>When the exact theory is known and readily computable, it's hard to see science/engineering value in "learned" surrogates that merely approximate the symmetries.
I completely agree, which is why the approach I am taking is to only utilize surrogates to think which are unknown or do not have an exact theory. I don't think surrogates will be more efficient than methods developed that exploit specific properties of the problem. In fact, I think the recent proof of convergence for PINNs simultaneously demonstrates this might be an issue (there was no upper bound to the proved convergence rate, but the one they could prove was low order).
>More generally, it is disheartening to see trends that would conflate discretization errors with modeling errors, lest it bring back the chaos of early turbulence modeling days that prompted this 1986 Editorial Policy Statement for the Journal of Fluids Engineering. https://jedbrown.org/files/RoacheGhiaWhite-JFEEditorialPolic....
Agree, this is a difficult issue with approaches that augment numerical approaches with data-driven components. There are ways to validate these trained components independent of the training data (i.e. by using other data), but validation will always be more difficult.
