A different angle from the sibling comment's: the hash functions used here are meant to approximate pseudorandom functions, which are functions that, while deterministic, literally have no structure at all relating their inputs to their outputs in any way.
Hash functions used in the real world haven't been proven to have this property (and there are various theoretical limits on how readily they could be proven to have it, and maybe on the extent to which they could actually have it), but in order to be widely adopted, a hash function has to pass every available statistical test for approximating pseudorandomness, and also has to resist mathematical analysis aimed at finding useful structure. That means that ordinary human intelligences fail to find a practical recipe for predicting properties of the output from properties of the input.
In the same way, we would expect that deep learning systems fail to find such recipes too.
On the other hand, it's not absolutely impossible that there are some kinds of regularities that a deep learning system might discover. If so, they would be considered very serious flaws in the hash function in question. But deployed cryptographic primitives have sometimes had problems like this. The best example that I know of is the RC4 cipher
https://en.wikipedia.org/wiki/RC4#Security
where there have been a series of statistical biases (which are often forms of correlation between input and output, which should not exist if RC4 approximated pseudorandomness well). Some of these were apparently discovered experimentally by researchers with some kind of hypothesis testing tools, as opposed to based on theoretical abstract reasoning about the mathematics of RC4. This makes me think that some kinds of deep learning systems might also have been able to discover those correlations, although I'm not sure that they would have been the most efficient methods for doing so. (An interesting test might be to try to use deep learning to find new correlations in RC4 that aren't yet known -- which seems plausible since researchers have repeatedly found new ones over time.)
I think there are interesting problems about what kinds of correlations and structures deep learning systems can or can't learn efficiently, and whether those are the kinds of correlations and structures that are likely to exist as genuine flaws within deployed hash functions. I definitely don't know enough about the mathematics of deep learning to appreciate how to begin answering this question; I only know that if it turned out to be useful in some case, it would mean that the application of human intelligence and existing statistical tools to assessing hash functions' security had dramatically fallen down on the job.