The underlying assumption of all of modern cryptography is that one-way functions exist (implying P != NP), and further that the algorithms we use in practice are actually instances of these theoretically hard problems. So there provably always exists an algorithm that breaks any given PRNG, it will just always take you very long to compute and/or a lot of queries to the PRNG. ANNs are nothing special in that regard, they are just algorithms and have the same theoretical lower bound on their runtime.

So yes, an absurdly large ANN will break any PRNG, but so will other absurdly long-running algorithms, some of which are quite trivial: Just try every possible seed until you get the same sequence as what you observe, repeat until only one candidate seed is left.

EDIT: To add to this, you seem to be referring to the universal approximation theorems for ANNs. These theorems state that for any function (subject to some conditions not relevant here) and any arbitrary approximation ratio, there exists a finite ANN that approximates the function to within the desired approximation ratio. It says nothing about whether it's possible to train such an ANN, merely that it exists. Which in this case is a fairly trivial results, you could feasibly create a look-up table of PRNG sequences to seeds and encode that as an enormous ANN. But finding/training this ANN is prohibitively expensive.