and apparatus for supervised neural learning of time dependent trajectories exploits the concepts of adjoint operators to enable computation of the gradient of an objective functional with respect to the various parameters of the network architecture in a highly efficient manner specifically it combines the advantage of dramatic reductions in computational complexity inherent in adjoint methods with the ability to solve two adjoint systems of equations together forward in time not only is a large amount of computation and storage saved but the handling of real time applications becomes also possible the invention has been applied it to two examples of representative complexity which have recently been analyzed in the open literature and demonstrated that a circular trajectory can be learned in approximately 200 iterations compared to the 12000 reported in the literature a figure eight trajectory was achieved in under 500 iterations compared to 20000 previously required the trajectories computed using our new method are much closer to the target trajectories than was reported in previous studies application_number US 96986892 A description we formalize a neural network as an adaptive dynamical system whose temporal evolution is governed by the following set of coupled nonlinear differential equations equ1 where u n represents the output of the nth neuron u n 0 being the initial state and t nm denotes the strength of the synaptic coupling from the m th to the n th neuron the constants k.