  Consider the differential forms $A^*(L)$ on a Lagrangian submanifold $L
\subset X$. Following ideas of Fukaya-Oh-Ohta-Ono, we construct a family of
cyclic unital curved $A_\infty$ structures on $A^*(L),$ parameterized by the
cohomology of $X$ relative to $L.$ The family of $A_\infty$ structures
satisfies properties analogous to the axioms of Gromov-Witten theory. Our
construction is canonical up to $A_\infty$ pseudoisotopy. We work in the
situation that moduli spaces are regular and boundary evaluation maps are
submersions, and thus we do not use the theory of the virtual fundamental
class.
