The application of signal flow graphs to the learning process of neura
l networks is presented. By introducing the so-called adjoint graph, n
ew insight into the mechanism of learning phenomena of the weights in
neural networks has been obtained. The derived updating formulas are v
alid for both feedforward and recurrent neural networks and are especi
ally useful from the hardware implementation point of view of the self
-learning networks. The presented numerical experiments confirmed the
usefulness of the presented approach.