Empirical evaluated SDE modelling for dimensionality-reduced systems and its predictability estimates

Naoto Nakano, Masaru Inatsu, Seiichiro Kusuoka, Yoshitaka Saiki

    Research output: Contribution to journalArticlepeer-review


    This paper develops and validates a method of empirical modelling for a dimensionality-reduced system of a nonlinear dynamical system based on the framework of the stochastic differential equation (SDE). Following the mathematical theorem corresponding to some inverse problem of the probability theory, we derive the empirically evaluating formulae for the drift vector and diffusion matrix. Focusing on a low-dimensional dynamical system of the Lorenz system, we empirically reconstruct an SDE that approximates the original time-series on the projected 2-dimensional plane. The distribution of the ensemble variance of solutions generated by the numerical SDE well agrees with that of the trajectories of the projected time-series, which indicates the ability of the SDE modelling to represent local predictability. Moreover, we also compare our SDE constructing method with the conventional Mori–Zwanzig projected operator method, which is used to derive a generalised Langevin equation for dimensionality-reduced systems, to assess the applicability of the obtained SDE model derived by the presented method.

    Original languageEnglish
    Pages (from-to)553-589
    Number of pages37
    JournalJapan Journal of Industrial and Applied Mathematics
    Issue number2
    Publication statusPublished - Jul 1 2018


    • Dimensionality reduction
    • Inverse problem
    • Nonlinear dynamical systems
    • Predictability
    • Stochastic differential equation

    ASJC Scopus subject areas

    • Engineering(all)
    • Applied Mathematics


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