An end-to-end deep learning approach for extracting stochastic dynamical systems with a-stable Levy 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 alpha-stable Levy noise from only random pairwise data. Our innovations include (1) designing a deep learning approach to learn both drift and diffusion coefficients for Levy induced noise with alpha 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 alpha-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. Published under an exclusive license by AIP Publishing.