Classifcation of non-linear regressions based on their shapes

I have a data set of thousands of individual y ~ x relationships that can have varying shapes in their relationships. For example, they can follow an exponential, asymptotic, logistic or hump-shaped (with different skewness) pattern. I actually don’t know all the shapes the relationships can take, these are just a few patterns that I hypothesise to emerge.

I need a way to classify each of these relationships into one of the non-linear patterns without looking at each individual relationship. I was wondering whether there is way to do this in machine learning. I am willing to sit down and classify maybe some of the patterns myself but would like to avoid classifying all of them by hand.

Ultimately, I’m not interested into prediction. The regression should only serve as a way to describe different non-linear patterns and classify each individual into a category based on the shape of the regression.

What I am doing so far is fitting non-linear regressions with different equations to each relationship and decide which of the non-linear regressions (exponential, asymptotic etc) is fitting best by evaluating the residual variance and the p-value. However, I was wondering whether there is a way to do this with machine learning instead.