Choosing best predictors neural networks
A very quick way is to run some Tree-based ML model on your data, such as Random Forest or XGBoost. Tree-based models can return importance coefficients, estimating the relative explanatory power of each variable. You can implement a very large and deep ensemble of trees (we don't really care about overfitting at this point) so they return you the three strongest predictors. You can then take them and feed them into a Neural Network.

Another, more time consuming method is to run the model multiple timesa and substitute each variable, in alternation, with random noise with the same mean and standard deviation of the original variable. This perturbation method will tell you how much performance decreases when one variable is replaces by noise. This is accurate but very time consuming.