Recent work has shown that machine learning (ML) models can be trained to accurately forecast the dynamics of unknown chaotic dynamical systems. Such ML models can be used to produce both short-term predictions of the state evolution and long-term predictions of the statistical patterns of the dynamics (``climate''). Both of these tasks can be accomplished by employing a feedback loop, whereby the model is trained to predict forward one time step, then the trained model is iterated for multiple time steps with its output used as the input. In the absence of mitigating techniques, however, this technique can result in artificially rapid error growth, leading to inaccurate predictions and/or climate instability. In this article, we systematically examine the technique of adding noise to the ML model input during training as a means to promote stability and improve prediction accuracy. Furthermore, we introduce Linearized Multi-Noise Training (LMNT), a regularization technique that deterministically approximates the effect of many small, independent noise realizations added to the model input during training. Our case study uses reservoir computing, a machine-learning method using recurrent neural networks, to predict the spatiotemporal chaotic Kuramoto-Sivashinsky equation. We find that reservoir computers trained with noise or with LMNT produce climate predictions that appear to be indefinitely stable and have a climate very similar to the true system, while reservoir computers trained without regularization are unstable. Compared with other types of regularization that yield stability in some cases, we find that both short-term and climate predictions from reservoir computers trained with noise or with LMNT are substantially more accurate. Finally, we show that the deterministic aspect of our LMNT regularization facilitates fast hyperparameter tuning when compared to training with noise.
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