The algebra of derivations of quasi-modular forms from mirror symmetry

We study moduli spaces of mirror non-compact Calabi-Yau threefolds enhanced with choices of differential forms. The differential forms are elements of the middle dimensional cohomology whose variation is described by a variation of mixed Hodge structures which is equipped with a flat Gauss-Manin connection. We construct graded differential rings of special functions on these moduli spaces and show that they contain rings of quasi-modular forms. We show that the algebra of derivations of quasi-modular forms can be obtained from the Gauss--Manin connection contracted with vector fields on the enhanced moduli spaces. We provide examples for this construction given by the mirrors of the canonical bundles of $\mathbb{P}^2$ and $\mathbb{F}_2$.