# ML to approximate a dampened harmonic oscillator from a related curve

I want to approximate the output of a Monte Carlo simulation that takes a probability density function sampled at x-axis points $0,1,2,3,\ldots,n$ and outputs what tends to look like a dampened harmonic oscillator.

Four examples appear below. The blue dots are the sampled pdf (or pmf), and the heights of the black lines give the curves I want to approximate.

My inclination is that I would be able to use Machine Learning to either (a) provide second-order differential equations that approximate the output of the Monte Carlo simulation, or (b) bypass the dampened harmonic idea and approximate the output of the MC simulation directly.

Needless to say, I can generate tons of learning data using the MC simulation. I should also mention that once the ML has done its training, it needs to perform much faster than the MC simulation!

I'm hoping this will be a good introduction to ML algorithms for me, but I'm not sure where I would start. Hints about learning resources and software frameworks to do my research in are welcome.

Can this be solved with ML? If so, would I bother with the oscillator stuff?

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.
• @AlejandroErickson, OK, then you might need to use more general features of the distribution as the inputs to $f$: the mean, the variance, the location of the first quartile, the location of the third quartile, a histogram (one feature per bar in the histogram, indicating the height of that bar), that sort of thing.