In UI design we often talk about "mental models" that users need to form. I'm wondering if there is a particular relationship between that mental model and the actual information model that the system implements. Is the mental model a strict subset? Do they just overlap?
Just break down the two terms to see the difference and, consequently, the relationship.
Note from @PhilipW's link in the comments:
Mental models are representations in the mind of real or imaginary situations.
From A Gentle Introduction to Mental Models (Phil Johnson-Laird and Ruth Byrne, May 2000)
In the context of your question, a mental model is just jargon for "the picture I have in my head of how I expect this thing to work". An "information model" or domain model is the way the designer of the system intends for it to work.
For designers, it's important to recognise that whatever you design, however well you design it, people will end up forming their own picture in their heads of how they think it works. And that can differ pretty substantially from what you as a designer had intended.
Great designers therefore are able to intuit and predict the ways that their domain model will end up being translated by users of the system into different pictures. Being able to do that is usually a function of experience and talking to hundreds of people to learn the patterns that lead to that translation taking place.
This is why domain-driven design is a great methodology: it standardises that approach and helps you learn to map domain models to mental models more effectively.
See Mental model vs conceptual model: what's the difference? for a similar question.
Mental models are about goals and tasks, and the links between them.
Information models are about objects and meta data, and the relationships between them. In many ways the info model is the designer's view of the system.
Ideally, they do overlap. See Don Norman here: http://jnd.org/dn.mss/design_as_communication.html
If you use correct idioms in UI to explain the relationship between the information and how the users may think about it (their mental model), its easier to overlap them. The more the implementation or information models resemble the user's mental model the better. Alan Cooper talks about it extensively in About Face 3