STWCR: Weak collocation regression for revealing hidden stochastic dynamics from single trajectory data

Revealing hidden stochastic dynamics from observation data is of great importance in applications. Traditional methods usually require a large number of independent trajectories to obtain the probability distributions at several time snapshots. This paper proposes a fast and accurate regression method to obtain the stochastic dynamics from a single trajectory. If the stochastic system possesses a stationary distribution, we can utilize the ergodicity and the Weak Collocation Regression method, which is a fast method based on the weak form and sparse regression to solve inverse problems of partial differential equations related to probability density functions, to extract the score function of the stationary distribution from the single trajectory data without high cost. Then a novel expression of the drift term in terms of the score function and the diffusion term can be established, based on which we can extract the drift and diffusion terms through the Weak Collocation Regression method again. Our method has a high efficiency and low requirements for the trajectory data. Its outstanding performance has been verified through extensive numerical experiments, including non-gradient drift, multi-scale and high-dimensional problems.