Proof of Concept for a Low-Order Fokker–Planck Formulation Using the Lorenz System

Abstract We propose a formulation for estimating the drift vector and the diffusion tensor in a reduced Fokker–Planck equation from the governing equations, under the assumption that the high modes follow mutually independent normal distributions, whose mean and variance conditioned on a specific low-mode state are inferred from finite time-series data. We performed a long numerical integration of the three-dimensional Lorenz system and then applied principal component (PC) decomposition to separate the two gravest PC modes from the remaining modes. The conditional mean and variance of the high mode, PC3, were locally estimated using a self-organising map constructed in the low-mode phase space. The resulting drift and diffusion fields showed reasonable agreement with empirical estimates computed directly from the data, capturing the rotational dynamics around unstable equilibria of the Lorenz model and the enhanced diffusion in the inter-lobe transition region. These results suggested that the proposed framework served as a proof of concept for reconstructing local drift and diffusion coefficients of reduced Fokker–Planck models for nonlinear dynamical systems under a closure assumption.