there is no support from StataCorp in this so you will have to depend on user written commands. In my opinion, at the moment, Matlab is the best tool for this (although expensive), followed by R.
There is currently no program written to perform neural network modeling in Stata. However a colleague of mine is currently writing a program for running regression trees. I am attaching his presentation from the 2014 Stata User's Conference in Boston
I made a simple Google search (https://www.google.com/search?client=safari&rls=en&q=neural+networks+in+stata&ie=UTF-8&oe=UTF-8) regarding NN and Stata. It has a rather limited support and you might, as colleagues suggested, use Matlab or R. Additionally Mathematica has excellent NN embedded toolbox. R as you well know is free and presents itself as a natural solution. Here please find a comparison of a number of professional packages and what they support.
I just remembered that there is another user-written command in Stata that you may find helpful. chaid performs Chi-square automated interaction detection, and thus it operates much as any other decision tree/regression tree approach. You can download the program from within Stata by typing: ssc install chaid