By extending the extreme learning machine by additional control inputs, we achieved almost complete reproduction of bifurcation structures of dynamical systems. The learning ability of the proposed neural network system is striking in that the entire structure of the bifurcations of a target one-parameter family of dynamical systems can be nearly reproduced by training on transient dynamics using only a few parameter values. Moreover, we propose a mechanism to explain this remarkable learning ability and discuss the relationship between the present results and similar results obtained by Kim et al.