The main goal of mechanics is finding solutions for important variables of interest, from which every other dynamical variable can be easily computed. In fluid mechanics, the main objective for a given problem is finding a velocity field (once you have this, you're done). Since the NS equations are non-linear partial differential equations, finding exact solutions is impossible in most cases. Some notable non-trivial exceptions exist [1]. Navier-Stokes is extremely successful in describing flows in many real physical situations. It is believed, but has not been experimentally confirmed, that Navier-Stokes successfully describes turbulent flow. Most physical theories have a fairly well known domain in which they are applicable and not applicable (Newtonian physics breaks down at velocities near the speed of light). Currently, we don't even know if solutions to NS always exist, never mind make physical sense [2]. With this in mind, we have to acknowledge the uncertainty of the correctness of NS in describing an arbitrary flow. It is backed by well-grounded theory (parts of the equations can be derived from conservation laws) and works under many different conditions. This comment landed up longer than I expected, but my main point is that the question can we "reliably simulate fluid behavior" for an arbitrary fluid - is largely up in the air [3].

However, we should restrict our attention to flows that we very strongly believe are correctly described by NS. From the original paper:

>> The main problem of our method is the same as the weakness of all machine learning approaches; the learning methods are not capable to extrapolate the model far outside the data observed during training.

The training data are existing numerical simulations of specific fluid flows. The learning algorithm learned fluid flow dynamics from simulations. This is extremely impressive. It also achieves a significant speed up in simulation time - which is also very impressive. However, as the authors say, it does not generalise well. So, in short the answer to your first question is probably "sometimes yes, but in general not really".

edit: I should add that the simulations in the paper are decidedly based on Navier-Stokes.

[1] https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_flow_...

[2] https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...

[3] https://en.wikipedia.org/wiki/Turbulence