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 made something like this (minmax search with heuristic, only expanded one child node with the biggest heuristic value, I hope you ment to say the same :D) and yes, it was able to search a lot better and faster, the only problem with this was that (and this comes from the game itself too) you have to think at least 3-4 steps ahead to get a "real life" stategy, and i couldn't make a good heuristic function for this (well this maybee my fault :))
the other thing i tried is that the algorithm didn't use all of the usable operators on a game state, just a few ones (i think only in a 6 point or so radius in every point), and at the beginning of the game it was useful, but as the game advance, it used some operators more then just once simply because there were a lot points selected, so it did more harm then it helped...