Detecting Stochastic Governing Laws with Observation on Stationary Distributions

Mathematical models for complex systems are often accompanied with uncertainties. The goal of this paper is to extract a stochastic differential equation governing model with observation on stationary probability distributions. We develop a neural network method to learn the drift and diffusion terms of the stochastic differential equation. We introduce a new loss function containing the Hellinger distance between the observation data and the learned stationary probability density function. We discover that the learnt stochastic differential equation provides a fair approximation of the data-driven dynamical system after minimizing this loss function during the training method. The effectiveness of our method is demonstrated in numerical experiments.