Agree that utilizing AI to mine PoW of existing crypto currencies doesn't make any sense.
(for ex sha256 used in btc is designed so that it's computationally irreducible and so can't use ai to find a nonce faster than a miner that is just randomly trying out nonces).

But the thing I am experimenting is not using AI to mine existing PoWs.
It's to design a new PoW so that the forward pass (the hash, equivalent to sha256 in btc) is similar to a forward pass (inference) in a deep neural network.
Afaik not obvious that this is not technically feasible (let me know why if that's wrong).
The point of making the forward pass similar to inference in a neural net is so that miners that are optimized for that pow are also optimized at dnn inference.
Afaik such a pass can be deterministic, not approximate or probabilistic.
Mining is then finding inputs to the neural net such that the output satisfies some target difficulty.
Afaik, it should be possible to make the layers of the net so that it satisfies the same properties as a hash (evenly spread out etc).
In fact many existing hashes and ciphers are also based on rounds of diffusion (can be linear and permutations) and confusion (for ex s-boxes).

At the moment the way it works is model weights are generated randomly from the block header (256x256*64 weights). And then miners can search for nonces such that the output of the network satisfies target (for example by building a pytorch model initialized with random weights derived from the block header and then doing batched inference in the mining loop until find the right output).
The non-linearities (activation layers) between rounds are such that rounds are not easily reversible, evenly spread out etc. (same properties as a one way hash)

So the PoW doesn't involve any approximations or probalistic solutions, but just in experimenting with new pow such that computing the forward pass (hash) involves the same kind of operations as those needed for deep learning inference (so that optimized miners are also useful for that task).
The hash used in that PoW won't be any better than sha256, but should have similar properties and difference is that would require miners to be good at work that looks like deep learning inference. (matmults, accessing weights etc).