The confirmed world record for memorizing the digits of $\pi$ goes to a Chinese graduate student named Lu Chao, who claims he has memorized up to 100,000 digits (although for the record breaking attempt, only got to just shy of 68,000).
Memorizing the digits of $\pi$ seems to be a thing people actually do for fun, and while I'm not here to judge their definition of fun, I am curious how one could memorize so many digits of what seems to be a random string of numbers. I know $\pi$ isn't random (quite the opposite), but there doesn't seem any correlation between digits, and it boils right down to rote memorization than any kind of actual maths involved.
Is there a mathematical approach to calculate a given digit of $\pi$, and this is the method these people use? I have trouble memorizing seven digit phone numbers, so it comes to me as a surprise that people can memorize tens of thousands of a digit sequence without error.
How to memorize things is not a mathematical question, BUT something in the posting is a mathematical question.
If there were patterns in the decimal expansion of $\pi$ that can be undrestood by understanding some mathematical idea, then understanding of those could help one memorize the digits.
Here's the relevant mathematical finding: Such patterns do emerge in some things like the continued-fraction expansion of $e$ and some yet more elaborate things. But so far no one has identified such patterns in the decimal expansion of $\pi$. It behaves statistically like a sequence of random digits in which all digits are equally likely to appear at any point, and the digits that came before a particular digit do not alter the probability that any particular digit will appear. And it is conjectured (no one has been able to prove it so far) that $\pi$ is a "normal number", i.e. one that has just those statistical properties in the long run. The short run may be another question, but I don't think any sensible patterns appear there either.
Consequently mathematics can't offer any help here other than that you're essentially just memorizing random digits. However, no one can yet rule out the possibility that some future discovery will reveal some patterns that could be used.
I'll have to leave it to others to cite the relevant literature.