Here are a few thoughts I haven't formulated before:

It seems clear enough to me that training AIs on copyrighted works is typically or commonly a fair use under existing law, because the AIs can and commonly do learn non-copyrightable elements and aspects of those works. It's very obvious from enormous numbers of examples that current AI systems are capable of learning much more abstract features of human culture (grammar, concepts, facts, cultural tropes, and many others).

A human being doesn't violate copyright in learning from a copyrighted work, including when that human being is later more able to produce other works based on that learning (e.g. reading fantasy novels and learning concepts, tropes, or vocabulary that one uses to produce other fantasy novels; reading a newspaper and learning facts that one incorporates into an essay; learning artistic techniques or stylistic conventions from studying existing artworks and using them when producing new artworks). Current AI systems are (amazingly) becoming capable of all of these things and may do them in ways that are somewhat akin to how human beings do them. (although I guess Jaron Lanier would object "that's what they want you to think")

But there are also examples in existing copyright doctrine where people accidentally repeat enough of a prior work to get in trouble for infringement -- most often with song composition (like George Harrison's "My Sweet Lord") because relatively small pieces of melody (which a person might easily memorize) may be considered copyrightable.

If human beings had much more accurate memories, copyright would be quite a bit more intrusive (and/or quite a bit less effective) because, following any exposure to some kinds of works, we could use our own memories to reproduce those entire works from scratch for our own use or pleasure without obtaining authorized copies from elsewhere.

Computers do have such accurate memories, and machine learning systems, which are optimized for things like maximum likelihood estimation, can and do reproduce both copyrightable and non-copyrightable elements of works that they've been trained on. After all, the maximum likelihood continuation of a fragment of a text or a song is ... the complete original work. And the ability to reproduce the complete original work would, other things being equal, reduce loss in training. After all, that's something someone might specifically ask for, and if the system could oblige, it would be doing a better job of providing what the user wanted.

It's relatively foreseeable that machine learning systems would potentially be able to reproduce both copyrightable and non-copyrightable elements of various works, because the distinction between the two isn't especially clear from an algorithmic or mechanical point of view. (For instance, facts aren't copyrightable, but the notion of what constitutes a "fact" for this purpose is a culturally-bound legal notion and not at all straightforward to make precise.)

But if you had a human author or artist or scholar or programmer who was "trained on" exposure to an enormous body of works, and that person had an exceptional eidetic memory, you could imagine that he or she would be perfectly capable of recreating many of those works from memory (and that other people might request such recreations). (Again, in music in particular, it's already routine that someone could have unambiguously copyrightable material memorized and be subject to copyright restrictions on performing songs. Like if a singer or band performs a cover from memory.)

If you wanted to avoid this ability then you might need to build in an explicit notion of copyright that limits the accuracy or level of detail inside of the model in some way. This is tricky because (1) I don't think people have really tried to do this much so far, (2) copyright applies very differently to different categories of work, (3) it obviously wouldn't satisfy critics even if it mitigated the most extreme examples of "regurgitation", and (4) it would be kind of weird because you would be intentionally limiting the quality and extent of learning that the system was allowed to do. (I imagine Jaron Lanier getting mad again about my repeated comparison between human learning and machine learning, and between human memory and machine memory)

Some of the weirdness in point (4) is that accurate prediction is usually cool / great / impressive / accepted as an appropriate goal or capability, but if it's too accurate in certain contexts, it may be deemed a copyright infringement. Like if you said "what word comes next? FOUR SCORE AND SEVEN YEARS AGO OUR FATHERS", there's a clear correct answer and knowing it requires having a certain text memorized. OK, if you said "what word comes next? MR. AND MRS. DURSLEY OF NUMBER FOUR PRIVET DRIVE WERE PROUD TO SAY" ... same thing, but Bloomsbury Publishing may be unhappy if you have a system that can get all such questions right.