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tinymind
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Here's a resource-efficient Neural-Network library that I made1 specifically for MCUs, which I think you will find pretty interesting. Here's a simple xor-circuit NN example and here's a more advanced NN that predicts handwritten digits2 on an arduino UNO. Those combined with this research, I believe that will answer many of your questions and get you started.
Take a look at TinyMind: https://github.com/danmcleran/tinymind