Detecting the maximum likelihood transition path from data of stochastic dynamical systems

In recent years, data-driven methods for discovering complex dynamical systems in various fields have attracted widespread attention. These methods make full use of data and have become powerful tools to study complex phenomena. In this work, we propose a framework for detecting dynamical behaviors, such as the maximum likelihood transition path, of stochastic dynamical systems from data. For a stochastic dynamical system, we use the Kramers-Moyal formula to link the sample path data with coefficients in the system, then use the extended sparse identification of nonlinear dynamics method to obtain these coefficients, and finally calculate the maximum likelihood transition path. With two examples of stochastic dynamical systems with additive or multiplicative Gaussian noise, we demonstrate the validity of our framework by reproducing the known dynamical system behavior.