Data-driven approach for extracting the most probable exit trajectory of stochastic dynamical systems with non-Gaussian Lévy noise

The burgeoning data-driven techniques endow large potential to predict fairly practical or complex dynamical systems in various fields through massive data. Lévy noise, a more universal and intricate fluctuation model comparing with Gaussian white noise, is widely employed in many non-Gaussian cases to mimic bursting or hopping. In this manuscript, we present a systematic data-driven method to identify the most probable exit trajectory of a system that is perturbed both by Gaussian white noise and non-Gaussian Lévy noise. The main theoretical and numerical conceptions involve a set of extended Kramers-Moyal formulas and the Kolmogorov forward equation in classic dynamical systems theory as well as a supervise learning theory to solve the fitting problems by using the Cross Validation. We then give two examples to show the feasibility in detail, and do a brief bifurcation analysis for the most probable exit trajectory. The above approach will serve as a numerical correspondence to as well as verification for the relative theoretical research, and provide a referential resolution to the numerical identification of more transition indicators of this complex system, which is more general than the Gaussian diffusion process.