An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable Lévy noise

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.