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A Proposal on Machine Learning via Dynamical Systems

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We discuss the idea of using continuous dynamical systems to model general high-dimensional nonlinear functions used in machine learning. We also discuss the connection with deep learning.

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  1. Fan, J., Gijbels, I.: Local Polynomial Modeling and Its Applications. Chapman & Hall, London (1996)

    MATH  Google Scholar 

  2. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction a, Springer Series in Statistics, second edition, (2013)

  3. LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436–444 (2015)

    Article  Google Scholar 

  4. Han, J., E, W.: in preparation

  5. Li, Q., Tai, C., E, W.: in preparation

  6. Almeida, L.B.: A learning rule for asynchronous perceptrons with feedback in a combinatorial environment. In: Proceedings ICNN 87. San Diego, IEEE (1987)

  7. LeCun, Y.: A theoretical framework for back propagation. In: Touretzky, D., Hinton, G., Sejnouski, T. (eds.) Proceedings of the 1988 connectionist models summer school, Carnegie-Mellon University, Morgan Kaufmann, (1989)

  8. Pineda, F.J.: Generalization of back propagation to recurrent and higher order neural networks. In: Proceedings of IEEE conference on neural information processing systems, Denver, November, IEEE (1987)

  9. Recht, B.:

  10. E, W., Ming, P.: Calculus of Variations and Differential Equations, lecture notes, to appear

  11. He, K., Zhang, X., Ren, S., Sun, J.: Identity mapping in deep residual networks. (July, 2016) arXiv:1603.05027v3

  12. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley, New York (1992)

    Google Scholar 

  13. Stroock, D.W., Varadhan, S.R.S.: Multi-Dimensional Diffusion Processes. Springer, Berlin (2006)

    MATH  Google Scholar 

  14. Wang, C., Li, Q., E, W., Chazelle, B.: Noisy Hegselmann–Krause systems: phase transition and the 2R-conjecture. In: Proceedings of 55th IEEE Conference on Decision and Control, Las Vegas, (2016) (Full paper at arXiv:1511.02975v3, 2015)

  15. Tabak, E.G., Vanden-Eijnden, E.: Density estimation by dual ascent of the log-likelihood. Commun. Math. Sci. 8(1), 217–233 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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This is part of an ongoing project with several collaborators, including Jiequn Han, Qianxiao Li, Jianfeng Lu and Cheng Tai. The author benefitted a great deal from discussions with them, particularly Jiequn Han. This work is supported in part by the Major Program of NNSFC under Grant 91130005, ONR N00014-13-1-0338 and DOE DE-SC0009248.

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Correspondence to Weinan E.

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Dedicated to Professor Chi-Wang Shu on the occasion of his 60th birthday.

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E, W. A Proposal on Machine Learning via Dynamical Systems. Commun. Math. Stat. 5, 1–11 (2017).

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