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.Comment: 52 figure

Data-driven method to learn the most probable transition pathway and stochastic differential equations

Transition phenomena between metastable states play an important role in complex systems due to noisy fluctuations. In this paper, the physics informed neural networks (PINNs) are presented to compute the most probable transition pathway. It is shown that the expected loss is bounded by the empirical loss. And the convergence result for the empirical loss is obtained. Then, a sampling method of rare events is presented to simulate the transition path by the Markovian bridge process. And we investigate the inverse problem to extract the stochastic differential equation from the most probable transition pathway data and the Markovian bridge process data, respectively. Finally, several numerical experiments are presented to verify the effectiveness of our methods