Fang C, Lu Y, Gao T, Duan J. An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable Lévy noise.
CHAOS (WOODBURY, N.Y.) 2022;
32:063112. [PMID:
35778145 DOI:
10.1063/5.0089832]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2022] [Accepted: 05/17/2022] [Indexed: 06/15/2023]
Abstract
Recently, extracting data-driven governing laws of dynamical systems through deep learning frameworks has gained much attention in various fields. Moreover, a growing amount of research work tends to transfer deterministic dynamical systems to stochastic dynamical systems, especially those driven by non-Gaussian multiplicative noise. However, many log-likelihood based algorithms that work well for Gaussian cases cannot be directly extended to non-Gaussian scenarios, which could have high errors and low convergence issues. In this work, we overcome some of these challenges and identify stochastic dynamical systems driven by α-stable Lévy noise from only random pairwise data. Our innovations include (1) designing a deep learning approach to learn both drift and diffusion coefficients for Lévy induced noise with α across all values, (2) learning complex multiplicative noise without restrictions on small noise intensity, and (3) proposing an end-to-end complete framework for stochastic system identification under a general input data assumption, that is, an α-stable random variable. Finally, numerical experiments and comparisons with the non-local Kramers-Moyal formulas with the moment generating function confirm the effectiveness of our method.
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