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