As a physicist, my understanding of mirror symmetry is very limited, and perhaps the most "mathematical" literature I have read on mirror symmetry is the book of M. Gross. In the genus-0 Gromov-Witten invariant case, the computation of the instanton expansion of the Yukawa coupling on the complex moduli side plays an essential role. In the case of genus-$g$ Gromov-Witten invariant case, what "physics" quantity on the complex side will play the same role as Yukawa coupling in the genus zero case?
There are many references on this, and perhaps the most famous one is the BCOV paper written in 1993, which is however non-deterministic. Any updated references are welcomed.