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

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.