How to analyze a curve?

Question

Mathematics

The essence of the question is this:
There is a set of curves (graphs) from which you can find out x and y. It is required to find graphics in this set that do not satisfy a certain form, which is difficult to set mathematically as a curve. How can this be done and where can I read about them?
I solved a similar problem by polynomial regression using the alglib library with adding the R2 count to it, but I want to know about other solution methods.

Answer 1

As far as I understand, bad is when the difference between the minimum and maximum is very large, good is when this difference is small (at least at the end of the chart). Smoothness also matters.
I see two solutions here:
1. Descriptive.
You yourself formulate the criteria for "goodness", and find a way to measure it. That is, we need a function L: "a set of functions" -> [0, 1], and if L(f) = 1, then the graph of f is good, if L(f) = 0, then the graph of f is bad.
2. Machine learning.
Here you still need to describe some function criteria, and build a feature vector for each function. For example, a feature vector may contain information (maximum value; minimum value; difference between max and min; some kind of smoothness index (maybe calculate the derivative at each point, and make sure that the derivative does not change much); subplot area, etc. .).
Then, you collect a sample of graphs, for them you get a sample of feature vectors, and then you can use classification algorithms: logistic regression, neural networks, SVM, etc. The success of this approach depends on how well the feature vector you come up with describes your own criteria.
The result will be the same function L: "a set of functions" -> [0, 1], only, perhaps, you will not be able to comprehend its work (there will be no verbal criterion, as in paragraph 1), but it will also output is function good or bad.