Accurate information about the fission barrier is important for studying of the fission process. Fission barrier is needed for discovering the island of stability in superheavy region and searching of the superheavy elements. Furthermore, the astrophysical r-process is closely related to the fission barrier of the neutron-rich nuclei. In this study, by using artificial neural network (ANN) method, we have estimated the fission barrier heights of the Rf, Db, Ra and Ac nuclei covering 230 isotopes. For inner barrier calculation, we have used Rf and Db nuclei and the barrier heights have been determined between nearly 1 MeV and 7 MeV. The related mean square error value has been obtained as 0.108 MeV. For outer barrier calculation, we have used Ra and Ac nuclei and the heights have been determined between nearly 8 MeV and 28 MeV. The related mean square error has been obtained as 0.407. The results of this study indicate that ANN is capable for the estimations of inner and outer fission barrier heights.