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I mentioned the example of bAbI. If you already know how to go from the input data to the answer, you can express the process as a parametric formula and have your network learn the parameters. Most of the work was done by you, not by your network.
Another way to inject prior knowledge is by adding to your network differentiable operations that you have identified as building blocks of the solution (e.g. if you know the solution involves cosines, you can add cos ops to your architecture).
And you say there are 10,000 papers (approx.) doing this?
Nearly every "architecture" paper does this.
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