  Drawing on ergodic theory, we introduce a novel training method for machine
learning based forecasting methods for chaotic dynamical systems. The training
enforces dynamical invariants--such as the Lyapunov exponent spectrum and
fractal dimension--in the systems of interest, enabling longer and more stable
forecasts when operating with limited data. The technique is demonstrated in
detail using the recurrent neural network architecture of reservoir computing.
Results are given for the Lorenz 1996 chaotic dynamical system and a spectral
quasi-geostrophic model, both typical test cases for numerical weather
prediction.
