Solving a non linear ODE with unknown parameter

Question

khaoula Oueslati 2022-4-14

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评论： khaoula Oueslati 2022-4-19

Hello ! I am working on solving an ODE equation with an unknown kinetic parameter A. I have been using python and deep learning to solve the equation and also determine the value of A , however the loss function is always in the order of 10**4 and the paramter A is wrong , I tried with different hyperparamters but it´s not working. this is the ODE equation : dDP/dt=-k1*([DP]^2) and k1=k= Ae^(1/R(-E/(T+273))) , A is in the order of 10**8, I have DP(t) data.

I am stuck and I would like to know what´s the best way to solve this using matlab ? or is there any examples similar to my problem ?

Any help is highly appreciated !

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Answer 1

Torsten 2022-4-19

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编辑：Torsten 2022-4-19

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%time points
ts=[1 2 3 4 5 6 7 8];
DP=[1000 700.32 580.42 408.20 317.38 281.18 198.15 100.12];
p0 = 1e1;
p = fminunc(@(p)fun(p,ts,DP),p0)
E = 111e3;
R = 8.314;
T = 371;
A = p*exp(E/(R*T))
plot(ts,DP)
hold on
plot(ts,1./(1/DP(1)+ A*exp(-E/(R*T))*(ts-ts(1))));
function obj = fun(p,ts,DP)
  DP_model = 1./(1/DP(1)+ p*(ts-ts(1)));   
  obj = sum((DP-DP_model).^2)
end

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Torsten 2022-4-19

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the loss value is :loss value is 0.35787 and A value is 1.08e10 and the ground_truth A value is 7.8e8

I am not sure the source of this mismatch.

I don't know either. Maybe T or E were different. The fit at least is perfect.

btw , how did you find the DP_model expression ? is it some appoximation ? or after integration we get that expression of the solution ?

If you don't trust in my pencil-and-paper solution, here is MATLAB code to solve the differential equation:

syms Dp(t) k1 t0 Dp0
eqn = diff(Dp,t) == -k1*Dp^2;
cond = Dp(t0) == Dp0;
DpSol(t) = dsolve(eqn,cond)

khaoula Oueslati 2022-4-19

Thanks a lot @Torsten ! it's not that I don't trust in the expression , I just wanted to know how to find it so that I can explain it to my academic supervisor. About the mismatch , it's probably due to some variation in the data or E value , I will further look into that and thanks again !

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Answer 2

Torsten 2022-4-14

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Your ODE for D_p gives

D_p = 1/(1/D_p0 + k1*(t-t0))

where D_p0 = D_p(t0).

Now you can apply "lsqcurvefit" to fit the unknown parameter A.

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khaoula Oueslati 2022-4-16

Thanks Torsten for your help! I will try it !

khaoula Oueslati 2022-4-19

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Hello @Torsten , I didnt know how to implement using Lsqcurvefit and since I am not really familiar with matlab , I am struggling to use it for my problem. I tried to use fminunc but erros keep coming up and I didnt know how to tackle them.

this is my code using fminunc

function dDP= Mymodel(t,DP,A)
%known parameters
E=111e3;
R=8.314;
T=371;
dDP= -(A*(exp(-E/R*T)))*(DP^2);
end
function obj = objective(A)
DP0=1000;
%time points
ts=[1 2 3 4 5 6 7 8];
[t,DP]= ode45(@(t,DP)Mymodel(t,DP,A),ts,DP0);
DP_measured=[1000 700.32 580.42 408.20 317.38 281.18 198.15 100.12];
A=(DP-DP_measured).^2;
obj=sum(A);
end
A0=1e8;
fun = @objective;
[A,fval]=fminunc(fun,A0);
disp(['A:' num2str(A)])
These errors come up:
odearguments(odeIsFuncHandle,odeTreatAsMFile, solver_name, ode, tspan, y0, options, varargin);
Error in objective (line 5)
[t,DP]= ode45(@(t,DP)Mymodel(t,DP,A),ts,DP0);
 
optimization_DP
Error using fminunc
Supplied objective function must return a scalar value.
Error in optimization_DP (line 3)
[A,fval]=fminunc(fun,A0);
 

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Answer 3

Sam Chak 2022-4-14

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编辑：Sam Chak 2022-4-14

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Hi @khaoula Oueslati

This governing equations are given and you have acquired the

data.

The objective is want to find A.

From the

data, you can possibly estimate for

. Next,

can be determined from the differential equation:

Now, if R, E and T are known, then

can be determined from the algebraic equation:

Please verify this.

If the

data is uniformly distributed, then you can use this method to estimate

.

t = -pi:(2*pi/100):pi;
x = sin(t);                     % assume Dp is a sine wave
y = gradient(x)/(2*pi/100);     % estimate dotDp, a cosine wave is expected
plot(t, x, 'linewidth', 1.5, t, y, 'linewidth', 1.5)
grid on
xlabel('t')
ylabel('x(t) and x''(t)')
legend('x(t) = sin(t)', 'x''(t) = cos(t)', 'location', 'northwest')

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khaoula Oueslati 2022-4-14

Thanks a lot Sam ! I will try it, but the thing is , I need to be able to find that parameter using an optimization algorithm and not from an algebraic equation and I tried with python but I didnt figure it out.

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