Identifying stochastic governing equations from data of the most probable transition trajectories.

Extracting the governing stochastic differential equation model from elusive data is crucial to understand and forecast dynamics for various systems. We devise a method to extract the drift term and estimate the diffusion coefficient of a governing stochastic dynamical system, from its time-series data for the most probable transition trajectory. By the Onsager–Machlup theory, the most probable transition trajectory satisfies the corresponding Euler–Lagrange equation, which is a second-order deterministic ordinary differential equation (ODE) involving the drift term and diffusion coefficient. We first estimate the coefficients of the Euler–Lagrange equation based on the data of the most probable trajectory, and then calculate the drift and diffusion coefficient of the governing system. These two steps involve sparse regression and optimization for a loss function with parameters. We select the estimators from all the results with different parameters by comparing the errors caused by the difference between the two sides of the second-order ODE for the most probable transition trajectory. We finally illustrate our method, especially for parameter selection, with examples to verify the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]