Is there a formula for the inverse of Hadamard product?

Say $A$ and $B$ are two square, positive-semidefinite matrices. Is there an expression in terms of matrix product, transpose, and inverse for the Hadamard product $A∘B$?

For example, "$(A∘B)^{-1} = A^{-1} ∘ B^{-1}$" (which is not true).

Edit: I understand that $A∘B$ may not be invertible, but is there any expression if invertibility is given?

There won't be such a formula. In fact, if $A$ and $B$ are invertible, we can't guarantee that $A\circ B$ will even have an inverse. In particular, consider $$ A = \pmatrix{1&0\\0&1}, \quad B = \pmatrix{0&1\\1&0} $$ we note that $A=A^{-1}$ and $B = B^{-1}$, but $A\circ B = 0$.

It is often, however, useful to consider the Hadamard product as a submatrix of the Kronecker product. For the Kronecker product, we have $$ (A \otimes B)^{-1} = A^{-1} \otimes B^{-1} $$

The Hadamard (aka element-wise) inverse $A^{\theta}$ is such that $A\circ A^\theta = 1$.

If any element of $A$ is zero, then the Hadamard inverse is undefined.

It satisfies a product rule similar to that of the Kronecker product, i.e. $$ (A\circ B)^\theta = A^\theta\circ B^\theta $$