The second paper you are referencing is not quite what you think it is. It is the application of a neural network to the processor itself, to assist in performing physical register renaming, at runtime, in order to achieve out-of-order execution. This is a very different problem than a compiler's register allocator (for one you're dealing with hundreds of potential registers vs like, 16-30.)

On that topic, NN for branch prediction are, in some sense, nothing new; perceptron and perceptron-based designs have existed for at least 20 years. But I'm not aware of anything concrete or specific in current BP designs as of recently (beyond marketing hype) but I haven't kept up with it; maybe some variation of TAGE with model-assistance is out there, but I'm not sure.

I do not know what a training set or neural network model for performing register allocation on real-world programs would look like at this moment.