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#x1f4da;2 运行结果
#x1f389;3 参考文献
#x1f468;#x1f4bb;4 Matlab代码 #x1f4a5;1 概述
本代码说明了“最小二乘支持向量机”在学习偏微分方程 #xff08;PDE#xff09; 解方面的应用。提供了一个示例#xff0c… 目录 1 概述
2 运行结果
3 参考文献
4 Matlab代码 1 概述
本代码说明了“最小二乘支持向量机”在学习偏微分方程 PDE 解方面的应用。提供了一个示例并将获得的结果与精确的解决方案进行比较。
2 运行结果 主函数部分代码
clc; clear all; close all
warning(off,all) a00;
b01;
n11;
h(b0-a0)/n;
[X1,Y1]meshgrid(a0h:h:b0-h);
W[];
for i1:size(X1,2) Z[X1(:,i),Y1(:,1)]; W[W ; Z];
end
subplot(2,3,1)
plot(W(:,1),W(:,2),o)
hold on
[X,Y]meshgrid(a0:h:b0);
W2[];
for i1:size(X,2) Z[X(:,i),Y(:,1)]; W2[W2 ; Z];
end
L1[];
for i1:n1 L1[L1 ; W2(i,:)];
end
L2[];
for in*(n1)1:size(W2,1) L2[L2 ; W2(i,:)];
end
L3[L1(:,2) L1(:,1)];
L4[L2(:,2) L2(:,1)]; plot(L1(:,1),L1(:,2),s)
plot(L2(:,1),L2(:,2),o)
plot(L3(:,1),L3(:,2),p)
plot(L4(:,1),L4(:,2),)
title(Training points,Fontsize,14)
xlabel(x)
ylabel(y) %% f(s,v) exp(-s).*(s-2v.^36*v); % right hand side of the given PDE
gamma10^14; % the regularization parameter
sig0.95; % kernel bandwidth KKernelMatrix(W,RBF_kernel,sig);
xW(:,1);
yW(:,2);
xx1x*ones(1,size(x,1));
xx2x*ones(1,size(x,1));
cof12*(xx1-xx2)/(sig);
xx3y*ones(1,size(y,1));
xx4y*ones(1,size(y,1));
cof22*(xx3-xx4)/(sig);
Kxx(-2/sig)*K (cof1.^2) .* K;
Kyy(-2/sig)*K (cof2.^2) .* K;
Kx2x2( ( 12/(sig^2) - (12/sig)* (cof1.^2) (cof1.^4) ) .*K);
Ky2y2( ( 12/(sig^2) - (12/sig)* (cof2.^2) (cof2.^4) ) .*K);
Kx2y2( ( 4/(sig^2) - (2/sig)* (cof1.^2) - (2/sig)* (cof2.^2) (cof1.^2).*(cof2.^2) ) .*K);
Ky2x2( ( 4/(sig^2) - (2/sig)* (cof1.^2) - (2/sig)* (cof2.^2) (cof1.^2).*(cof2.^2) ) .*K); K1T Kx2x2 Kx2y2 Ky2x2 Ky2y2;
msize(K1T,1);
%******************************************************************* KL1KernelMatrix(W,RBF_kernel,sig,L1); L1b1xL1(:,1)*ones(1,size(x,1));
L1b2xx*ones(1,size(L1(:,1),1));
cofL1x-2*(L1b1x-L1b2x)/(sig);
L1b1yL1(:,2)*ones(1,size(y,1));
L1b2yy*ones(1,size(L1(:,2),1));
cofL1y-2*(L1b1y-L1b2y)/(sig);
KL1xx(-2/sig)*KL1 (cofL1x.^2) .* KL1;
KL1yy(-2/sig)*KL1 (cofL1y.^2) .* KL1;
KL1T KL1xx KL1yy; %************************************************* KL2KernelMatrix(W,RBF_kernel,sig,L2);
L2b1xL2(:,1)*ones(1,size(x,1));
L2b2xx*ones(1,size(L2(:,1),1));
cofL2x-2*(L2b1x-L2b2x)/(sig);
L2b1yL2(:,2)*ones(1,size(y,1));
L2b2yy*ones(1,size(L2(:,2),1));
cofL2y-2*(L2b1y-L2b2y)/(sig);
KL2xx(-2/sig)*KL2 (cofL2x.^2) .* KL2;
KL2yy(-2/sig)*KL2 (cofL2y.^2) .* KL2;
KL2T KL2xx KL2yy; %************************************************* KL3KernelMatrix(W,RBF_kernel,sig,L3);
L3b1xL3(:,1)*ones(1,size(x,1));
L3b2xx*ones(1,size(L3(:,1),1));
cofL3x-2*(L3b1x-L3b2x)/(sig);
L3b1yL3(:,2)*ones(1,size(y,1));
L3b2yy*ones(1,size(L3(:,2),1));
