Much physicist math can't be made rigorous so easily! Which isn't to say that much of it doesn't still have great value.

However the math in AI papers is indeed different. For example, Kingma and Ba's paper self-presents as having a theorem with a rigorous proof via a couple of lemmas proved by a chain of inequalities. The key thing is that the mathematical details are purportedly all present, but are just wrong.

This isn't at all like what you see in physics papers, which might just openly lack detail, or might use mathematical objects whose existence or definition remain conjectural. There can be some legitimate problems with that, but at least in the best cases it can be very visionary. (Mirror symmetry is a standard example.) By contrast I'm not sure what 'spirit' is even possible in a detailed couple-page 'proof' that its authors probably don't even fully understand. In most cases, the 'theorem' isn't remotely interesting enough as pure mathematics and is also not of any serious relevance to the empirical problem at hand. It just adds an impressive-looking section to the paper.

I do think it's possible that in the future there will be very interesting pure mathematics inspired by AI. But it hasn't been found yet, and I'm very certain it won't come from reconsidering these kinds of badly-written theorems and proofs.