I am currently trying to find a machine learning algorithm that can predict about 5 - 15 parameters used in a mathematical model(MM). The MM has 4 different ordinary differential equations(ODE) and a few more will be added and thus more parameters will be needed. Most of the parameters can be measured, but others need to be guessed. We know all the 15 parameters, but we want the computer to guess 5 or even 10. To test if parameters are correct, we fill in the parameters in the MM and then calculate the ODEs with a numerical method. Subsequently we calculate the error between the calculations of the model with the parameters we know(and want to guess) and the calculated values of the MM for which we guessed the parameters. Calculating the values of the models ODEs is done multiple times, the ODEs represent one minute in real time and we calculate for 24 hours, thus 1440 calculations.
Currently we are using a particle filter to gues the variables, this works okay but we want to see if there are any better methods out there to gues parameters in a model. The particle filter takes a random value for a parameter which lies between a range we know about the parameter, e.g. 0,001 - 0,01. this is done for each parameter that needs to be guessed.
If you can run a lot of full simulations (tens of thousands) you can try black-box optimization. I'm not sure if black-box is the right approach for you (I'm not familiar with particle filters). But if it is, CMA-ES is a clear match here and easy to try.
You have to specify a loss function (e.g. the total sum of square errors for a whole simulation) and an initial guess (mean and sigma) for your parameters. Among black-box algorithms CMA-ES is a well-established baseline. It is hard to beat if you have only few (at most a few hundreds) continuous parameters and no gradient information. However anything less black-box-ish that can e.g. exploit the ODE nature of your problem will do better.