Extracting stochastic governing laws by non-local Kramers–Moyal formulae

Abstract

With the rapid development of computational techniques and scientific tools, great progress of data-driven analysis has been made to extract governing laws of dynamical systems from data. Despite the wide occurrences of non-Gaussian fluctuations, the effective data-driven methods to identify stochastic differential equations with non-Gaussian Lévy noise are relatively few so far. In this work, we propose a data-driven approach to extract stochastic governing laws with both (Gaussian) Brownian motion and (non-Gaussian) Lévy motion, from short bursts of simulation data. Specifically, we use the normalizing flows technology to estimate the transition probability density function (solution of non-local Fokker–Planck equations) from data, and then substitute it into the recently proposed non-local Kramers–Moyal formulae to approximate Lévy jump measure, drift coefficient and diffusion coefficient. We demonstrate that this approach can learn the stochastic differential equation with Lévy motion. We present examples with one- and two-dimensional decoupled and coupled systems to illustrate our method. This approach will become an effective tool for discovering stochastic governing laws and understanding complex dynamical behaviours. This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.

Data-driven prediction in dynamical systems: recent developments

“…Lu et al . [ 192 ] present a new data-driven approach to extract stochastic governing laws with both (Gaussian) Brownian motion and (non-Gaussian) Lévy motion from short bursts of simulation data. The normalizing flow technique is used to estimate the transition probability density function from data and approximate Lévy jump measure, drift coefficient and diffusion coefficient of a stochastic differential equation using non-local Kramers–Moyal formulae.…”

On the anti-missile interception technique of unpowered phase based on data-driven theory

“…Then we can learn the components of the vector field b 3 (x) = Ψ(x)c 3 and b 4 (x) = Ψ(x)c 4 with the coefficients c 3 =[−0.0018, −0.0054, 0.0266, −0.0183, 0.0019, 0.0061, −0.0020, − 0.0677, 0.0025, 0, −0.0029, 0.0079, 0, 0.0025, 0.0029, 0, 0.0190, −0.0024, 0, 0.0021], c 4 =[−0.1559, 0.0134, 0.3722, 0, −0.0316, 0.0039, −0.0018, −0.3626, − 0.0097, 0.0391, 0.0159, 0.0018, 0.0010, −0.0030, 0.0022, 0, 0.1232, 0.0024, −0.0295, 0.0012]. (17) The paths integrated by the learned and true model are compared in Figure 2. It is seen that the results also agree well except more data information, which implies that the algorithm is very robust against environmental noise.…”

“…Then Li and Duan [13,14] made further investigation to propose nonlocal Kramers-Moyal formulas and accordingly devised a novel data-driven method to extract stochastic dynamical systems with (Gaussian) Brownian motion and (non-Gaussian) Lévy motion from sample path data. Additionally, there are also some data-driven methods based on neural networks to learn dynamical models from time-series data [15][16][17][18].…”

Learning effective SDEs from Brownian dynamic simulations of colloidal particles

“…For non-Gaussian stochastic differential equations, modifications of the traditional Kramers–Moyal expansion have also been proposed. 40,41 In ref. 42 the authors proposed a stochastic physics-informed neural network framework (SPINN) that minimizes the distance between the predicted moments of the network (drift and diffusivity) from moments computed with Kramers–Moyal.…”