  We consider the problem of data-assisted forecasting of chaotic dynamical
systems when the available data is in the form of noisy partial measurements of
the past and present state of the dynamical system. Recently there have been
several promising data-driven approaches to forecasting of chaotic dynamical
systems using machine learning. Particularly promising among these are hybrid
approaches that combine machine learning with a knowledge-based model, where a
machine-learning technique is used to correct the imperfections in the
knowledge-based model. Such imperfections may be due to incomplete
understanding and/or limited resolution of the physical processes in the
underlying dynamical system, e.g., the atmosphere or the ocean. Previously
proposed data-driven forecasting approaches tend to require, for training,
measurements of all the variables that are intended to be forecast. We describe
a way to relax this assumption by combining data assimilation with machine
learning. We demonstrate this technique using the Ensemble Transform Kalman
Filter (ETKF) to assimilate synthetic data for the 3-variable Lorenz system and
for the Kuramoto-Sivashinsky system, simulating model error in each case by a
misspecified parameter value. We show that by using partial measurements of the
state of the dynamical system, we can train a machine learning model to improve
predictions made by an imperfect knowledge-based model.
