I'm an author of one of the arXiv papers above. One thing to consider is that the approximative power of a given parametric function is not the only criterium. Being able to optimize that function efficiently is as important. Neural networks excel in this. So the comparison you ask for most likely won't appear, because any other parametric ansatz with tens of thousands (or more) parameters would be impossible to optimize. At the least that's the case in quantum Monte Carlo, the domain of our paper. As for "provable", I also don't think that will appear. All the exact theorems about neural networks are way too abstract to be applicable to practical problems.
