ML to approximate a dampened harmonic oscillator from a related curve
I would suggest that you try to learn the function $f$ that maps the parameters of the pmf to the parameters of the dampened harmonic.

For each example, you can presumably derive the parameters of the Weibull distribution for the pmf.  You can also presumably derive the parameters of the dampened harmonic (by curve-fitting, or other means).  So, each example gives you one input-output pair for $f$.

Now, try to learn $f$ from these input-output pairs.  You should start by visualizing the function $f$, to try to understand how each output of $f$ depends on the inputs of $f$, and then you can probably select an appropriate (nonlinear) regression model.  k-nearest neighbors regression might be worth a try in this context.