  We consider in dimension four weakly convergent sequences of approximate
biharmonic maps to a Riemannian manifold with bi-tension fields bounded in
$L^p$ for $p>\frac43$. We prove an energy identity that accounts for the loss
of hessian energies by the sum of hessian energies over finitely many
nontrivial biharmonic maps on $\mathbb R^4$. As a corollary, we obtain an
energy identity for the heat flow of biharmonic maps at time infinity.
