The Schrödinger equation, the fundamental basis of wavefunction-based methods, cannot be solved in closed form per se. One can, however, solve it by using a finite number of closed-form basis sets (and the corresponding closed-form Slater determinants) and letting the basis set tend toward the complete basis set; this is similar to how most mathematical functions, that cannot be written in closed form per se, can be written as limits of sequences of closed-form functions.
It is currently unknown whether the exact exchange-correlation functional in DFT can be written in closed form or not. However, it should be able to be written as some limit of some sequence closed-form formulae, since the end result, i.e. the solution, is (vide supra) and the exchange-correlation functional can be written in a closed form expression when the end result is explicitly included as a term. (I'm a mathematics student, so I know what I'm talking about)
Now comes my main question- is my reasoning correct; i.e. is there any explicit series of closed-form "approximate DFT functionals' that mathematically converges to the exact potential?
UPDATE: After reading the answer that a sequence of neural networks can converge to the exact potential, which was not the type of answer that I wanted, I'm editing this post to re-ask a more intended question.