Java / AI / ANN

so, here is my problem:

i'm currently developing a two player logic game in university, and i'm facing a huge problem. The game is a state space representated, and I alredy have Alpha-Beta pruning and MinMax algorithm implemented and they are working fine.
The root of the problem is that my gametree is extra large. It is a 20*20 square grid field, which is actualy a 20*20 JRadioButtons. In every turn, a player can almost select any buttons (not the ones that are already selected and a few other ones), so my searching algoritms create a huge gametree, with only a depth of 4-5 (which is really low) there could be billions of treenodes, and it takes forever to go through all of them, even the A-B pruning fails.
So someone gave me the idea that I should use neural networks, so that way I would be able to do something in real time, but i have no idea how should I even start it and what should I do. I've read a lot about ANN, but still, I have no clue how to begin this. I've  only found a game which implements an ANN for the AI, but it's in C++ and I could't understand it perfectly.

So, any ideas what should I do, where should I begin or WTF to do?

thx

if you want a pic of the game I can show you 1 if you guys didn't understood a word what I just wrote :D

i've worked with ANN before, but it had nothing to do with this (it just took a lot of 3D points (like from a 3D scanner), and it draw the approximated Bezier surface, not that big of a deal), and that was pretty simple and I thought when I heard the word neural network that i could do this easily, but i couldn't be more mistaken
the reason the person who recommended ANN said that the network could take a huge benefit from the fact that the game tree is so large and could be really efficient, i don't know if that is rigth or not (well my gut says yes), but if everyone is talking me out of it I belive that it's realy that difficult