I would be interested in collecting a bunch of examples of mathematical modeling of progress. I think there are probably several of these here, but I don't expect to be able to find all of them myself. I'm also interested to know about any models like this elsewhere.

I was reading the LessWrong 2018 books, and the following posts stuck out to me:

• 1960: The Year The Singularity Was Cancelled
• Hyperbolic Growth
• Takeoff Speeds

The Year The Singularity Was Cancelled talks about a model which predicted world population quite well, by supplementing a basic population equation with a simple mathematical model of technological progress. To summarize: population carrying capacity is assumed to increase due to technological progress. Technological progress is modeled as proportional to the population: a particular population $p$ leads carrying capacity $c$ to have a derivative $pc$. (This reflects the idea that carrying capacity multiplies the carrying capacity; if it added to the carrying capacity, we might make the derivative equal $p$ instead.) Population should typically remain close to the carrying capacity; so, we could simply assume that population equals carrying capacity. We then expect hyperbolic growth, IE something like $c=\frac{1}{s-t}$; here, $s$ is the year of the (population) singularity. This model is a decent fit to the data until the year 1960, which is of course the subject of the post.

One of my thoughts after reading this was: wouldn't it make more sense to avoid the assumption that population equals carrying capacity? Population growth can't be greater than exponential. The hyperbolic model doesn't make any sense, and the assumption that population equals carrying capacity appears to be the culprit. 

It would make more sense to, instead, use more typical population models (which predict near-exponential growth when population is significantly below carrying capacity, tapering off near carrying capacity). I don't yet know if this has been done in the literature. However, it's commonly said that around the time of the industrial revolution, humankind escaped the Malthusian trap, because progress outpaced birthrates. (I know the demographic transition is a big player here, but let's ignore it for a moment.) If we were modeling this possibility, it makes sense that progress would stop accelerating so much around this point: once progress is increasing the carrying capacity faster than the population can catch up, we no longer expect to see population match carrying capacity.

This would imply that population transitions from hyperbolic growth to exponential growth, some time shortly before the singularity of the hyperbola. Which approximately matches what we observe: a year where the singularity was "cancelled".

However, in the context of AI progress in particular, this model seems naive. Human birthrates cannot keep pace with the resources progress provides. However, AI has no such limitation. Therefore, we might expect progress to look hyperbolic again at some point, when AI starts contributing significantly to progress. (Indeed, one might have expected this from computers alone, without saying the words "AI" -- computers allow "thinking power" to increase, without the population actually increasing.)

Some of the toy mathematical models Paul Christiano discusses in Takeoff Speeds might be used to add AI to the projection.

So, I'm interested in:

• Any ideas you have about this, especially in the form of equations.
• Links to mathematical models of population growth, perhaps slightly more detailed than the one Scott Alexander discusses.
• Mathematical models of GDP growth, along the same lines.
• Mathematical models of AI progress, such as what Paul Christiano discusses. I'm sure there are a number of essays about that posted here, but again, I don't expect to dig through all of them myself; what things do you think are most relevant?
• Mathematical models of progress generally, especially anything which uses a slightly less simplistic model of progress. For example, it's sometimes claimed that the explosive progress of the industrial revolution was due to technological progress starting to really build on itself, providing better tools for making progress.
• Informal discussions of these same topics, which nonetheless discuss critical features which could be made mathematical. For example, Against GDP as a metric for timelines and takeoff speeds can be seen as a challenge for this kind of modeling; someone putting together mathematical models of the sort I'm discussing might want to address those challenges.
• Less importantly, data to fit curves to, mathematical tools (like Guesstimate) which seem particularly useful for what I'm trying to do, etc.

Related Question: Any response to Paul Christiano on takeoff speeds?