I’m way out of my depth here but if I’m reading this right a flow model would not base outputs on a prompt but instead, after it’s sampling/inverse flow pipeline (if I got that right?) is activated it would somehow just output whatever image it’s statistically sampling at the moment. Is that even close?

It's also been a while since I had to flex my math muscles and they are quite flabby. I recommend reading things like this slow, they are written by people who know the field forward and backward and aren't supposed to be trivial. If you aren't in the field you got to be a little bit of an active reader.

For example, I also didn't remember why you needed a determinant in the equation you linked so I made a simple example and saw the determinant was there to keep the density normalized, (i.e the indefinite integral of the density = 1).

In this case, the simple example I used was a distribution q0(x) = 1 where x>=0 and x <= 1 (zero elsewhere) and scaling transformation y = ax

People who have not been exposed to math heavy content on a routine basis often have the unrealistic expectation of understanding everything on first read. That almost never happens unless the paper is very close to the reader’s own field of expertise. Denser papers, in fact, may take days of effort to fully internalize even for someone mathematically inclined and accustomed to reading papers.

$q_0$ and $x$ are explicitly defined.

This is "misleading" because it's presented as a blog post but really it should be read as a paper or book section. Take your time. Take a pen and paper out. Reread things multiple times. Go back to definitions. I don't mean mean that dismissively, math is just read very very very differently than natural text. ( I'm sure there are just brilliant people who can read math like a novel but for me, and I assume most people but maybe I'm just dumb, that's not the case )

Here q_0 is some density, whatever you want. x is a random variable distributed according to q_0

Per the text: x is a sample (see the tilde symbol) from the density q0; the Jacobian comes from the change of variables in the density. The typesetting could use some work but the definitions are in the snippet you pasted. I agree that people are often sloppy when writing though this is not such a case. If you want to get into this area of research, it still helps to read a bunch, especially the earlier works or review papers, and you tend to pick things up gradually. Some basic math background is required, probably at advanced undergrad level.

it would be nice if there was some way, like a map of what concepts you needed to know, that you could follow, so if you did want to get into this you could ram up from nothing (or at least high school maths)

There is! Start with a textbook. Most books (and practically all good ones) will include a preface or note from the author at the front saying “we wrote this with the intent of being accessible to people who know XYZ.” The best even say “if you don’t know that, we refer you to these books for those prereqs.” It’s a fantastic set of resources, but people seem to always want to consume a million blog posts instead of a couple books!

Textbooks will get you foundations, and then for more up to date stuff, you need to read academic papers. Start with survey papers and literature reviews on your topic, and when you can’t find one, start with any old (new) paper on the topic and read the related work section (all papers have them). You’ll have to learn to distinguish the “we’re building off of XYZ” from “ABC is kind of like our stuff,” because XYZ is the ‘prereq’ work. Then just go recursively until you hit stuff you understand from the textbook foundations.

Most academic resources contain pointers to their prereqs, so use them!

If you specifically want pointers where to start, this is my field so I’d be happy to point you to a path if you like.

Both x and q0 are clearly defined there. In fact the post is very well written. Are you sure you are ready to learn “this stuff”?

... it says exactly what they are, q0 is an arbitrary density on R^d, and x is a realization from that density, p1 is the transformed density. This is intro graduate level probability.