Nonlocal Kramers-Moyal formulas and data-driven discovery of stochastic dynamical systems with multiplicative Lévy noise

TL;DR

This work addresses identifying stochastic dynamics driven by multiplicative Lévy noise, where jumps and heavy tails challenge traditional methods. It develops nonlocal Kramers-Moyal formulas that connect short-time transition statistics to drift, diffusion, Lévy-kernel, and state-dependent noise intensity, and then delivers a data-driven pipeline to recover all components from trajectory data. The framework includes learning algorithms for the Lévy motion, drift, and diffusion (with convergence and error guarantees) and is validated on Maier-Stein and high-dimensional Rössler networks, demonstrating accurate parameter recovery and scalability. Collectively, the approach provides a principled, interpretable, and scalable toolbox for discovering SDEs with non-Gaussian, state-dependent fluctuations across domains such as climate science, neuroscience, epidemiology, finance, and biology.

Abstract

Traditional data-driven methods, effective for deterministic systems or stochastic differential equations (SDEs) with Gaussian noise, fail to handle the discontinuous sample paths and heavy-tailed fluctuations characteristic of Lévy processes, particularly when the noise is state-dependent. To bridge this gap, we establish nonlocal Kramers-Moyal formulas, rigorously generalizing the classical Kramers-Moyal relations to SDEs with multiplicative Lévy noise. These formulas provide a direct link between short-time transition probability densities (or sample path statistics) and the underlying SDE coefficients: the drift vector, diffusion matrix, Lévy jump measure kernel, and Lévy noise intensity functions. Leveraging these theoretical foundations, we develop novel data-driven algorithms capable of simultaneously identifying all governing components from data and establish convergence results and error analysis for the algorithms. We validate the framework through extensive numerical experiments on prototypical systems. This work provides a principled and practical toolbox for discovering interpretable SDE models governing complex systems influenced by discontinuous, heavy-tailed, state-dependent fluctuations, with broad applicability in climate science, neuroscience, epidemiology, finance, and biological physics.

Figures (4)

• Figure 1: Schematic diagrams for illustrating the proposed data-driven framework. (a) Input data, including sample path data and the constructed datasets $(\mathbbm{Z},\mathbbm{X})$. (b) The algorithm for learning Lévy jump measure and intensity functions, including data preprocessing and Lévy noise parameter identification such as non-Gaussian index $\alpha$, skewness parameter $\beta$, and noise intensity function $\sigma(x)$. (c) The algorithm for learning the drift coefficient $b(x)$. (d) The algorithm for learning the diffusion coefficient $a(x)$. (e) The underestimated non-Gaussian stochastic dynamical systems.
• Figure 2: Comparison between learned and true coefficient functions of the Maier-Stein system. (a,b) The approximated values, sparse regression results, neural network results, and true functions of the Lévy noise intensity functions $\sigma_1(x_1)$ and $\sigma_2(x_2)$, denoted by blue stars, red points, green curves, and orange circles, respectively. (c,e) The learned drift coeffcient $\tilde{b}(x)$. (d,f) The true drift coeffcient $b(x)$. (g,i,k) The learned diffusion coeffcient $\tilde{a}(x)$. (h,j,l) The true diffusion coeffcient $a(x)$.
• Figure 3: The influence of hyperparameters such as data volume $M$ and time step $h$ on approximation errors of Lévy noise parameters $\alpha$, $\beta$, and $\sigma$, with $N_c$ and $\varepsilon$ fixed. The blue line with circles and the brown line with stars represent the first and second dimensions, respectively. (a-c) The approximation errors of Lévy noise parameters about data volume $M$ with $h=0.001$ fixed. The red dashed lines indicate the function $20 \times M^{-0.5}$. (d-f) The approximation errors of Lévy noise parameters about data volume $M$ with $Mh=10^4$ fixed. (g-i) The approximation errors of Lévy noise parameters about $h^{-1}$ with $M=10^7$ fixed. (j) The graph of the function $f(x) = \sqrt{x} + \frac{1}{x}$.
• Figure 4: Comparison between the Lévy noise intensity functions $\sigma_{i,j}(x_{i,j})$, $i=1,2,3,4,5$, $j=1,2,3$ of approximated values, sparse regression results, neural network results, and true functions of the Rössler system, denoted by blue stars, red points, green curves, and orange circles, respectively.