What are some methodologies for performing feature selection for simple feed-forward neural networks? In multiple linear regression there is an F-test which can be used to evaluate whether or not a covariate has a meaningful impact on a model. This is typically done through either a forward selection or backwards selection algorithm. Does such a meaningful process exist for neural networks as well? The only reason I ask is because working with neural nets is an inherently stochastic process, so I do not know how I should try and get accurate bounds for the F-statistic.
I usually try to do the following process, I don't know whether it has name or not:


Try to find features which an expert can say what the label is without hesitating. 
Plot correlation matrix for your sample to investigate whether each feature has correlation with others or not. I do this process whenever the input features are so much. Then try to reduce the correlated features or try to apply PCA. Consider the point that PCA alone does not care about the labels, so it may find principal components that your data would be so difficult to be separated but not necessarily.