An already established model for a given system described by nonlinear differential equations have 8 state variables. I have collected the input-output data from the standard model.
Now, I have modified another nonlinear ODE model for the same system having 6 state variables. The response of both the models are coming similar.
Now, I want to update the parameters of the new model using ANN. How can I use ANN to fit/update the parameters of the new model by using the input-output data obtained from the old model?
Dear Anirudh,
Indeed there have nested enough situations to be able to find a good way and to find a solution to this problem, I propose the following attached files.
Best regards
You want to fit an 8 parameter ODE model to a 6 parameter ODE model.
A simple method is as follows.
You need to generate data with the 8 Parameter model and use something like excel's solver to fit that data to the 6 parameter model.
see
http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwiZk6Xk27bOAhXoBcAKHYZKClAQFggoMAA&url=http%3A%2F%2Fpeople.chem.ucsb.edu%2Flaverman%2Fleroy%2FChem116%2FPDF116CL%2FSolver.pdf&usg=AFQjCNFriOOHaXoM4DFgFN08qhg8WVLchA
https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwiZk6Xk27bOAhXoBcAKHYZKClAQtwIILTAB&url=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3D3Fd4ukzkxps&usg=AFQjCNH4OLvkibMBGrm7rf0Mc7U-70ARlg
http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwiF04XT4bbOAhWpC8AKHUnLC_wQFggkMAE&url=http%3A%2F%2Fsherrytowers.com%2F2013%2F01%2F29%2Fneiu-lecture-vi-fitting-the-parameters-of-an-sir-model-to-influenza-data%2F&usg=AFQjCNFEPY8ueMTMLQNqxQKrX8l_-H1WKA
You will need to set out the 6 parameter model as a numerical solution to the 6 parameter ODE to give out some calculated data (or use the analytical solution to the 6 parameter ODE instead). Then calculate the total mean squared error. This you do by subtracting the output of the 8 parameter model from the 6 parameter model and squaring. Add up all the squared values to give the sum squared. You minimise this sum squared value by varying the parameters of the 6 parameter model using excel solver.
@Loopinder Sood
Thank you sir for your suggestion. Can you say little more about this. Actually, the nonlinear ODE model with 8 parameters has 6 state variables (system of 6 1st order ODEs) and the nonlinear model with 6 parameters have 3 state variables (system of 3 1st order ODEs).
How to utilize input-output data fitting of the 1st model to estimate the parameters of 2nd model in MS Excel?
Do the following
1) use the initial conditions of the first system (6 state variable system) and the parameter values of the first system to get out a solution either numerically (eg by Runge Kutta, or Euler etc) or analytically if you know the analytical solution to the first system
2) I take it least one of the 3 state variables of the second system are present in the first system. It is these state variables that are common to both that will be used to fit the second system to the first. You want these common variables to have similar values in both systems.
3) so take the second system and make initial guesses to the initial conditions of the second system and make initial guesses to the parameter values of the second system. (You probably will want the initial conditions of the common variables to be the same as in the first system). Use these guesses to get a solution as given by the second system using either a numerical method or analytically
4) the solution as given by the second system needs to be very close to the solution given by the first system. So use MS Excel solver to minimise the difference between the two solutions, by using solver to minimise the sum square error for the data points (which I have explained before and sent you links for)
5) solver needs to be set up to vary the parameters of the second system and may be also the initial conditions of the second system so that the solutions of the second system match the solutions of the first system
Hi Anirudh,
In short, from your 2nd post it seems that you would lke to fit input-output data (obtained from a 6th-order model) using a 3rd-order model. This is a standard problem. I assume you already have the data. Next step is to write a regresion equation (this is your 3rd-order model). Using this equation and the data, you can write a set of equations which might be linear in the parameters (even if your regression equation is nonlinear). Then just solve it using least squares. If your regression eqution is not linear in the parameters, then you will need a nonlinear estimator. From the estimated 3rd-orde ODE you may then rewrite it in state space. Notice that as your data are input-output, there is need to fret about the states. Now if you require that the states of the estimated model correspond to 3 (out of 6) of the original model, then things will get more involved.
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