  In this paper, we study numerically flat holomorphic vector bundles over a
compact non-K\"ahler manifold $(X, \omega)$ with the Hermitian metric $\omega$
satisfying the Gauduchon and Astheno-K\"ahler conditions. We prove that
numerically flatness is equivalent to numerically effectiveness with vanishing
first Chern number, semistablity with vanishing first and second Chern numbers,
approximate Hermitian flatness and the existence of a filtration whose
quotients are Hermitian flat. This gives an affirmative answer to the question
proposed by Demailly, Peternell and Schneider.
