Solving an ODE with best-fit adjustment to empirical observations

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Mario Trevino 2011-4-8

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https://ww2.mathworks.cn/matlabcentral/answers/5054-solving-an-ode-with-best-fit-adjustment-to-empirical-observations

Hi everybody, I want to numerically solve an ODE but I want to fit the constants to experimental data following that ODE. Any simple example out there?

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Answer 1

Teja Muppirala 2011-4-8

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If I understood correctly, you want to find the parameters for an ODE such that the ODE fits some experimental data. If you have the Optimization Toolbox this is easy to program. You basically put the ODE solver inside the cost function for your optimization

Say your ODE is :

y' = A*y*(B-y)

And you want to find A, B, and y(0). This is some sample code that shows how it's done. Save it and run it.

function xout = do_optimization
% The true parameters
y0_true = 1;
A_true = 2;
B_true = 3;
T = (0:.01:1)';
ytrue =3./(1 + 2*exp(-6*T)) + 0.1*randn(size(T));
hold on;
plot(T,ytrue);
h = plot(T,nan*ytrue,'r');
set(h,'tag','solution');
x0 = [0.4 3.9 1.2];   % Just some Initial Condition
ub = [5 5 5];         % Upper bounds
lb = [0 0 0];         % Lower bounds
F = @(x) COST(x,T,ytrue);
xout = fmincon(F,x0,[],[],[],[],lb,ub); %<-- FMINCON is the optimizer
legend({'Experimental Data','Fitted Data'});
function COST = COST(x,T,ytrue)
y0 = x(1);
A = x(2);
B = x(3);
% The cost function calls the ODE solver.
[tout,yout] = ode45(@dydt,T,y0,[],A,B);
COST = sum((yout - ytrue).^2);
h = findobj('tag','solution');
set(h,'ydata',yout);
title(['y0 = ' num2str(y0) '   ' 'A = ' num2str(A) '    B = ' num2str(B)]);
drawnow;
function yprime = dydt(t,y,A,B)
yprime = A*y*(B-y);