MATLAB: Warning: Derivative finite-differencing step was artificially reduced to be within bound constraints. This may adversely affect convergence. Increasing distance between bound constraints, in dimension 3, to be at least 2e-08 may improve results.

lsqcurvefitMATLABode45

I'm trying to use lsqcurvefit to fit three coefficiencients to a system of coupled equations that I solve using ode45. The data I am fitting comes from the below code:
function [N T] = rk4method(k,t)
n0=[1e20, 0];
[T,Nv]=ode45(@DifEq,t,n0);
%
function dN=DifEq(t,n)
dndt = zeros(2,1);
dndt(1)= -((8.89e+6)+(1.75e+7)).*n(1) + (1.91e+6).*n(2) - k(1).*n(1).^2 - k(2).*n(1).*n(2) + 0.25.*k(3).*n(2).^2;
dndt(2)= (1.75e+7).*n(1)-(1.91e+6).*n(2)-(3.43e+5).*n(2)-1.25.*k(3).*n(2).^2;
dN=dndt;
end
N = Nv(:,1);
T = T;
end
I use this to create my data:
[ndat t] = rk4method([5.1e-12 1.12e-12 5.12e-14], [0 2e-6]);
Here's my fitting code:
n = ndat;
k0=[1e-11 1e-13 1e-15];
lb=[1e-16 1e-16 1e-18];
ub=[1e-8 1e-8 1e-8];
%tried but doesn't help: 'FunctionTolerance',1e-8,'FiniteDifferenceType','central',
options = optimoptions('lsqcurvefit','Diagnostics','on','Display','iter-detailed');
[k,Resnorm]=lsqcurvefit(@rk4method,k0,t,n,lb,ub,options);
k
Nfin = rk4method(k,t);
figure(1)
hold on;
plot(t, log10(ndat(:,1)), '*r');
plot(t, log10(Nfin), '-k');
function [N T] = rk4method(k,t)
n0=[1e20, 0];
[T,Nv]=ode45(@DifEq,t,n0);
function dN=DifEq(t,n)
dndt = zeros(2,1);
dndt(1)= -((8.89e+6)+(1.75e+7)).*n(1) + (1.91e+6).*n(2) - k(1).*n(1).^2 - k(2).*n(1).*n(2) + 0.25.*k(3).*n(2).^2;
dndt(2)= (1.75e+7).*n(1)-(1.91e+6).*n(2)-(3.43e+5).*n(2)-1.25.*k(3).*n(2).^2;
dN=dndt;
end
N = Nv(:,1);
T = T;
end
lsqcurvefit isn't actually changing the value of the parameters k from its initial values k0, with Warning: Derivative finite-differencing step was artificially reduced to be within bound constraints. This may adversely affect convergence. Increasing distance between bound constraints, in dimension 3, to be at least 2e-08 may improve results.
I've tried tweaking my ub to be greater, however this doesn't result in convergence.

Best Answer

Well, there are technical issues with applying optimization toolbox solvers to ODEs. You will want to be thoroughly familiar with the material on this page,
https://www.mathworks.com/help/optim/ug/optimizing-a-simulation-or-ordinary-differential-equation.html
Also, as the warning message tells you, your bounds are very tight in absolute terms. It would be advisable to change the units of your coefficients so that they, and their bounds, are on the order of 1 instead of 1e-8. Even though the tolerance parameters for lsqnonlin (e.g. StepTolerance, FiniteDifferenceStepSize) and the other toolbox solvers have some ability to divine the scale of your parameters and of your cost function, I find that it's usually best not to rely on that.
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