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Drawing on ergodic theory, we introduce a novel training method for machinelearning based forecasting methods for chaotic dynamical systems. The trainingenforces dynamical invariants--such as the Lyapunov exponent spectrum andfractal dimension--in the systems of interest, enabling longer and more stableforecasts when operating with limited data. The technique is demonstrated indetail using the recurrent neural network architecture of reservoir computing.Results are given for the Lorenz 1996 chaotic dynamical system and a spectralquasi-geostrophic model, both typical test cases for numerical weatherprediction.

Constraining Chaos: Enforcing dynamical invariants in the training of recurrent neural networks