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Although I can't fully explain it in any logical notation, I think I can see how neural networks can save memory. They're not actually "storing data" in them, but they are storing "patterns of data flows". This requires way less storage capacity compared to saving every single image or text as would have been the case if this was a look-up table function.
But fact remains, that when using your brain, you can in fact recover all "data", so this representation in neural networks is only representing the external structure.