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I want to make a model with Neural Network. I have one dependent variable Y and many independent variables X_i. I want to choose for example only 5 vairables who describe my model most. One of the methods I think I can use, is Stepwise regression. Can I do that? And what method else I could use to choose only some of independent variables?

If you have a large number of potential independent variables, stepwise method may be a poor choice. Search for LASSO or LAR methods...