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       目录

💥1 概述

📚2 运行结果

🎉3 参考文献

👨‍💻4 Matlab代码

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💥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')

a0=0;

b0=1;

n=11;

h=(b0-a0)/n;

[X1,Y1]=meshgrid(a0+h:h:b0-h);

W=[];

for i=1: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 i=1:size(X,2)

    Z=[X(:,i),Y(:,1)];

    W2=[W2 ; Z];

end

L1=[];

for i=1:n+1

    L1=[L1 ; W2(i,:)];

end

L2=[];

for i=n*(n+1)+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-2+v.^3+6*v); % right hand side of the given PDE

gamma=10^14; % the regularization parameter

sig=0.95;  % kernel bandwidth

K=KernelMatrix(W,'RBF_kernel',sig);

x=W(:,1);

y=W(:,2);

xx1=x*ones(1,size(x,1));

xx2=x*ones(1,size(x,1));

cof1=2*(xx1-xx2')/(sig);

xx3=y*ones(1,size(y,1));

xx4=y*ones(1,size(y,1));

cof2=2*(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;

m=size(K1T,1);

%*******************************************************************

KL1=KernelMatrix(W,'RBF_kernel',sig,L1);

L1b1x=L1(:,1)*ones(1,size(x,1));

L1b2x=x*ones(1,size(L1(:,1),1));

cofL1x=-2*(L1b1x'-L1b2x)/(sig);

L1b1y=L1(:,2)*ones(1,size(y,1));

L1b2y=y*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;

%*************************************************

KL2=KernelMatrix(W,'RBF_kernel',sig,L2);

L2b1x=L2(:,1)*ones(1,size(x,1));

L2b2x=x*ones(1,size(L2(:,1),1));

cofL2x=-2*(L2b1x'-L2b2x)/(sig);

L2b1y=L2(:,2)*ones(1,size(y,1));

L2b2y=y*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;

%*************************************************

KL3=KernelMatrix(W,'RBF_kernel',sig,L3);

L3b1x=L3(:,1)*ones(1,size(x,1));

L3b2x=x*ones(1,size(L3(:,1),1));

cofL3x=-2*(L3b1x'-L3b2x)/(sig);

L3b1y=L3(:,2)*ones(1,size(y,1));

L3b2y=y*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;

%*************************************************

KL4=KernelMatrix(W,'RBF_kernel',sig,L4);

L4b1x=L4(:,1)*ones(1,size(x,1));

L4b2x=x*ones(1,size(L4(:,1),1));

cofL4x=-2*(L4b1x'-L4b2x)/(sig);

L4b1y=L4(:,2)*ones(1,size(y,1));

L4b2y=y*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;

%*************************************************

KL1L1=KernelMatrix(L1,'RBF_kernel',sig,L1);

KL2L1=KernelMatrix(L2,'RBF_kernel',sig,L1);

KL3L1=KernelMatrix(L3,'RBF_kernel',sig,L1);

KL4L1=KernelMatrix(L4,'RBF_kernel',sig,L1);

%*************************************************

KL1L2=KernelMatrix(L1,'RBF_kernel',sig,L2);

KL2L2=KernelMatrix(L2,'RBF_kernel',sig,L2);

KL3L2=KernelMatrix(L3,'RBF_kernel',sig,L2);

KL4L2=KernelMatrix(L4,'RBF_kernel',sig,L2);

%************************************************

KL1L3=KernelMatrix(L1,'RBF_kernel',sig,L3);

KL2L3=KernelMatrix(L2,'RBF_kernel',sig,L3);

KL3L3=KernelMatrix(L3,'RBF_kernel',sig,L3);

KL4L3=KernelMatrix(L4,'RBF_kernel',sig,L3);

%************************************************

KL1L4=KernelMatrix(L1,'RBF_kernel',sig,L4);

KL2L4=KernelMatrix(L2,'RBF_kernel',sig,L4);

KL3L4=KernelMatrix(L3,'RBF_kernel',sig,L4);

KL4L4=KernelMatrix(L4,'RBF_kernel',sig,L4);

%************************************************

A= [K1T+1/gamma*eye(m) , KL1T , KL2T, KL3T , KL4T , zeros((n-1)^2,1) ;....

