  The reconstruction of the dynamics of an observed physical system as a
surrogate model has been brought to the fore by recent advances in machine
learning. To deal with partial and noisy observations in that endeavor, machine
learning representations of the surrogate model can be used within a Bayesian
data assimilation framework. However, these approaches require to consider long
time series of observational data, meant to be assimilated all together. This
paper investigates the possibility to learn both the dynamics and the state
online, i.e. to update their estimates at any time, in particular when new
observations are acquired. The estimation is based on the ensemble Kalman
filter (EnKF) family of algorithms using a rather simple representation for the
surrogate model and state augmentation. We consider the implication of learning
dynamics online through (i) a global EnKF, (i) a local EnKF and (iii) an
iterative EnKF and we discuss in each case issues and algorithmic solutions. We
then demonstrate numerically the efficiency and assess the accuracy of these
methods using one-dimensional, one-scale and two-scale chaotic Lorenz models.
