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!
I think your question depends almost entirely on the actual problem that you're trying to represent, i.e. it depends on (your) expert knowledge.
Yes, data generated this way should be considered synthetic. However there's no general requirement to add noise: adding noise (as well as how much and in which way) is normally meant to make synthetic data more realistic, i.e. closer to real data.
In general with synthetic data it's important to show that the data is representative enough of the real phenomenon it represents. Again this depends on expert knowledge but most of the time it's essential to make the distribution of the data realistic as well. So yes, if you know the true probability distribution it's certainly a good idea to make the synthetic data follow it.