This is argument is complete nonsense. Navier Stokes has a rigorous derivation based on extremely high-fidelity assumptions, such as conservation of mass, momentum, and energy. We understand these assumptions, and we understand regimes in which using N-S would result in catastrophe (such as rarefied gases, relativistic velocities, etc.)
Neural networks require data. The Navier stokes does not need to be 'trained'. Deep networks have very little a-priori knowledge baked-in (from a baysian perspective there are priors such as translation invariance that are intrinsic). They are admittedly extremely useful, because they are high dimensional (and so are universal approximators) and can be trained efficiently.
Furthermore, you can develop an intuitive approach to many fluid flows. I can provide a much better estimate of the drag profile for a given wing geometry than an untrained person. No such analog is possible with deep nets, which as significantly more opaque in terms of dynamics and non-linear response.
The only way his comments make any sense is if you assume he isn't talking about physical models, like Navier stokes, and instead consider turbulence models, such as RANS or LES. These are parameterized models, and are used for turbulence modeling. They also have little physical intuition. However, this is not the same as saying we do not have high confidence in physical-based systems such as Navier stokes.
Source: I have a PhD in CFD and several ML publications.