Yeah. Guarantees about robustness are pretty useful.

That said, even if you didn't trust it, it might be possible to use the approximate alleged solution spat out by the neural net as an initial solution to a trusted solver with guarantees. In some cases you might be able to get a guaranteed result from the proper solver with the neural net solution used to accelerate convergence

Depends on the problems, but in many situations yes. For PDE's you can plug in the proposed solution and calculate the error. Usually you'd do this in a discretized version of the PDE, but since neural nets are symbolically differentiable you could instead approximate (by sampling) the average error from the real PDE itself.

"trust, but verify".

You can get decent error estimates for lots of these sorts of systems, with a manageable amount of extra computation.