A Machine Learning Framework for Computing the Most Probable Paths of Stochastic Dynamical Systems
This work addresses a key issue in understanding rare events in nonlinear systems across various scientific fields, representing an incremental improvement over existing methods like the shooting method.
The authors tackled the problem of computing most probable paths in stochastic dynamical systems, which is challenging in high dimensions, by developing a machine learning framework that reformulates a boundary value problem and uses a neural network, achieving efficacy and accuracy in prototypical examples with Gaussian and non-Gaussian noise.
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) Lévy noise. This novel approach is effective in exploring the internal mechanisms of rare events triggered by random fluctuations in various scientific fields.