I may be revealing my lack of mathematical understanding, but in some of the simple examples I feel the data looks exponential yet a polynomial is used to fit the data. Can an exponential model be used in a regression model?
Yes you can!
See the details here:
However, I would take a Feature Engineering approach and if I suspected the data to be exponential take the log(feature) so that a linear model will be a better fit.
it really depends on the system you are modelling. In general, there are systems where you can use an exponential regression model, see e.g. this example here: Bias and variance , tradeoff - #2 by Christian_Simonis
But also when modelling e.g. chancer growth, exponential fits are used, see also this toolbox.
When it comes to limited data I agree with you: then it can sometimes be hard to see then if it’s a polynomial or exponential model that would be suitable, but the longer your data set (e.g. the longer your time series) the stronger it’s visible what is suitable and what not.
In the end you should really incorporate your domain knowledge which model to use and how to design your features most effectively. Also doing some research on the system characteristics you are modelling is highly recommended.
Maybe as some outlook for the future: In case you have limited data but still rich domain knowledge, I can also recommend to think about a hybrid approach, combining a domain model with a data-driven (probabilistic) approach, see also this Github Repo.
Thank you both for your responses. Your comments and links have helped a great deal.