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I mean, based just on the abstract that's not entirely unlike what they're doing, except that they're using GP emulation.

Yes, and modern methods would be really easy to incorporate because of awesome open source libraries.

If you got really lucky, there could be public data from telescopes that had already recorded whatever regions of space were of interest, and evidence of planet nine might have been disregarded as noise.

Is this something NNs could do? The output might just be a probability field of orbits, which would not tell you anything more than known analytical methods would.

People are being conditioned to apply NNs to every problem, regardless of their suitability. Having to only think about a single tool makes everything easier.