  A model-based approach to forecasting chaotic dynamical systems utilizes
knowledge of the physical processes governing the dynamics to build an
approximate mathematical model of the system. In contrast, machine learning
techniques have demonstrated promising results for forecasting chaotic systems
purely from past time series measurements of system state variables (training
data), without prior knowledge of the system dynamics. The motivation for this
paper is the potential of machine learning for filling in the gaps in our
underlying mechanistic knowledge that cause widely-used knowledge-based models
to be inaccurate. Thus we here propose a general method that leverages the
advantages of these two approaches by combining a knowledge-based model and a
machine learning technique to build a hybrid forecasting scheme. Potential
applications for such an approach are numerous (e.g., improving weather
forecasting). We demonstrate and test the utility of this approach using a
particular illustrative version of a machine learning known as reservoir
computing, and we apply the resulting hybrid forecaster to a low-dimensional
chaotic system, as well as to a high-dimensional spatiotemporal chaotic system.
These tests yield extremely promising results in that our hybrid technique is
able to accurately predict for a much longer period of time than either its
machine-learning component or its model-based component alone.
