  This work presents a hybrid modeling approach to data-driven learning and
representation of unknown physical processes and closure parameterizations.
These hybrid models are suitable for situations where the mechanistic
description of dynamics of some variables is unknown, but reasonably accurate
observational data can be obtained for the evolution of the state of the
system. In this work, we propose machine learning to account for missing
physics and then data assimilation to correct the prediction. In particular, we
devise an effective methodology based on a recurrent neural network to model
the unknown dynamics. A long short-term memory (LSTM) based correction term is
added to the predictive model in order to take into account hidden physics.
Since LSTM introduces a black-box approach for the unknown part of the model,
we investigate whether the proposed hybrid neural-physical model can be further
corrected through a sequential data assimilation step. We apply this framework
to the weakly nonlinear Lorenz model that displays quasiperiodic oscillations,
the highly nonlinear Lorenz model, and two-scale Lorenz model. The hybrid
neural-physics model yields accurate results for the weakly nonlinear Lorenz
model with the predicted state close to the true Lorenz model trajectory. For
the highly nonlinear Lorenz model and the two-scale Lorenz model, the hybrid
neural-physics model deviates from the true state due to the accumulation of
prediction error from one time step to the next time step. The ensemble Kalman
filter approach takes into account the prediction error and updates the
diverged prediction using available observations in order to provide a more
accurate state estimate. The successful synergistic integration of neural
network and data assimilation for low-dimensional system shows the potential
benefits of the proposed hybrid-neural physics model for complex systems.
