Will current encryption always remain secure? Is there mathematical proof? Current encryption algorithms, such as AES, RSA, elliptic curve, etc. work based on known mathematical problems. I am specifically interested in the RSA.

Will such security always remain secure?

One way of breaking them would be due to a revolutionary increase computing speeds, making brute-force a viable option. This could be due to invention of quantum computers, neural-network-based hardware, etc. Such technology is at a very early phase, and will take some years to develop.

The other means is to find loopholes in the algorithm itself, using mathematics. Has it been mathematically proved that these algorithms can not be cracked, or is it a possibility? If so, how likely is it that cracking the RSA via a mathematical weakness will be possible in the near future?

This should be an important question, since being able to crack such a system potentially could let you blackmail companies, and even governments, since it is their data which is at risk.

Note: I first asked this question on Security SE, a user there told me to post it here.
"Will such security always remain secure?"

No. $\:$ In particular, quantum computers will break RSA and elliptic curves.

Has it been mathematically proved that these algorithms can not be cracked, or is it a possibility?

It is a possibility, since a very-practical algorithm for the Boolean satisfiability problem
would be enough to break essentially all complexity-based cryptography.

(There might still be a secure complexity-based proof-of-work system,

although it would need a super-constant number of rounds of interaction.)

If so, how likely is it that cracking the RSA via a

mathematical weakness will be possible in the near future?

Does Shor's algorithm constitute "a mathematical weakness" in RSA?