Is there any NP-hard problem which was proven to be solved in polynomial time or at least close to polynomial time?
By definition, if you were to find a polynomial time algorithm for an NP-hard (or NP-complete) problem, then $P=NP$. So, short answer is - no.
However, its possible to think instead of solving the problems fully, to approximate a solution, or to solve them randomly. There are attempts at attacking from those points of view, but they are not perfect at all. Randomization isn't known to help &quot;as-is&quot;, and many NP-hard problems are known that are also hard to approximate (finding an approximation will yield $P=NP$)
There are also attemps taking a look at notions much stronger than the usual turing machine. For instance, the quantum computer can be considered one. Another obvious example is the non-deterministic turing machines, in which - by definition - $NP$ would be in polynomial time.