Agree in general, though I have read about cases where a neural network was reverse engineered to create a solution that did not require one.

I get what you're saying, but this is not immediately obvious.

For example, control systems also adapt to data and are often robust to noise, and they're understood mathematically very well.

I don't necessarily disagree, but you would need some more justification.

To be a little pedantic, I’d use the word approximate rather than solve.

To be even more pedantic, I'd use the phrase approximate the solution ;-)

Nice