I am a statistician, and neural nets are in essence nonlinear regression models that predict some target variable Y based on a complex function of several input variables, say a vector X. So they are usually used to identify systematic relationships between some target and many possible predictors.

These papers describe a creative but non-standard use that I'm not sure has much use in POW mining. In mining, Y is a randomly generated target number that has to be hashed with some function, with solutions reflecting randomly generated numbers hashed through the same function then compared to see if they are < Y. There is nothing particularly special about using neural nets for the hashing, because one is just trying still by luck to obtain a solution smaller than the target. There are no systematic regression relationships among variables to be modeled with the nnet. If there were systematic predictors of the target it would actually be much easier to hit it.

In proof of useful work, real models with real data might be substituted, but it is a different issue than using nnets to construct a hash function which seems computationally wasteful.