Learning a trajectory using adjoint functions and teacher forcing

A new methodology for faster supervised temporal learning in nonlinear neural networks is presented. It builds upon the concept of adjoint operators, to enable a fast computation of the gradients of an error functional with respect to all parameters of the neural architecture, and exploits the concept of teacher forcing to incorporate information regarding the desired output into the activation dynamics. The importance of the initial or final time conditions for the adjoint equations (i.e., the error propagation equations) is discussed. A new algorithm is presented, in which the adjoint equations are solved simultaneously (i.e., forward in time) with the activation dynamics of the neural network. We also indicate how teacher forcing can be modulated in time as learning proceeds. The algorithm is illustrated by examples. The results show that the learning time is reduced by one to two orders of magnitude with respect to previously published results, while trajectory tracking is significantly improved. The proposed methodology makes hardware implementation of temporal learning attractive for real-time applications.