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Prediction of dynamic systems driven by Lévy noise based on deep learning

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Abstract

Predicting strongly noise-driven dynamic systems has always been a difficult problem due to their chaotic properties. In this study, we investigated the prediction of dynamic systems driven by strong noise intensities, which proves that deep learning can be applied in diverse fields. This is the first study that uses deep learning algorithms to predict dynamic systems driven by strong noise intensities. We examined the effect of hyperparameters in deep learning and introduced an improved algorithm for prediction. Several numerical examples are presented to illustrate the performance of the proposed algorithm, including the Lorenz system and the Rössler system driven by noise intensities of 0.1, 0.5, 1, and 1.25. All the results suggest that the proposed improved algorithm is feasible and effective for predicting strongly noise-driven dynamic systems. Furthermore, the influences of the number of neurons, the spectral radius, and the regularization parameters are discussed in detail. These results indicate that the performances of the machine learning techniques can be improved by appropriately constructing the neural networks.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request. For part of the code of the IRC algorithm, we have made it public at https://github.com/Lukas22eee/IRC-algorithm.

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Acknowledgements

The research was supported by the National Natural Science Foundation of China (NNSFC) (Grant No. 11902234), Natural Science Basic Research Program of Shaanxi (Program No. 2020JQ-853), Shaanxi Provincial Department of Education Youth Innovation Team Scientific Research Project (Program No. 22JP025), and the Young Talents Development Support Program of Xi’an University of Finance and Economics. T.K. has been supported by the National Science Centre, Poland, OPUS Programme (Project No. 2021/43/B/ST8/00641).

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Authors and Affiliations

  1. School of Statistics, Xi’an University of Finance and Economics, Xi’an, 710100, People’s Republic of China

    Zi-Fei Lin, Yan-Ming Liang & Jia-Li Zhao

  2. Xi’an University of Finance and Economics, China (Xi’an) Institute for Silk Road Research, Xi’an, 710100, People’s Republic of China

    Zi-Fei Lin & Jiao-Rui Li

  3. Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924, Lodz, Poland

    Tomasz Kapitaniak

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  1. Zi-Fei Lin
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  2. Yan-Ming Liang
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  4. Jiao-Rui Li
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Correspondence to Zi-Fei Lin.

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Lin, ZF., Liang, YM., Zhao, JL. et al. Prediction of dynamic systems driven by Lévy noise based on deep learning. Nonlinear Dyn 111, 1511–1535 (2023). https://doi.org/10.1007/s11071-022-07883-9

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  • DOI: https://doi.org/10.1007/s11071-022-07883-9

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