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Recent work has shown that machine learning (ML) models can be trained to accurately forecast the dynamics of unknown chaotic dynamical systems. Short-term predictions of the state evolution and long-term predictions of the statistical patterns of the dynamics (``climate'') can be produced by employing a feedback loop, whereby the model is trained to predict forward one time step, then the model output is used as input for multiple time steps. In the absence of mitigating techniques, however, this technique can result in artificially rapid error growth. In this article, we systematically examine the technique of adding noise to the ML model input during training 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 regularization techniques 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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Machine learning-based statistical closure models for turbulent dynamical systems
We propose a machine learning (ML) non-Markovian closure modelling framework for accurate predictions of statistical responses of turbulent dynamical systems subjected to external forcings. One of the difficulties in this statistical closure problem is the lack of training data, which is a configuration that is not desirable in supervised learning with neural network models. In this study with the 40-dimensional Lorenz-96 model, the shortage of data is due to the stationarity of the statistics beyond the decorrelation time. Thus, the only informative content in the training data is from the short-time transient statistics. We adopt a unified closure framework on various truncation regimes, including and excluding the detailed dynamical equations for the variances. The closure framework employs a Long-Short-Term-Memory architecture to represent the higher-order unresolved statistical feedbacks with a choice of ansatz that accounts for the intrinsic instability yet produces stable long-time predictions. We found that this unified agnostic ML approach performs well under various truncation scenarios. Numerically, it is shown that the ML closure model can accurately predict the long-time statistical responses subjected to various time-dependent external forces that have larger maximum forcing amplitudes and are not in the training dataset. This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.
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- Award ID(s):
- 2207328
- PAR ID:
- 10422615
- Date Published:
- Journal Name:
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 380
- Issue:
- 2229
- ISSN:
- 1364-503X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1098/rsta.2021.0205
