Summary: We consider the problem of data-assisted forecasting of chaotic dynamical systems when the available data are 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 to assimilate synthetic data for the three-variable Lorenz 1963 system and for the Kuramoto-Sivashinsky system, simulating a 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.
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