First of all for the general case, some back of the envelope estimates suggest to me that 1/distance is a better weighting than my original 1/distance**2.
As for your specific function, you may find it worthwhile to do some transformations first. For instance if I understand your description, then log(F(x,y,z)) is roughly of the form K*log(log(x))/(y*z). So log(F(x,y,z))*y*z/log(log(x)) is roughly a constant.
This is good because the estimator that I provided is going to give the best results when approximating functions that are roughly constant. (Cubic splines, etc, give very good results at approximating functions that locally look like lowdegree polynomials.) And estimating this "rough constant" gives you (after reversing the above calculation) your original function F.
In general a judicious application of general theory and specific knowledge about your situation is more effective than abstract theory by itself...
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