  This paper explores the problem of training a recurrent neural network from
noisy data. While neural network based dynamic predictors perform well with
noise-free training data, prediction with noisy inputs during training phase
poses a significant challenge. Here a sequential training algorithm is
developed for an echo-state network (ESN) by incorporating noisy observations
using an ensemble Kalman filter. The resultant Kalman-trained echo-state
network (KalT-ESN) outperforms the traditionally trained ESN with least square
algorithm while still being computationally cheap. The proposed method is
demonstrated on noisy observations from three systems: two synthetic datasets
from chaotic dynamical systems and a set of real-time traffic data.
