  This work presents a hybrid analysis and modeling approach to the 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 devise an effective methodology based on
a recurrent neural network to model the unknown dynamics and sequential data
assimilation to correct the hybrid model in the online deployment phase. We
apply this framework to the weakly nonlinear Lorenz model that displays
quasiperiodic oscillations, the highly nonlinear chaotic Lorenz model,
\textcolor{rev1}{and two-scale Lorenz model}. The hybrid neural-physics model
is successful in forecasting the weakly nonlinear Lorenz model with the
predicted state close to the true Lorenz model trajectory. For the highly
nonlinear Lorenz model \textcolor{rev1}{ 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 wrong prediction to achieve an accurate forecast with the
hybrid-neural physics model for the highly nonlinear chaotic Lorenz model. The
successful synergistic integration of neural network and data assimilation for
low-dimensional system shows the potential benefits of such hybrid methods for
complex dynamical systems.
