Data model with more outputs than inputs?

I am working on parametric studies in physics simulations, i.e. I vary some real input parameters (e.g. x0,x1,x2,x3) and get an output with a larger size (e.g. y0,y1 … y100). Assuming that I have a database of some thousand different input parameters and corresponding outputs, is there a good way to build a model that can give a prediction for the output at a new position?

I have looked into various techniques, but so far I couldn’t find a method that seems promising for this kind of problem. I found a lot of stuff on classification problems and for one output it seems that Gaussian process regression would be suitable. But I am pretty lost when it comes to floating-point input and an output that is larger in dimensions than the input. Any suggestions or references to papers addressing this kind of problem?

Edit: I understand that my description might have been a bit too abstract, so here’s the concrete problem: I have a laser physics simulation with a some input parameters that I am changing, e.g. the coefficients of the spatial wavefront and coefficients of the spectral phase. These change the spatio-temporal intensity distribution in focus and I am looking at the resulting dynamics. One observable is the momentum distribution of heated electrons (which is a histogram of ~100 bins). I can run each of these simulations fairly quickly (on a scale of 5 minutes each), but my parameter space is rather large with 5-10 different values to tune. So I am looking for techniques to perform a multi-dimensional regression to see what the influence of each parameter is within the parameter range. Ideally I would like to be able to make a reasonable prediction how the momentum distribution should look like at an unexplored position based on the knowledge of previous simulations …