  We consider the asymmetric simple exclusion process in $d\ge 3$ with open
boundaries. The particle reservoirs of constant densities are modeled by birth
and death processes at the boundary. We prove that, if the initial density and
the densities of the boundary reservoirs differ for order of $\epsilon$ from
1/2, the density empirical field, rescaled as $\epsilon^{-1}$, converges to the
solution of the initial-boundary value problem for the viscous Burgers equation
in a finite domain with given density on the boundary.
