Network structures are reconstructed from dynamical data by respectively naive mean field (nMF) and Thouless-Anderson-Palmer (TAP) approximations. TAP approximation adds simple corrections to the nMF approximation, taking into account the effect of the focused spin on itself via its influence on other neighboring spins. For TAP approximation, we use two methods to reconstruct the network: (a) iterative method; (b) casting the inference formula to a set of cubic equations and solving it directly. We investigate inference of the asymmetric Sherrington-Kirkpatrick (aS-K) model using asynchronous update. The solutions of the set of cubic equations depend on temperature T in the aS-K model, and a critical temperature Tc≈2.1 is found. The two methods for TAP approximation produce the same results when the iterative method is convergent. Compared to nMF, TAP is somewhat better at low temperatures, but approaches the same performance as temperature increases. Both nMF and TAP approximation reconstruct better for longer data length L, but for the degree of improvement, TAP performs better than nMF.