the developed surrogate-model approach is shown to be useful for efficient aerodynamic data modeling and accurate uncertainty analysis with low computational cost. An efficient aerodynamic shape optimization is also realized with the variable fidelity Kriging model. Faster reduction of an aerodynamic objective function is achieved by utilizing the derivative-enhanced variable-fidelity-model approach in which low-fidelity information as well as adjoint-derivative information are simultaneously utilized to construct a surrogate model.

Hierarchical Kriging Model for Variable-Fidelity Surrogate Modeling

The efficiency of building a surrogate model for the output of a computer code can be dramatically improved via variable-fidelity surrogate modeling techniques. In this article, a hierarchical kriging model is proposed and used for variable-fidelity surrogate modeling problems. Here, hierarchical kriging refers to a surrogate model of a highfidelity function that uses a kriging model of a sampled lower-fidelity function as a model trend. As a consequence, the variation in the lower-fidelity data is mapped to the high-fidelity data, and a more accurate surrogate model for the high-fidelity function is obtained. A self-contained derivation of the hierarchical kriging model is presented. The proposed method is demonstrated with an analytical example and used for modeling the aerodynamic data of an RAE 2822 airfoil and an industrial transport aircraft configuration. The numerical examples show that it is efficient, accurate, and robust. It is also observed that hierarchical kriging provides a more reasonable mean-squared-error estimation than traditional cokriging. It can be applied to the efficient aerodynamic analysis and shape optimization of aircraft or any other research areas where computer codes of varying fidelity are in use.

Improving variable-fidelity surrogate modeling via gradient-enhanced kriging and a generalized hybrid bridge function

Variable-fidelity surrogate modeling offers an efficient way to generate aerodynamic data for aero-loads prediction based on a set of CFD methods with varying degree of fidelity and computational expense. In this paper, direct Gradient-Enhanced Kriging (GEK) and a newly developed Generalized Hybrid Bridge Function (GHBF) have been combined in order to improve the efficiency and accuracy of the existing Variable-Fidelity Modeling (VFM) approach. The new algorithms and features are demonstrated and evaluated for analytical functions and are subsequently used to construct a global surrogate model for the aerodynamic coefficients and drag polar of an RAE 2822 airfoil. It is shown that the gradient-enhanced GHBF proposed in this paper is very promising and can be used to significantly improve the efficiency, accuracy and robustness of VFM in the context of aero-loads prediction.