  Motivated by recent progress in data assimilation, we develop an algorithm to
dynamically learn the parameters of a chaotic system from partial observations.
Under reasonable assumptions, we rigorously establish the convergence of this
algorithm to the correct parameters when the system in question is the classic
three-dimensional Lorenz system. Computationally, we demonstrate the efficacy
of this algorithm on the Lorenz system by recovering any proper subset of the
three non-dimensional parameters of the system, so long as a corresponding
subset of the state is observable. We also provide computational evidence that
this algorithm works well beyond the hypotheses required in the rigorous
analysis, including in the presence of noisy observations, stochastic forcing,
and the case where the observations are discrete and sparse in time.
