The inputs could be copyrighted and the weights could be copyrighted if creating the weights from the inputs is (legally) regarded as a transformative use. And I think it could reasonably be considered to be transformative - the weights don't look anything like the input data.
Disclaimer: IANAL. So far as I know, no court has ruled on whether this qualifies as a transformative use. I take no position on how the courts will actually rule. I merely say that they could regard this as transformative use. (But see jerf's "creativity" argument for another hurdle that weights must pass to be copyrightable.)
Transformative use doesn't necessarily mean copyrightable.
Google's thumbnails are a purely mathematical transformation on images (no copyright themselves), and yet are considered a transformative use.
I believe that trained models are similarly a purely mathematical transformation of {data}, but is transformative in what that can be used for going forward.
"Can" bearing a lot of weight in that sentence.
It's how the human, with agency, uses the model that may be a derivative or copyright infringing use - not the model itself nor necessarily the output.
The output of a generative AI may be similar enough to an existing work that it is derivative of that work. It is possible to construct a prompt that infringes on an existing work even if that work wasn't part of the training data.
For that case, consider you drew a picture. That picture that you just drew isn't part of any training data. I could presumably look at it and describe it with sufficient detail that something similar enough would be generated... and that may be considered a derivative work. The same test could be applied to me describing it to someone on Fiverr with the same outcome.
If I were to publish that work by the generative AI or Fiverr - who would be infringing on copyright? me? or the black box that may be AI or Fiverr that created a picture based on my prompts?
Another way to look at it is if a thing reproduced a data subjectively resembling originals and then you used it anyhow, then its non-transformative use, and methods used is just extra details.
The inputs could be copyrighted and the weights could be copyrighted if creating the weights from the inputs is (legally) regarded as a transformative use. And I think it could reasonably be considered to be transformative - the weights don't look anything like the input data.
Disclaimer: IANAL. So far as I know, no court has ruled on whether this qualifies as a transformative use. I take no position on how the courts will actually rule. I merely say that they could regard this as transformative use. (But see jerf's "creativity" argument for another hurdle that weights must pass to be copyrightable.)