Engineering Guess – Do The Numerical Iteration Process Have Some Real World Meaning. Efficiency?

The process of finding Pi is ?Iterative. So when I was trying to come up with a differential equation for Pi. I wondered. Is there an equation that looked like, dp/dn = g(p,n). Where g(p,n)->0 when n is large. That is. When the iteration has processed a couple of times the update to p is zero. Like a constant.

From this I guessed that since differential equations can be solved iteratively. The iteration part has some physical meaning. Not just some numerical problem.

So I wondered. Could the problem be related to efficiency. I guess is that machine learning could connect error curves in FEA software with real world ?efficiency results.