  We introduce the notion of symplectic flatness for connections and fiber
bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a
flatness condition that enables the twisting of the differential complex
associated with the $A_\infty$-algebra. The symplectic flatness condition
arises from twisting the $A_\infty$-algebra of differential forms constructed
by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a
compatible metric, the symplectic flat connections represent a special subclass
of Yang-Mills connections. We further study the cohomologies of the twisted
differential complex and give a simple vanishing theorem for them.
