What are the approaches to predict sequence of $\pi$ numbers?

You would probably have to pack recursive structures into finite-dimensional real vectors and there have been such attempts. The finite precision limits goes as far as the recursion can go.

The limitation of feedforward neural networks is restricted to finite input and output spaces, so recurrent may be more suitable for this task as in theory can process arbitrarily long strings of numbers, but it has much more practical difficulties than feedforward network.

These kind of methods are open to debate.

Source: SAS FAQ

References:

• Blair, 1997; Pollack, 1990; Chalmers, 1990; Chrisman, 1991; Plate, 1994; Hammerton, 1998; Hadley, 1999

Pseudo-random number generators are specifically defined to defeat any form of prediction via 'black box' observation. Certainly, some (e.g. linear congruential) have weaknesses, but you are unlikely to have any success in general in predicting the output of a modern RNG. For devices based on chaotic physical systems (e.g. most national lotteries), there is no realistic possibility of prediction.

"Patterns or statistical association" is a much weaker criterion than 'prediction'. Some very recent work has applied topological data analysis to visualize patterns within the infamous Randu RNG.