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

Transition phenomena between metastable states play an important role in complex systems due to noisy fluctuations. We introduce the Onsager–Machlup theory and the Freidlin–Wentzell theory to quantify rare events in stochastic differential equations. By the variational principle, the most probable transition pathway is the minimizer of the action functional, which is governed by the Euler–Lagrange equation. In this paper, the physics-informed neural networks (PINNs) are presented to compute the most probable transition pathway through computing the Euler–Lagrange equation. The convergence result for the empirical loss is obtained for the forward problem. Then we investigate the inverse problem to extract the stochastic differential equation from the most probable transition pathway data. Finally, several numerical experiments are presented to verify the effectiveness of our methods.