A Basic Recurrent Neural Network Model
  We present a model of a basic recurrent neural network (or bRNN) that includes a separate linear term with a slightly "stable" fixed matrix to guarantee bounded solutions and fast dynamic response. We formulate a state space viewpoint and adapt the constrained optimization Lagrange Multiplier (CLM) technique and the vector Calculus of Variations (CoV) to derive the (stochastic) gradient descent. In this process, one avoids the commonly used re-application of the circular chain-rule and identifies the error back-propagation with the co-state backward dynamic equations. We assert that this bRNN can successfully perform regression tracking of time-series. Moreover, the "vanishing and exploding" gradients are explicitly quantified and explained through the co-state dynamics and the update laws. The adapted CoV framework, in addition, can correctly and principally integrate new loss functions in the network on any variable and for varied goals, e.g., for supervised learning on the outputs and unsupervised learning on the internal (hidden) states. 
Fathi M. Salem
cs.NE stat.ML