A differential recurrent neural network (RNN) is described that handles dependencies that go arbitrarily far in time by allowing the network system to store states using recurrent loops without adversely affecting training. The differential RNN includes a state component for storing states, and a trainable transition and differential non-linearity component which includes a neural network. The trainable transition and differential non-linearity component takes as input, an output of the previous stored states from the state component along with an input vector, and produces positive and negative contribution vectors which are employed to produce a state contribution vector. The state contribution vector is input into the state component to create a set of current states. In one implementation, the current states are simply output. In another implementation, the differential RNN includes a trainable OUT component which includes a neural network that performs post-processing on the current states before outputting them.