    KL1T' , KL1L1' , KL2L1' , KL3L1' , KL4L1' , ones(n+1,1) ;...

    KL2T' , KL1L2' , KL2L2' , KL3L2' , KL4L2' , ones(n+1,1) ;...

    KL3T' , KL1L3' , KL2L3' , KL3L3' , KL4L3' , ones(n+1,1) ;...

    KL4T' , KL1L4' , KL2L4' , KL3L4' , KL4L4' , ones(n+1,1) ;...

    zeros((n-1)^2,1)' , ones(n+1,1)' , ones(n+1,1)' , ones(n+1,1)' , ones(n+1,1)' , 0 ];

B=[f(W(:,1),W(:,2)); L1(:,2).^3 ; (1+L2(:,2).^3)*exp(-1)  ;  L3(:,1).*exp(-L3(:,1)) ; exp(-L4(:,1)).*(L4(:,1)+1) ; 0 ];

result=A\B;

alpha=result(1:m);

beta1=result(m+1:m+n+1);

beta2=result(m+n+2:m+2*n+2);

beta3=result(m+2*n+3:m+3*n+3);

beta4=result(m+3*n+4:m+4*n+4);

b=result(end);

%% Result for training points

yhat= (Kxx' + Kyy')* alpha + KL1 * beta1 + KL2* beta2 + KL3* beta3 + KL4* beta4 +b;

yexa=@(p,q) exp(-p).*(p+q.^3);

yexact=yexa(W(:,1),W(:,2));

Error1= yexact- yhat;

MAX_Absolute_error_training=max(abs(yhat-yexact));

RMSE_training=sqrt(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')

NError=reshape(Error1,size(X1,1),size(Y1,1));

Xn=linspace(0,1,n-1);

Yn=linspace(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

h=colorbar;

set(h,'fontsize',14);

title('Absolute errors for training set','Fontsize',14)

%% Result for test points

a0=0;

b0=1;

n=31;

h=(b0-a0)/n;

[X2,Y2]=meshgrid(a0+h:h:b0-h);

WT=[];

for i=1: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')

Kt=KernelMatrix(W,'RBF_kernel',sig,WT);

xt=WT(:,1);

yt=WT(:,2);

xx1t=x*ones(1,size(xt,1));

xx2t=xt*ones(1,size(x,1));

cof1t=-2*(xx1t-xx2t')/(sig);

xx3t=y*ones(1,size(yt,1));

xx4t=yt*ones(1,size(y,1));

cof2t=-2*(xx3t-xx4t')/(sig);

Ktestxx=(-2/sig)*Kt + (cof1t.^2) .* Kt;

Ktestyy=(-2/sig)*Kt + (cof2t.^2) .* Kt;

KKlte1=KernelMatrix(WT,'RBF_kernel',sig,L1);

KKlte2=KernelMatrix(WT,'RBF_kernel',sig,L2);

KKlte3=KernelMatrix(WT,'RBF_kernel',sig,L3);

KKlte4=KernelMatrix(WT,'RBF_kernel',sig,L4);

Ytest= (Ktestxx' + Ktestyy')* alpha + KKlte1 * beta1 + KKlte2* beta2 + KKlte3* beta3 + KKlte4* beta4 + b;

yextest=yexa(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')

yextest=yexa(WT(:,1),WT(:,2));

MAX_Absolute_error_test=max(abs(Ytest-yextest));

RMSE_test=sqrt(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 ;

Ytnew=reshape(Ytest,size(X2,1),size(Y2,1));

Ytexa=reshape(yextest,size(X2,1),size(Y2,1));

NError=reshape(Error,size(X2,1),size(Y2,1));

Xn=linspace(0,1,n-1);

Yn=linspace(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

h=colorbar;

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