We formulate a data-driven method for constructing finite volume discretizations of a dynamical system's underlying Continuity / Fokker-Planck equation. A method is employed that allows for flexibility in partitioning state space, generalizes to function spaces, applies to arbitrarily long sequences of time-series data, is robust to noise, and quantifies uncertainty with respect to finite sample effects. After applying the method, one is left with Markov states (cell centers) and a random matrix approximation to the generator. When used in tandem, they emulate the statistics of the underlying system. We apply the method to the Lorenz equations (a three-dimensional ordinary differential equation) and a modified Held-Suarez atmospheric simulation (a Flux-Differencing Discontinuous Galerkin discretization of the compressible Euler equations with gravity and rotation on a thin spherical shell). We show that a coarse discretization captures many essential statistical properties of the system, such as steady state moments, time autocorrelations, and residency times for subsets of state space.

翻译：我们提出了一种数据驱动方法，用于构建动力系统底层连续性/福克-普朗克方程的有限体积离散格式。该方法允许灵活划分状态空间，可推广至函数空间，适用于任意长度的时序数据序列，对噪声具有鲁棒性，并能量化有限样本效应带来的不确定性。应用该方法后，可得到马尔可夫状态（单元中心）及生成元的随机矩阵近似。两者结合使用时，能模拟底层系统的统计特征。我们将该方法应用于洛伦兹方程（三维常微分方程）和修正的赫尔德-苏亚雷斯大气模拟（薄球壳上含重力与旋转的可压缩欧拉方程的磁通差分间断伽辽金离散格式）。结果表明，粗离散化捕捉了系统的许多基本统计特性，如稳态矩、时间自相关以及状态空间子集的驻留时间。