# Creating a training dataset from analytical solution

I am currently redesigning an inverse problem on an experimental technique, but I am having doubts about how to create a training dataset. Here is the problem I am trying to solve:

I have already created a model in order to solve the forward problem, such that if I input $$x_1,x_2,...,x_n$$, the model solves a generalized eigenvalue problem and returns the expected output $$y_1,y_2,..,y_m$$.

Thus, what I am trying to do is to generate random $$(\vec{x},\vec{y})$$ pairs in order to create a training data set that will allow me to solve the inverse problem. Considering this, I am confused as if the random $$(\vec{x},\vec{y})$$ pairs should be considered as synthetic data (since they come from the actual analytical solution). If so, should I generate the training data by following a probability distribution and adding noise?

Thanks in advance for the help!