Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas

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 nonlocal Fokker-Planck equation) from data, and then substitute it into the recently proposed nonlocal Kramers-Moyal formulas 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 behaviors.

Extracting stochastic dynamical systems with α-stable Lévy noise from data

“…The identification of non-Gaussian stochastic systems via nonlocal Kramers-Moyal formulas has a rather high precision, despite the requirements of huge amounts of data with the magnitude of at least 10 6 and an limitation of choosing an appropriate dictionary of basis functions [20]. We employed an advanced technique named normalizing flows to improve this method in [25]. Normalizing flows is a generative model in deep learning field, which can be used to express probability density using a prior probability density and a series of bijective transformations [30].…”

“…Two specific transformations are used in [25], i.e. neural spline flows and real-value non-volume preserving transformations (real NVP).…”

“…In order to improve the complexity of transformation, we take the N = 5 compositions T N θ as our final transformation, where the neural network is a full connected neural network with 3 hidden layers and 32 nodes each layer. In addition, we choose the interval [−B, B] = [−3, 3] and K = 5 bins, see [13,25] for more details.…”

“…So far, there are a few methods proposed recently. The first method is by nonlocal Kramers-Moyal formulas to express the Lévy jump measure, drift coefficient and diffusion coefficient by transition probability density (solution of nonlocal Fokker-Planck equation) or sample paths (solution of stochastic differential equation) [20,21], which can be also realized via normalizing flows [25]. The second method is to learn the nonlocal Fokker-Planck equation corresponding to the stochastic differential equation by neural networks [8].…”

Data-driven modeling of a four-dimensional stochastic projectile system

“…[9][10][11] There are also some data-driven methods based on neural networks to learn dynamical systems from sample paths. [12][13][14][15] Additionally, some researchers are devoted to developing techniques to extract the dynamical behav-iors such as mean exit time [16,17] and most probable path. [18,19] Compared with the Koopman operator method, the neural network method and many other methods for system identification, the sparse learning based on the Kramers-Moyal formulas used in this study has the advantages that its computation speed is very fast and it is easy to program.…”