cofL3y-2*(L3b1y-L3b2y)/(sig);
KL3xx(-2/sig)*KL3 (cofL3x.^2) .* KL3;
KL3yy(-2/sig)*KL3 (cofL3y.^2) .* KL3;
KL3T KL3xx KL3yy; %************************************************* KL4KernelMatrix(W,RBF_kernel,sig,L4);
L4b1xL4(:,1)*ones(1,size(x,1));
L4b2xx*ones(1,size(L4(:,1),1));
cofL4x-2*(L4b1x-L4b2x)/(sig);
L4b1yL4(:,2)*ones(1,size(y,1));
L4b2yy*ones(1,size(L4(:,2),1));
cofL4y-2*(L4b1y-L4b2y)/(sig);
KL4xx(-2/sig)*KL4 (cofL4x.^2) .* KL4;
KL4yy(-2/sig)*KL4 (cofL4y.^2) .* KL4;
KL4T KL4xx KL4yy; %************************************************* KL1L1KernelMatrix(L1,RBF_kernel,sig,L1);
KL2L1KernelMatrix(L2,RBF_kernel,sig,L1);
KL3L1KernelMatrix(L3,RBF_kernel,sig,L1);
KL4L1KernelMatrix(L4,RBF_kernel,sig,L1); %************************************************* KL1L2KernelMatrix(L1,RBF_kernel,sig,L2);
KL2L2KernelMatrix(L2,RBF_kernel,sig,L2);
KL3L2KernelMatrix(L3,RBF_kernel,sig,L2);
KL4L2KernelMatrix(L4,RBF_kernel,sig,L2); %************************************************ KL1L3KernelMatrix(L1,RBF_kernel,sig,L3);
KL2L3KernelMatrix(L2,RBF_kernel,sig,L3);
KL3L3KernelMatrix(L3,RBF_kernel,sig,L3);
KL4L3KernelMatrix(L4,RBF_kernel,sig,L3); %************************************************ KL1L4KernelMatrix(L1,RBF_kernel,sig,L4);
KL2L4KernelMatrix(L2,RBF_kernel,sig,L4);
KL3L4KernelMatrix(L3,RBF_kernel,sig,L4);
KL4L4KernelMatrix(L4,RBF_kernel,sig,L4); %************************************************ A [K1T1/gamma*eye(m) , KL1T , KL2T, KL3T , KL4T , zeros((n-1)^2,1) ;.... KL1T , KL1L1 , KL2L1 , KL3L1 , KL4L1 , ones(n1,1) ;... KL2T , KL1L2 , KL2L2 , KL3L2 , KL4L2 , ones(n1,1) ;... KL3T , KL1L3 , KL2L3 , KL3L3 , KL4L3 , ones(n1,1) ;... KL4T , KL1L4 , KL2L4 , KL3L4 , KL4L4 , ones(n1,1) ;... zeros((n-1)^2,1) , ones(n1,1) , ones(n1,1) , ones(n1,1) , ones(n1,1) , 0 ]; B[f(W(:,1),W(:,2)); L1(:,2).^3 ; (1L2(:,2).^3)*exp(-1) ; L3(:,1).*exp(-L3(:,1)) ; exp(-L4(:,1)).*(L4(:,1)1) ; 0 ];
resultA\B;
alpharesult(1:m);
beta1result(m1:mn1);
beta2result(mn2:m2*n2);
beta3result(m2*n3:m3*n3);
beta4result(m3*n4:m4*n4);
bresult(end); %% Result for training points yhat (Kxx Kyy)* alpha KL1 * beta1 KL2* beta2 KL3* beta3 KL4* beta4 b;
yexa(p,q) exp(-p).*(pq.^3);
yexactyexa(W(:,1),W(:,2));
Error1 yexact- yhat;
MAX_Absolute_error_trainingmax(abs(yhat-yexact));
RMSE_trainingsqrt(mse(yhat-yexact));
fprintf(------- training set ------------------\n\n)
fprintf(Max Abs Error on training set%d\n,MAX_Absolute_error_training)
fprintf(RMSE on training set%d\n\n,RMSE_training) subplot(2,3,2)
plot3(W(:,1),W(:,2),yhat,pr)
hold all
plot3(W(:,1),W(:,2),yexact,sb)
title(Approximate and exact solution for training points,Fontsize,14)
