How to improve curve-fitting results using Genetic Algorithm?

Question

Faezeh Manesh 2020-7-6

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https://ww2.mathworks.cn/matlabcentral/answers/560342-how-to-improve-curve-fitting-results-using-genetic-algorithm

Hello all,

I am trying to solve a curve-fitting problem using multi-objective GA (I've tried lsqcurvefit previously which didn't work well so I switch to GA hoping get better results). What I am trying to do is to fit some curves to my experminetal data using the results of a differential equation. In fact, I have a parametric differential equation that I am trying to adjust those parametres and add a number of them together to obtain my experimental curve. I have written the following codes for this task:

My parametric differential equation is incorporated into the following function:

    function denaturate = firstorderDSC(T,n,D,C,A)
    
           dndT=A*exp(D)*exp(C*1e4/T)*(1-n);
           
          denaturate=dndT;
    end

In which A, D, and C needs to be determined through curve-fitting (either GA or any other curve-fitting function)

I've also introduced another function to add-up the result of the differential equation (because I need to add multiple curves to form my final curve)

function dndt = firstorderDSC1(B,xx,m)
%a matrix that include the result of differential equation 
dndt1=zeros(size(xx,1),1);
for i=1:m
% solve the differential equation with diferent initial values
odefun = @(xx,n) firstorderDSC(xx,n,B(i),B(m+i),B(2*m+i));
options = odeset('AbsTol', 1e-8,'RelTol',1e-8);%,'OutputFcn',@odeplot);
[temp,num] = ode23s(odefun, xx, 0, options);
%add-up the result of differential equation with different initial values
  for k=1:size(num,1)
      dndt1(k,1) = firstorderDSC(temp(k),num(k),B(i),B(m+i),B(2*m+i))+dndt1(k,1);
  end   
   dndt=dndt1; 
end
end

Finally, my main program is as follows:

clc
clear all;
close all;
%My experimental data (my curves should be fitted to these data)
[d,s,r] = xlsread('DSC.csv');
xx = d(:,2);
yy = d(:,3);
%Initial guesses for the parameters of my differential equation @firstorderDSC
B1=[198.1498,183.8,23.9814,13,132.4,203.0657,221.0482,29.9336];
m=size(B1,2);
B2=[-6.7629,-6.226,-0.8756,-0.5292,-4.55,-7.0564,-7.5,-1.1];
m2=size(B2,2);
B3=[1,1,1,1,1,1,1,1];
m3=size(B3,2);
  
B0=[B1,B2,B3];
m0=size(B0,2);
FitnessFunction = @(B) norm(yy-firstorderDSC1(B,xx,m)); %Fittness function
numberOfVariables = m0; % Number of decision variables
lb =[0,0,0,0,0,0,0,0,-7,-7,-7,-7,-7,-7,-7,-7,0,0,0,0,0,0,0,0]; % Lower bound
ub=[200,200,200,200,200,200,200,200,0,0,0,0,0,0,0,0,10,10,10,10,10,10,10,10]; % Upper bound
A = []; % No linear inequality constraints
b = []; % No linear inequality constraints
Aeq = []; % No linear equality constraints
beq = []; % No linear equality constraints
options = optimoptions(@gamultiobj,'CreationFcn',{@gacreationlinearfeasible},'FunctionTolerance',1e-4,...
         'MaxGenerations',300,'MaxStallGenerations',200,'PopulationSize',500,'MutationFcn',{@mutationadaptfeasible},...
         'CrossoverFraction',0.8,'CrossoverFcn',{@crossoverintermediate});
[B,Fval,exitFlag,Output] = gamultiobj(FitnessFunction,numberOfVariables,A,b,Aeq,beq,lb,ub,options);
%Create plot
figure;
plot(xx,firstorderDSC1(B,xx,m),xx,yy,'LineWidth',2)
legend('Experimental data (Gaussian)','Differential equation','real DSC')
ylim([-0.02 0.2])

What GA does is make all curves flat and keep only one curve which doesn't fit to my experimental data well (I have also attached B matrix that GA obtained). I don't know what should I do to improve the result of GA. I have also attached my experimetal data "DSC.csv".