Robust ensemble of metamodels based on the hybrid error measure

Abstract

Metamodels have been widely used as an alternative for expensive physical experiments or complex, time-consuming computational simulations to provide a fast but accurate analysis. However, challenge remains in the prior determination of the most suitable metamodel for a particular case because of the lack of information about the actual behavior of a system. In addition, existing studies on metamodels have largely restricted on solving deterministic problems (e.g., data from finite element models), whereas some real-life engineering problems (e.g., data from physical experiment) are stochastic problems with noisy data. In this work, a robust ensemble of metamodels (EMs) is proposed by combining three regression stand-alone metamodels in a weighted sum form. The weight factor is adaptively determined according to the hybrid error metric, which combines global and local error measures to improve the accuracy of the EMs. Furthermore, three typical individual metamodels that can filter noise are selected to construct the EMs to extend their application in practical engineering problems. Three well-known benchmark problems with different levels of noise and three engineering problems are used to verify the effectiveness of the proposed EMs. Results show that the proposed EMs have higher accuracy and robustness than the individual metamodels and other typical EMs in major cases.

Key words： metamodel; ensemble of metamodels; hybrid error measure; stochastic problem

Acknowledgements

This work was supported by the National Key R&D Program of China (Grant No. 2017YFD0400405)

References

1 XuH, Liu L, ZhangM. Adaptive surrogate model-based optimization framework applied to battery pack design. Materials & Design, 2020, 195 : 108938–

2 TaoJ, Sun G, GuoL. Application of a PCA-DBN-based surrogate model to robust aerodynamic design optimization. Chinese Journal of Aeronautics, 2020, 33( 6): 1573– 1588

3 YuanY, JonesA, SetchfieldR. Robust design optimisation of underplatform dampers for turbine applications using a surrogate model. Journal of Sound and Vibration, 2021, 494 : 115528–

4 BoucheritS, BerkaniS, YalleseM A. Modeling and optimization of cutting parameters during machining of austenitic stainless steel AISI304 using RSM and desirability approach. Periodica Polytechnica Mechanical Engineering, 2020, 65( 1): 10– 26

5 ZhaoY, QuR Z, XingZ X. Identifying groundwater contaminant sources based on a KELM surrogate model together with four heuristic optimization algorithms. Advances in Water Resources, 2020, 138 : 103540–

6 Vu-BacN, LahmerT, ZhuangX. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100 : 19– 31

7 Myers R H, Montgomery D C, Anderson-Cook C M. Response Surface Methodology: Process and Product Optimization Using Designed Experiments. Hoboken: John Wiley & Sons, 2016

8 DynN, Levin D, RippaS. Numerical procedures for surface fitting of scattered data by radial functions. SIAM Journal on Scientific and Statistical Computing, 1986, 7( 2): 639– 659

9 SacksJ, WelchW J, MitchellT J. Design and analysis of computer experiments. Statistical Science, 1989, 4( 4): 433– 435

10 MartinJ D, SimpsonT W. Use of Kriging models to approximate deterministic computer models. AIAA Journal, 2005, 43( 4): 853– 863

11 BuscemaM. Back propagation neural networks. International Journal of the Addictions, 1998, 33( 2): 233– 270

12 ClarkeS M, GriebschJ H, SimpsonT W. Analysis of support vector regression for approximation of complex engineering analyses. Journal of Mechanical Design, 2005, 127( 6): 1077– 1087

13 Gunn S R. Support Vector Machines for Classification and Regression. ISIS Technical Report, 1998

14 GoelT, HaftkaR T, ShyyW. Ensemble of surrogates. Structural and Multidisciplinary Optimization, 2007, 33( 3): 199– 216

15 AcarE, Rais-RohaniM. Ensemble of metamodels with optimized weight factors. Structural and Multidisciplinary Optimization, 2009, 37( 3): 279– 294

16 AcarE. Various approaches for constructing an ensemble of metamodels using local measures. Structural and Multidisciplinary Optimization, 2010, 42( 6): 879– 896

17 SanchezE, PintosS, QueipoN V. Toward an optimal ensemble of kernel-based approximations with engineering applications. Structural and Multidisciplinary Optimization, 2008, 36( 3): 247– 261

18 FerreiraW G, SerpaA L. Ensemble of metamodels: Extensions of the least squares approach to efficient global optimization. Structural and Multidisciplinary Optimization, 2018, 57( 1): 131– 159

19 PalarP S, ShimoyamaK. Efficient global optimization with ensemble and selection of kernel functions for engineering design. Structural and Multidisciplinary Optimization, 2019, 59( 1): 93– 116

20 YinH, Fang H, WenG. On the ensemble of metamodels with multiple regional optimized weight factors. Structural and Multidisciplinary Optimization, 2018, 58( 1): 245– 263

21 BadawyM F, MsekhM A, HamdiaK M. Hybrid nonlinear surrogate models for fracture behavior of polymeric nanocomposites. Probabilistic Engineering Mechanics, 2017, 50( 4): 64– 75

22 Burges C J C, Schölkopf B. Improving the accuracy and speed of support vector machines. In: Advances in Neural Information Processing Systems 9. Cambridge: MIT Press, 1996, 375−381

23 Vapnik V, Golowich S E, Smola A. Support vector method for function approximation, regression estimation and signal processing. In: Advances in Neural Information Processing Systems 9. Cambridge: MIT Press, 1996, 281−287

24 ParkJ S. Optimal Latin-hypercube designs for computer experiments. Journal of Statistical Planning and Inference, 1994, 39( 1): 95– 111

25 OlajideJ O, IgbekaJ C, AfolabiT J. Prediction of oil yield from groundnut kernels in an hydraulic press using artificial neural network (ANN). Journal of Food Engineering, 2007, 81( 4): 643– 646

26 HuangS, HuY, Li F. Optimization of mechanical oil extraction process from Camellia oleifera seeds regarding oil yield and energy consumption. Journal of Food Process Engineering, 2019, 42( 6): e13157–

27 Pérez-GálvezR, ChopinC, MastailM. Optimisation of liquor yield during the hydraulic pressing of sardine (Sardina pilchardus) discards. Journal of Food Engineering, 2009, 93( 1): 66– 71

28 Romuli S, Karaj S, Latif S, et al. Performance of mechanical co-extraction of Jatropha curcas L. kernels with rapeseed, maize or soybean with regard to oil recovery, press capacity and product quality. Industrial Crops and Products, 2017, 104: 81−90

29 HuangS, HuY M, LiF C. Multi-objective optimization of mechanical oil extraction from Camellia oleifera seeds using Kriging regression and NSGA-II. Journal of Food Process Engineering, 2020, 43( 12): e13549–

30 LamasD L, CrapisteG H, ConstenlaD T. Changes in quality and composition of sunflower oil during enzymatic degumming process. LWT-Food Science and Technology, 2014, 58( 1): 71– 76

31 ChenL, QiuH, Jiang C. Support vector enhanced Kriging for metamodeling with noisy data. Structural and Multidisciplinary Optimization, 2018, 57( 4): 1611– 1623