xlabel(x)
ylabel(y)
zlabel(u) NErrorreshape(Error1,size(X1,1),size(Y1,1));
Xnlinspace(0,1,n-1);
Ynlinspace(0,1,n-1);
subplot(2,3,3)
surface(Xn,Yn,NError)
shading interp
xlabel(y,Fontsize,14)
ylabel(x,Fontsize,14)
set(gca,Fontsize,20)
grid on
hcolorbar;
set(h,fontsize,14);
title(Absolute errors for training set,Fontsize,14) %% Result for test points a00;
b01;
n31;
h(b0-a0)/n;
[X2,Y2]meshgrid(a0h:h:b0-h);
WT[];
for i1:size(X2,2) Z[X2(:,i),Y2(:,1)]; WT[WT ; Z];
end
subplot(2,3,4)
plot(WT(:,1),WT(:,2),o)
title(Test points,Fontsize,14)
xlabel(x)
ylabel(y) KtKernelMatrix(W,RBF_kernel,sig,WT);
xtWT(:,1);
ytWT(:,2);
xx1tx*ones(1,size(xt,1));
xx2txt*ones(1,size(x,1));
cof1t-2*(xx1t-xx2t)/(sig);
xx3ty*ones(1,size(yt,1));
xx4tyt*ones(1,size(y,1));
cof2t-2*(xx3t-xx4t)/(sig);
Ktestxx(-2/sig)*Kt (cof1t.^2) .* Kt;
Ktestyy(-2/sig)*Kt (cof2t.^2) .* Kt;
KKlte1KernelMatrix(WT,RBF_kernel,sig,L1);
KKlte2KernelMatrix(WT,RBF_kernel,sig,L2);
KKlte3KernelMatrix(WT,RBF_kernel,sig,L3);
KKlte4KernelMatrix(WT,RBF_kernel,sig,L4);
Ytest (Ktestxx Ktestyy)* alpha KKlte1 * beta1 KKlte2* beta2 KKlte3* beta3 KKlte4* beta4 b;
yextestyexa(WT(:,1),WT(:,2)); subplot(2,3,5)
plot3(WT(:,1),WT(:,2),Ytest,pr)
hold on
plot3(WT(:,1),WT(:,2),yextest,sb)
title(Approximate and exact solution for test points,Fontsize,14)
xlabel(x)
ylabel(y)
zlabel(u)
yextestyexa(WT(:,1),WT(:,2));
MAX_Absolute_error_testmax(abs(Ytest-yextest));
RMSE_testsqrt(mse(Ytest-yextest));
fprintf(------- test set ------------------\n\n)
fprintf(Max Abs Error on test set%d\n,MAX_Absolute_error_test)
fprintf(RMSE on test set%d\n\n,RMSE_test)
fprintf(------- Finished -----------------------\n\n)
Error Ytest - yextest ;
Ytnewreshape(Ytest,size(X2,1),size(Y2,1));
Ytexareshape(yextest,size(X2,1),size(Y2,1));
NErrorreshape(Error,size(X2,1),size(Y2,1));
Xnlinspace(0,1,n-1);
Ynlinspace(0,1,n-1);
subplot(2,3,6)
surface(Xn,Yn,NError)
shading interp
xlabel(y,Fontsize,14)
ylabel(x,Fontsize,14)
set(gca,Fontsize,20)
grid on
hcolorbar;
set(h,fontsize,14);
title(Absolute errors for test set,Fontsize,14) 3 参考文献
[1] Mehrkanoon S., Falck T., Suykens J.A.K., Approximate Solutions to Ordinary Differential Equations Using Least Squares Support Vector Machines,IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 9, Sep. 2012, pp. 1356-1367.
[2] Mehrkanoon S., Suykens J.A.K.,LS-SVM approximate solution to linear time varying descriptor systems, Automatica, vol. 48, no. 10, Oct. 2012, pp. 2502-2511.
[3] Mehrkanoon S., Suykens J.A.K., Learning Solutions to Partial Differential Equations using LS-SVM,Neurocomputing, vol. 159, Mar. 2015, pp. 105-116.
4 Matlab代码
