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1

That equation is just an assumption that we make about the relationship between a response variable (aka dependent variable) $y$ and a predictor (aka independent variable) $x$, i.e. the response variable (target) is an unknown function $f$ of the predictor $x$ plus some noise $\epsilon$ due to e.g. measurement errors (caused e.g. by damaged sensors). So, if ...

3

Not necessarily. The neural network (or whatever else you use) is a model of what you are trying to do, and usually models are not able to perfectly model reality, as it is too complex. A noise term is generally used to represent that, ie the imperfection of the model's relationship with the actual world.

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