My guess is that what I'm asking for isn't that complex and could be done by a few serious mathematicians in a few years. The dynamics of tanh(Ax+b) is hardly more complex than Naiver-Stokes equation or the modern topology theory.
Naiver-Stokes is much simpler because it operates by itself. Of course turbulence is hard but even there we usually care about its coarse features, we'd be content to throw away almost all the information provided the calculation of the wing's lift works out OK.