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I use software for pollutant propagation on rivers that takes as input a set of parameters ($p_1,p_2,\ldots,p_n$) and creates an output file which is basically a matrix where on each row the concentration of the pollutant in various places along the river at a given timestamp is given.

\begin{array} {|r|r|r|r|r...|r|} \hline \text{TIME} &500\,\text{m}&1000\,\text{m}&1500\,\text{m}&2000\,\text{m}&2500\,\text{m}&...&25000\,\text{m} \\ \hline 2015/12/07 - \text{4:50:00}&0.75&0.71&0.6&0.58&0.55&...&0.12 \\ \hline \hline 2015/12/07- \text{4:55:00}&0.71&0.70&0.58&0.56&0.51&...&0.10 \\ \hline \end{array}

My question is: what models/theories are suitable to be used to predict, based on a set of given INPUT parameters, the OUTPUT that would be closest to what the software will give as a result?

Also how many pairs INPUT/OUTPUT will I need to have in order to minimize the error?

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    $\begingroup$ It's not very clear what your input parameters represent. Could you clarify this? It seems like they could be initial conditions for the pollution at various points along the river. $\endgroup$ – Tyler Olsen Oct 29 '15 at 16:06
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    $\begingroup$ If I understand your question it sounds like you have some software that models the pollutants in the river over time. If this is the case, why do you want to find a separate model that approximates what your software gives you already? Are you just trying to make a simplified model? Is this an interpolation problem? If you are trying to make a simplified model then do you know the water velocities at points in the river? $\endgroup$ – James Oct 29 '15 at 19:57
  • $\begingroup$ @TylerOlsen The input parameters are: startDate, endDate, Polutant Type, Pollutant Concentration, River Chainage and a set of pairs [Pollutant Quantity, DateTime] representing how much pollutant was thrown in the river and the moment when that happened. $\endgroup$ – Sorin Ciolofan Oct 31 '15 at 10:58
  • $\begingroup$ @James The software we have runs on a server and we need internet connection to connect to that software and run a simulation. Beside this, the server could happen to be down at some point when we need to run a simulation. So, as a backup solution we would want to "guess" , given an input i what is the closest output o that the software would give it would have the chance to run. We can use a historical archive of past N real simulations (pairs of input/output). Maybe I can use machine learning? $\endgroup$ – Sorin Ciolofan Oct 31 '15 at 11:02
  • $\begingroup$ It sounds like you're just trying to solve the transport equation for a scalar field (aka: convection-diffusion-reaction equation). This is a PDE that describes precisely your situation. If I had to guess what your software was doing, I'd say that it's solving a 1D version of this model, taking your input parameters as initial conditions and making assumptions about the river speed at every point. Check out this wiki page for more info: en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation $\endgroup$ – Tyler Olsen Oct 31 '15 at 15:10
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The question on the minimal # of data points from the second code, the "forward" solver, is a very broad one. It depends on the complexity of the underlying phenomenon.

In the simplest case, the flow is laminar and the pollutant particles are effectively "infinitesimal"; I guess you can approximate this with Fick diffusion (+ mass transport, so a convection-diffusion law) or something similar. The output sample then would then follow a Gaussian law, and there are results from math.statistics and optimal learning regarding the convergence rate of the estimators of such distributions as a function of sample size.

(well, without going too deep, the Markov and Chebyshev inequalities could be used: https://en.wikipedia.org/wiki/Chebyshev%27s_inequality#Finite_samples)

In a realistic case, the flow might be turbulent, so the statistics of pollutant transport become much more complex. I can't easily say, but a combination of statistical estimation and numerical PDEs (this sounds like an "uncertainty quantification" type of problem) will help you.

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