The 3 body problem (any classical mechanics where energy is conserved) takes place in a symplectic space:
https://en.wikipedia.org/wiki/Symplectic_geometry
If your integrator isn't built to respect the symmetries of symplectic space it is going to give qualitatively wrong answers as t -> infinity. A non-symplectic geometry might do a better job if you want to track the Earth around the sun for 1000 years, but if you use a non-symplectic integrator who knows where the Earth ends up in a billion years.
"Deep networks" are the new hypnotic induction phase to induce willing suspensions of disbelief. Maybe there is some "emperor's new clothes" going on. Usually if somebody makes a bogus claim people will think "the person making the claim is stupid", but if a person makes a bogus claim today involving a deep network, the reader tends to assume that they are too stupid to understand it.