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Is there any common/sound method to quantify (similar to T-test or F-test in regression models) the measures of influence and significance of terms in Artificial Neural Networks?

By terms I mean both the individual independent variables and interacting variables (e.g. terms like $x_i$ and $x_i x_j … x_k$ where $x_i$ are input variables).

In another word this question can be reduced to "feature selection" when we have only the Neural Network structure of the trained data.

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If you are specifically interested in neural networks, then this is what you do: you train the network with whatever many features you want, and then look at the magnitude of the weights in the first layer; the larger the magnitude is, the higher is the impact of that feature. But, if you're interested in knowing what features are the better ones to be constructed and then used in a model, then you're asking a question which is known as "structure learning". Structure learning problem is not completely solved yet, but there are many progresses towards some solutions. For example, this paper uses decision trees for structure learning: http://www.jmlr.org/papers/volume15/lowd14a/lowd14a.pdf

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