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

>>9
the rules are the following:
1. the players move after each other in rounds(duh :D)
2. every player can select any point in the field, except the ones that are already taken and the OTHER players sidepoints (he can altough pick the ones in the corner)
3. if there are two points which are connected via horse move(i don't know the correct english word for it, but the picture talks for itself), and if this line is not crossing any enemy lines, then a new line is drawn.
4. the first player who can create a continous line between his two base sides(green player: verticaly, blue player: horizontaly) wins

that is all