You can work out an adaptive estimator.  See our paper. 

Article Parameter identification for uncertain linear systems with p...

The problem you are addressing is an inverse problem (http://en.wikipedia.org/wiki/Inverse_problem).  The is a huge amount of literature in the field, but unfortunately, usually, if the direct problem is well posed (i. e. given the parameters (e.g coefficients, initial values, boundary values),  obtain the solution of the equation), the inverse problem (i. e. given the solution of the equation, obtain the parameters) is ill posed.

They usually involve the solution of an integral equation ( First kind Fredholm) which are ill posed. That means that a small change on the solution of the differential equation can lead to very different values of the parameters. There are also regularization techniques (Tichonov regularization) for dealing with such ill posed problems.

I am afraid there is no single technique that can be used for any problem and possibly you will have to develop one for your particular case.

Dear Rafael, I thought that if data are in much greater availability than the number of unknown parameters, then a few methods should be able to get estimates of the parameters, as it happens outside the field of differential equations, for example in least squares regression.

To be more precise, I have in mind arguments like the one suggested in the following papers attached. Zigang Pan says that adaptive estimators are one of the possible answers. Any more?

I've recently been working on a problem of this nature. I've been trying to estimate parameters for a Ca2+ extrusion and sequestration model from experimental data consisting of about 20,000 measurements of fluorescence as a function of time and 6 parameters. I'll share my thoughts, but I can't promise that you'll find them particularly insightful, let alone useful.

If your system of differential equations admits an analytical solution, then the problem obviously reduces to solving a nonlinear model. You also noted the obvious and common example of least squares regression. Of course these problems are already non-trivial in the sense that even with an analytical expression for the loss function, we have to resort to numerical optimization procedures. 

Most systems of practical interest will not admit an analytical solution. Conceptually this doesn't pose a very large hurdle. All we do is replace a loss function which is an analytical expression with a loss function that we have to compute with numerical methods. Thus as the abstract for the Xun et. a. paper suggests, "methods for estimating PDE parameters require repeatedly solving PDEs numerically under thousands of candidate parameter values." That might be a little bit of an exaggeration; it's probably not necessary to solve the system for "thousands of candidate parameter values," unless you have a very high dimensional parameter space or are foolish enough to try an exhaustive search of the parameter space.   

Although conceptually simple, the additional computational burden is significant. It takes significant computational resources to solve the associated numerical optimization problem when each iteration requires the numerical solution of a system of differential equations. 

However, it is difficult to over estimate the value of the conceptual simplicity. In part because the computational burden is so large, it is not difficult to find newer methods that seem to promise better performance, such as the recent paper you cited. I have found that in practice, the promised performance gains are irrelevant, as I am typically unable to implement the methods properly, but your mileage may vary.

Have you tried the least squares technique described in our paper "Parametric identification of the model with one predator and two prey species" available for download under my profile.

Prof. Tom Lahmer from Weimar is trying to solve those Problems. - You can find him here at RG.

You may want to check out "Practical Methods for Optimal Control and Estimation Using Nonlinear Programming" by JT Betts. In my experience the method he proposes works very well if you do not have too many states and/or parameters in your model.

You find in this reference list some methods for identifying differential equations parameters used in various domain

1- B. Calderhead, M Girolami, and N.D Lawrence.

Accelerating bayesian inference over nonlinear differential equations with

gaussian processes. In Advances in Neural Information Processing Systems 21 - Proceedings of

the 2008 Conference ,2009.

2- M.B. Elowitz and S. Leibler.

A synthetic oscillatory network of transcriptional regulators.

Nature ,403(6767):335–338,2000.

3-W.R. Esposito and C.A. Floudas.

Deterministic global optimization in nonlinear optimal control problems.

Journal of Global Optimization ,17:97–126,2000.

4- R. FitzHugh.

Impulses and physiological states in theoretical models of nerve membrane.

Biophysical Journal ,6:445–466,1961.

5- R. E. Fuguitt and J.E. Hawkins.

Rate of Thermal Isomerization of a-Pinene in the Liquid Phase.

J.A.C.S ,319(39),1947.

6- A. Gelman, F. Bois, and J. Jiang.

Physiological pharmacokinetic analysis using population modeling and

informative prior distributions.

Journal of the American Statistical Association ,91,1996.

7- O. Ghasemi, M. Lindsey, T. Yang, N. Nguyen, Y. Huang, and Y. Jin.

Bayesian parameter estimation for nonlinear modelling of biological

pathways. BMC Systems Biology ,5,2011.

8- H. Miao, X. Xia, A. S. Perelson, and H. Wu.

On identifiability of nonlinear ode models and applications in viral dynamics.

SIAM Review ,53:3–39,2011.

I think you may do sensitivity analysis which has been easier I feel. You can try sensitivity analysis proposed in many epidemic related papers. At present I don't have time give brief literature. But I know many experts have done sensitivity analysis for their problems. I also suggest this sensitivity analysis may solve your problem.

Many thanks for all your very useful replies. So far they've been extremely useful. I'll try and summarize what has been useful to me once have been able to see through more clearly.

Dear James,

please ask Hans-Georg Bock, University of Heidelberg . He has developed one of the best software for that purpose (direct multiple shooting). He also works on experimental design which might be also very usefull for you.

Best regards,

Hans Josef

James,

Another possible answer could be found in the solution methods of Inverse Problems of Differential Equations. These methods are different depending on the solution information you supposed to have from experimental data.

It could me some spectral data or other solution's functions. Usually these methods require regularization techniques.

Lev

If you are also interested in the theoretical point of view, you may want to look at this paper DOI: 10.1002/cpa.21453. This work was mainly motivated by the so called hybrid inverse problems. You can find similar problems and techniques from my contribution page.