Coupling neural networks to incomplete dynamical systems via variational data assimilation

The advent of the feed-forward neural network (N) model opens the possibili ty of hybrid neural-dynamical models via variational data assimilation. Suc h a hybrid model may be used in situations where some variables, difficult to model dynamically, have sufficient data for modeling them empirically wi th an N. This idea of using an N to replace missing dynamical equations is tested with the Lorenz three-component nonlinear system, where one of the t hree Lorenz equations is replaced by an N equation. In several experiments, the 4DVAR assimilation approach is used to estimate 1) the N model paramet ers (26 parameters), 2) two dynamical parameters and three initial conditio ns for the hybrid model, and 3) the dynamical parameters, initial condition s, and the N parameters (28 parameters plus three initial conditions). Two cases of the Lorenz model-( i) the weakly nonlinear case of quasiperiod ic oscillations, and (ii) the highly nonlinear, chaotic case-were chosen to test the forecast skills of the hybrid model. Numerical experiments showed that for the weakly nonlinear case, the hybrid model can be very successfu l, with forecast skills similar to the original Lorenz model. For the highl y nonlinear case, the hybrid model could produce reasonable predictions for at least one cycle of oscillation for most experiments, although poor resu lts were obtained for some experiments. In these failed experiments, the da ta used for assimilation were often located on one wing of the Lorenz butte rfly-shaped attractor, while the system moved to the second wing during the forecast period. The forecasts failed as the model had never been trained with data from the second wing.