Abstract
In the expensive structural optimization, the datadriven surrogate model has been proven to be an effective alternative to physical simulation (or experiment). However, the static surrogateassisted evolutionary algorithm (SAEA) often becomes powerless and inefficient when dealing with different types of expensive optimization problems. Therefore, how to select highreliability surrogates to assist an evolutionary algorithm (EA) has always been a challenging task. This study aimed to dynamically provide an optimal surrogate for EA by developing a brandnew SAEA framework. Firstly, an adaptive surrogate model (ASM) selection technology was proposed. In ASM, according to different integration criteria from the strategy pool, elite metamodels were recombined into multiple ensemble surrogates in each iteration. Afterward, a promising model was adaptively picked out from the model pool based on the minimum root of mean square error (RMSE). Secondly, we investigated a novel ASMbased EA framework, namely ASMEA, where the reliability of all models was updated in realtime by generating new samples online. Thirdly, to verify the performance of the ASMEA framework, two instantiation algorithms are widely compared with several stateoftheart algorithms on a commonly used benchmark test set. Finally, a realworld antenna structural optimization problem was solved by the proposed algorithms. The results demonstrate that the proposed framework is able to provide a highreliability surrogate to assist EA in solving expensive optimization problems.
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Abbreviations
 FEA:

finite element analysis
 CFD:

computational fluid dynamics
 EA:

evolutionary algorithm
 SAEA:

surrogateassisted evolutionary algorithm
 ASM:

adaptive surrogate model
 ASMEA:

ASMbased evolutionary algorithm
 PSO:

particle swarm optimization
 DE:

differential evolution
 ASMPSO:

particle swarm optimization based on ASM
 ASMDE:

differential evolution based on ASM
 GP:

Gauss process model
 KRG:

Kriging model
 PRS:

polynomial response surface model
 RBF:

radial basis function model
 SHEP:

Shepard model
 ANN:

artificial neural network
 SVM:

support vector machine
 RBNN:

radial basis neural network
 ELM:

extreme learning machine
 HFSS:

highfrequency simulation software
 WAS:

weighted average surrogate
 OWS:

optimum weight surrogate
 DOE:

design of experiment
 LHS:

Latin hypercube sampling
 RMSE:

root of mean square error
 AE:

absolute error
 PRESS:

prediction residual error sum of square
 SR:

successful run
 VTR:

value to reach
 n :

the number of metamodels
 k :

the number of elite metamodels
 D :

the number of variables
 NP :

the size of the population
 S _{t} :

the size of the strategy pool
 T _{1} :

the train set in the database
 T _{2} :

the test set in the database
 G or G _{m} :

the evolutional generation or maximum G
 FES or FES_{max} :

the function evaluations or maximum FES
 c_{1} and c_{2} :

learn rates in PSO
 F :

scale factor in DE
 Cr :

crossover rate in DE
 x_{min} and x_{max} :

variable space
 fun _{ASM} :

evaluation function based on ASM
 fun _{real} :

evaluation function based on simulation
 Gopt:

global optimum
 Lopt:

local optimum
 #G:

the number of Gopt
 #L:

the number of Lopt
 ASM1:

the ASMPSO algorithm
 ASM2:

the ASMDE algorithm
 N :

the number of sampling points
 S _{11} :

return less
 L, W, h, :

the length, width, height of the patch
 L _{1} :

edge distance
 τ:

scale factor in T_{1} and T_{2}
 ω:

inertia weight in PSO
 λ :

wavelength
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Funding
This research was supported by the National Natural Science Foundation of China (Grant Nos. 61803054 and 61876169) and the State Education Ministry and Fundamental Research Funds for the Central Universities (2019 CDJSK 04 XK 23).
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Appendices
Appendix1
(1) Kriging (KRG)
A general form of the Kriging model is as follows:
where is an unknown prediction function, is a known approximate function, and is the realization of a stochastic process with a zero mean and a nonzero covariance. The (i,j)th element of the covariance matrix z(x) is given as
where \( \sigma \frac{2}{z} \) is the process variance, and Rij is the correlation function between the ith and the jth data points. A Gaussian function is used as the correlation function in this paper, defined by
where θ_{k} is distinct for each dimension, and these unknown parameters are generally obtained by solving a nonlinear optimization problem. In this paper, the order of the global polynomial trend function is specified to be zero.
(2) Polynomial response surface (PRS)
It is relatively easy to establish a polynomial regression model using least squares. In general, the secondorder polynomial regression model can solve most problems well, which can be expressed as
where x_{i}(i = 1, 2, …, n) is design variable. yi is the corresponding predictive response. k is the number of variables, and is the fitting error. β_{1} is the parameter to be determined, which can be calculated by the least square method
where F is a Grammatrix based on sample x.
(3) Radial basis function (RBF)
RBF has been developed for the interpolation of scattered multivariate data. The model is defined by
where N denotes the number of training points, is the unknown coefficients, and is the multiquadric function. The multiquadric function is defined as
where r = ‖x − x_{i}‖ is the Euclidean distance of a point x from a given data point xi, and the parameter c > 0 is a constant in the multiquadric.
Appendix 2
Appendix 3
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Yu, M., Li, X. & Liang, J. A dynamic surrogateassisted evolutionary algorithm framework for expensive structural optimization. Struct Multidisc Optim 61, 711–729 (2020). https://doi.org/10.1007/s00158019023918
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DOI: https://doi.org/10.1007/s00158019023918
Keywords
 Evolutionary algorithm
 Adaptive surrogate model
 Expensive optimization
 Reliability