In 1936, Alan Turing gave the definition of computation which is still in use today. His model of computation has come to be known as a Turing machine (TM).
Turing proved that there is a problem, namely the Halting problem, that no TM can possibly solve.
Any AI using current technology is a practical implementation of a Turing machine; and is subject to the same limitations as explained in Turing's original paper. Even the fanciest, most advanced AI in use or even envisioned, is subject to the limitations Turing outlined.
Some approaches that don't help include:
Massive parallelism. Parallel computing is still subject to the limitations of Turing machines. You just execute each logic thread round-robin, exactly as conventional computers run many programs in parallel.
Introducing randomness. Computer scientists model this idea as nondeterministic automata. These also have the same computational power as Turing machines. The idea is that the Turing machine just executes every possible branch at each step.
Quantum computers. It's true that for some specialized problems there are quantum algorithms that offer substantial speedup over conventional computers. But speed is not a factor in determining what a computer can compute. Quantum computers have the same computational power as Turing machines.
Machine learning and "deep learning" techniques. Even the most impressive deep learning machine, AlphaZero, is a conventional program running on conventional hardware. It has no more power than a Turing machine.
Turing even studied [in his doctoral thesis, after writing his famous 1936 paper] what happens when you go beyond Turing machines. Suppose you have a TM, which we know can't solve the Halting problem. We could then introduce a black box apparatus called an oracle that solves the Halting problem. Such an idea is purely theoretical at present.
Turing showed that there is now a new problem that the augmented machine can't solve. And if we add an oracle for that, there will still be yet another unsolvable problem. We would have an endless hierarchy of oracle machines such that no matter how many oracles we add, there will always be an unsolvable problem. This phenomenon is closely related to Gödel's result that any sufficiently interesting axiomatic system [omitting the technical definition here] is necessarily incomplete. And even if you add a new axiom to plug the incompleteness, the augmented system must still be incomplete.
Now if we knew that a human could solve the Halting problem, we'd know that there's a problem we can solve that a computer can't. This would be a great breakthrough. However it's unknown if humans can solve the Halting problem. To date, we know of no noncomputable problem that can be solved by humans. So none of this analysis bears on the question of whether we ourselves might be nothing more than Turing machines or computations. Roger Penrose has speculated along these lines.