  In this paper, we consider the cohomology rings of some multiple weight
varieties of type A, that is, symplectic torus quotients for a direct product
of several coadjoint orbits of the special unitary group. Under some specific
assumptions, we prove the symplectic volumes of multiple weight varieties are
equal to the volumes of flow polytopes. Using differential equations satisfied
by the volume functions of flow polytopes, we give an explicit presentation of
the cohomology ring of the multiple weight variety of special type.
