  We consider a nonlinear SPDE approximation of the Dean-Kawasaki equation for
independent particles. Our approximation satisfies the physical constraints of
the particle system, i.e. its solution is a probability measure for all times
(preservation of positivity and mass conservation). Using a duality argument,
we prove that the weak error between particle system and nonlinear SPDE is of
the order $N^{-1-1/(d/2+1)}\log (N)$. Along the way we show well-posedness, a
comparison principle and an entropy estimate for a class of nonlinear
regularized Dean-Kawasaki equations with It\^o noise. Keywords: Dean-Kawasaki
equation, weak error analysis, Laplace duality
