Even if we were extremely generous would be a billion in hardware at the very least (I am assuming $500 manufacturing cost)
Theoretically, I can spend that much to make it's value go to $0. (yes, there are liquidity and other concerns, but you get the point)
The Market Cap of Bitcoin is $184,000,000,000
I have plenty of money to play with.
Moreover, to gain $184B by shorting Bitcoin you'd need to find someone willing to loan you $184B worth of Bitcoin, and then somehow sell all of it without crashing the price or drawing unwanted attention to your activities. Of course if the price doesn't react the way you expect you could end up paying a lot more than $184B to buy it all back to repay the loan. You shouldn't assume that even a successful (but expensive, and obviously temporary) 51% attack will drive the price down to zero.
That is, the attack you're proposing would require commandeering 15% of the world economy and more than doubling world electrical generation capacity.
Bribing or backdooring ASIC miners is a far more likely vulnerability.
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.
Most people think that this is impossible, but it has not been proven yet. Unless you had some reason for believing the technique would work, it's probably not worth the effort to try.
A sort-of precedent for that kind of problem which I mentioned in my sibling comment is the correlation weaknesses in RC4. While they're not the most powerful possible break of RC4, by any means, they are unanticipated flaws in the structure of RC4 specifically, and they might well have been discovered by software tools that can't solve NP problems in general. It seems to me that we don't have security proofs for symmetric cryptography at all, including for block cipher security properties as well as hash function security properties, and so while your observation is totally right in general, in any specific case it might just turn out that the cryptographic primitive we were using was weak in an unanticipated way that's specific to that class of functions.
Compare https://en.wikipedia.org/wiki/Random_oracle (although reading that article reminds me of how much I don't understand about this topic)!
As schoen pointed out, no common hash function, including the ones used in Bitcoin, has a proof of NP-completeness.
Moreover, none of those hash functions involve factoring numbers, and factoring numbers is also not known to be NP-complete, although it is also not known to be tractable in polynomial time. One reason commonly-used proof-of-work functions do not involve integer factorization is that, while integer factorization is not known to be doable in polynomial time, there are a number of algorithms that require subexponential time, so an integer-factorization-based proof-of-work witness would be much larger than an equivalent hash-function-based proof-of-work witness.
Also, it is not the case that efficient integer factorization would completely break all current encryption. Not only do no commonly-used hash functions depend on it, neither do any commonly-used symmetric ciphers (such as AES), and the currently-most-popular asymmetric cryptosystems also do not depend on the difficulty of integer factorization; instead they depend on the difficulty of the elliptic-curve discrete logarithm problem.
ECDLP is also not known to be NP-complete, but the currently-known algorithms for it are much worse than currently-known algorithms for integer factorization, so elliptic-curve cryptosystems require much smaller keys and less computation to resist the known attacks than integer-factorization-based cryptosystems.
As little as ten years ago, algorithms that could be broken by better integer factorization algorithms were relatively much more important than they are today, because elliptic-curve cryptography was much less widely used. Many vulgar accounts of the situation intended for the ignorant are not up to date.
Finally, it is not true that a proof that P=NP would "completely break all current encryption", for two reasons. First, it might not be a constructive proof — it might show that a polynomial-time algorithm for factoring integers, solving ECDLP, or computing hash preimages exists without actually telling you how to compute it. Second, even a constructive proof of P=NP might not provide an algorithm that was adequately efficient — if it takes O(n²) time to encrypt and decrypt, where n is the size of a key, but O(n⁸) time to break a message or a key, you might be adequately safe with, say, RSA-4096. But an O(n⁸) algorithm for 3-SAT would definitely be a constructive proof of P=NP.
(Shor's algorithm on a quantum computer can break RSA, because it does depend on integer factorization, in O(n³) time, if quantum computers can exist, which they probably can. This would not make it impossible to do RSA encryption securely, but it would require much larger keys than are currently used.)
However, your fundamental point is that a successful attack on Bitcoin's hashing algorithm, using artificial neural networks or anything else, would be very surprising and have major implications, because that proof-of-work scheme is designed to require exponential work, and as far as anyone knows, it does. And that fundamental point is correct, even though you have made a number of errors in your supporting points.
SO these machines are made to specifically mine bitcoin, other ASICs do other jobs, but individual units cannot transfer from one job to another and cannot be re-purposed.
Finally, you certainly cannot buy off the ASIC miners with a few million, not least because they own billions of dollars worth of ASICS that can't do anything but mine Bitcoin, and a successful state-orchestrated attack on Bitcoin would make all of them worthless. I don't think a state government taking down Bitcoin by its own rules is completely impossible, but you don't seem to have any grasp of the scales involved.
And in the end, after your massive investment, the community would just make some trivial change to the protocol and completely ignore the attack. Really, if this sort of thing was so easy then everyone would be doing it. You're certainly welcome to try.
So around $12 billion (plus the datacenter, operations, etc). You'd pretty much send BTC to being worth $0...and then you're stuck with $12 billion in useless hardware.