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

Area of Science:

• Dynamical Systems and Control Theory
• Machine Learning and Artificial Intelligence
• Stochastic Processes and Statistics

Background:

• Deep learning is increasingly used to extract governing laws from data in dynamical systems.
• Transferring deterministic to stochastic dynamical systems, particularly with non-Gaussian multiplicative noise, presents significant challenges.
• Existing log-likelihood methods struggle with non-Gaussian noise, leading to errors and convergence issues.

Purpose of the Study:

• To develop a novel deep learning framework for identifying stochastic dynamical systems driven by alpha-stable Lévy noise.
• To address the limitations of current methods in handling non-Gaussian multiplicative noise and complex noise intensities.
• To provide an end-to-end solution for stochastic system identification using general input data assumptions.

Main Methods:

• Designed a deep learning approach to learn both drift and diffusion coefficients for Lévy-induced noise across all alpha values.
• Enabled the learning of complex multiplicative noise without restrictions on noise intensity.
• Proposed a complete, end-to-end framework for stochastic system identification assuming alpha-stable random variables as input.

Main Results:

• Successfully identified stochastic dynamical systems driven by alpha-stable Lévy noise from random pairwise data.
• Demonstrated the ability to learn drift and diffusion coefficients for a wide range of alpha values.
• Validated the framework's effectiveness through numerical experiments and comparisons with non-local Kramers-Moyal formulas.

Conclusions:

• The proposed deep learning method effectively identifies stochastic dynamical systems with alpha-stable Lévy noise.
• The framework overcomes key challenges associated with non-Gaussian multiplicative noise and complex noise intensities.
• This approach offers a robust and generalizable solution for stochastic system identification in various scientific fields.