  The emergence of transition phenomena between metastable states induced by
noise plays a fundamental role in a broad range of nonlinear systems. The
computation of the most probable paths is a key issue to understand the
mechanism of transition behaviors. Shooting method is a common technique for
this purpose to solve the Euler-Lagrange equation for the associated action
functional, while losing its efficacy in high-dimensional systems. In the
present work, we develop a machine learning framework to compute the most
probable paths in the sense of Onsager-Machlup action functional theory.
Specifically, we reformulate the boundary value problem of Hamiltonian system
and design a neural network to remedy the shortcomings of shooting method. The
successful applications of our algorithms to several prototypical examples
demonstrate its efficacy and accuracy for stochastic systems with both
(Gaussian) Brownian noise and (non-Gaussian) L\'evy noise. This novel approach
is effective in exploring the internal mechanisms of rare events triggered by
random fluctuations in various scientific fields.
