Adaptive infill sampling criterion for multi-fidelity gradient-enhanced kriging model

Abstract

Multi-fidelity surrogate (MFS) method is very promising for the optimization of complex problems. The optimization capability of MFS can be improved by infilling samples in the optimization process. Furthermore, once the gradient information is provided, the gradient-enhanced kriging (GEK) can be utilized to construct a more accurate MFS model. However, for the existing infill sampling criterions, it is difficult to improve the optimization speed without sacrificing the optimization gains. In this paper, a novel infill sampling criterion named Adaptive Multi-fidelity Expected Improvement (AMEI) is proposed, in which the prediction accuracy and the optimization potential of the surrogate model are both considered. With a set of extra samples calculated, the AMEI determines which fidelity model for the new sample is to be added. Through two numerical examples and two engineering examples, it can be found that the AMEI always provides the best optimization result with the fewest analysis calls, and the robustness is also good. The optimization capability and efficiency of the AMEI have been demonstrated compared with traditional criterions.

Appendix Matlab code for the AMEI

%This part provides the codes of Hartmann 6 function in Section 3.2.2

%The parameters and the functions can be adjusted for other problems.

clear clc

dim=6;%dimension of this problem

mxn=50;%maximum iteration

t0=1*ones(1,dim);tl=0.00001*ones(1,dim);tu=20*ones(1,dim);%sets for GEK

Etol=0.0001;% the convergence condition

S1=lhsdesign(10,dim);%initial samples of low-fidelity model

S2=lhsdesign(5,dim);%initial samples of high-fidelity model

mt=lhsdesign(6,dim);%extra samples

xl=zeros(1,dim);xu=ones(1,dim);%lower and upper limitations of variables

n=size(mt,1);

global part y yc yt dmodel_low dmodel_Deta row

for i=1:size(S2,1)

[Asort,index]=sort(sum((S1-S2(i,:)).^2,2));

S1(index(1),:)=[];

end

S1=unique([S1;S2],'rows');

[y,yy]=fun_H(S2);%analysis code of high-fidelity model,y is response value, yy is the gradient information

[yc,yyc]=fun_L(S1);%analysis code of low-fidelity model

detay=max(y)-min(y);

infor=0;

Y1=[yc;yyc];Y2=[y;yy];

yh_t=fun_H(mt);yl_t=fun_L(mt);

for loop=1:mxn

[dmodel_low, ~] = dacefit(S1, Y1, @regpoly0,@corrgauss,t0,tl,tu);

[ty,~]= predictor(S2,dmodel_low);

[yt,yyt]=fun_L(S2);

[row,fval]=fmincon(@REG_FG,0.9,[],[],[],[],0.8,1.2);

d=y-row*yt;

dd=yy-row*yyt;

[dmodel_Deta, ~] = dacefit(S2,[d;dd],@regpoly0,@corrgauss,t0,tl,tu);

yL= predictor(mt,dmodel_low);

yD= predictor(mt,dmodel_Deta);

De=yh_t-row*yl_t;

rmse_RL=sqrt(1/n*sum((row*yL-yh_t).^2))/abs(mean(yh_t));

rmse_Deta=sqrt(sum((yD-De).^2)/n)/abs(mean(De));

part=1;

[news2,infor2]=ga(@AMEI,dim,[],[],[],[], xl,xu);

if -infor2>0.1*detay && rmse_RL<2*rmse_Deta && rmse_RL>rmse_Deta

infor=infor2;

[nyc,nyyc]=fun_L(news2);

S1=[S1;news2];yc=[yc;nyc];yyc=[yyc;nyyc];

Y1=[yc;yyc];

else

part=2;[news,infor]=ga(@AMEI,dim,[],[],[],[],xl,xu);

[nyc,nyyc]=fun_L(news);

S1=[S1;news];yc=[yc;nyc];yyc=[yyc;nyyc];

Y1=[yc;yyc];

[ny,nyy]=fun_H(news);

S2=[S2;news];y=[y;ny];yy=[yy;nyy];

Y2=[y;yy];

if -infor<Etol

break

end

end

end

function fun=REG_FG(row)

global yt y

fun=sqrt(sum((row*yt-y).^2));

end

function amei=AMEI(x)

global part y yc dmodel_low dmodel_Deta row

[n,~]=size(x);

[y1,s1]=predictor(x,dmodel_low);

[y2,s2]=predictor(x,dmodel_Deta);

if part==1 %update with AMEIL criterion

Ymin=min(yc); s=sqrt(s1);

Y=repmat(Ymin,n,1)-y1;

else %update with AMEIH criterion

Ymin=min(y); s=sqrt(row.^2.*s1+s2);

Y=repmat(Ymin,n,1)-y2-row.*y1;

end

id=find(s>0);

if isempty(id)==0

u=Y(id)./s(id);

PHI=normcdf(u,0,1);

phi=normpdf(u,0,1);

amei(id)=Y(id).*PHI+s(id).*phi;

end

id=find(s==0);

if isempty(id)==0

e=Y(id);

e(e<0)=0;

amei(id)=e;

end

amei=-amei;

end

function [y,dy]=fun_H(S)

for i=1:size(S,1)

x=S(i,:);

A=[10,3,17,3.5,1.7,8;0.05,10,17,0.1,8,14;3,3.5,1.7,10,17,8;17,8,0.05,10,0.1,14];

P=10^(-4)*[1312,1696,5569,124,8283,5886;2329,4135,8307,3736,1004,9991;

2348,1451,3522,2883,3047,6650;4047,8828,8732,5743,1091,381];

a=[1;1.2;3;3.2];

b=(repmat(x,4,[])'-P');

B=b.^2;

y(i,1)=1/(-1.94)*(2.58+a'*exp(-sum(A.*B',2)));

dy(1+(i-1)*size(S,2):i*size(S,2),1)=-1/1.94*(-2*a'.*A'.*b*exp(-sum(A.*B',2)));

end

end

function [y,dy]=fun_L(S)

for i=1:size(S,1)

x=S(i,:);

A=[10,3,17,3.5,1.7,8;0.05,10,17,0.1,8,14;3,3.5,1.7,10,17,8;17,8,0.05,10,0.1,14];

P=10^(-4)*[1312,1696,5569,124,8283,5886;2329,4135,8307,3736,1004,9991;

2348,1451,3522,2883,3047,6650;4047,8828,8732,5743,1091,381];

a=[0.5;0.5;2;4];

b=(repmat(x,4,[])'-P');

B=b.^2;

fv=(exp(-4/9)+exp(-4/9)*((-sum(A.*B',2)) +4)/9 );

y(i,1)=-1/1.94*(2.58+a'*(exp(-4/9)+exp(-4/9)*((-sum(A.*B',2)) +4)/9 ).^9);

dy(1+(i-1)*size(S,2):i*size(S,2),1)=2/1.94*A'.*b*(a.*fv.^8)*exp(-4/9);

end

end