A neural network is a function approximator. You can think of it as a high-tech cousin to piecewise linear fitting. If you want to fit the most complex phenomena ever with a single parameter - you are going to get the mean and should not be surprised if it isn't infinitely useful. To get a useful fit, you must couple the nature of the phenomena being modeled with the NN. If you are modeling a planar surface, then you are going to need more than one coefficient (typically 3 or 4 depending on your formulation).
One of the questions behind this question is "what is the basis of fingerprints". By basis I mean the heavily baggaged word from Linear Algebra and calculus that talks about vector spaces, span, and eigens. Once you know what the "basis" is then you can build a neural network to approximate the basis, and this neural network will give reasonable results.
So while I was looking for a paper on the basis, I found this:
And here you go, a good document of the basis of fingerprints:
http://math.arizona.edu/~anewell/publications/Fingerprint_Formation.pdf
Taking a very crude stab, you might try growing some variation on an narxnet (nonlinear autogregressive network with external inputs) link. I would grow it until it characterizes your set using some sort of doubling the capacity. I would look at convergence rates as a function of "size" so that the smaller networks inform how long convergence takes for the larger ones. That means it might take a very large network to make this work, but large networks are like the 787 - they cost a lot, take forever to build, and sometimes do not fly well.
If I were being clever, I would pay attention to the article by Kucken and formulate the inputs as some sort of a inverse modeling of a stress field.
Best of luck.