An optimal control method to compute the most likely transition path for stochastic dynamical systems with jumps

Area of Science:

• Complex Systems Dynamics
• Stochastic Processes
• Computational Physics

Background:

• Real-world phenomena often exhibit abrupt, intermittent behaviors best modeled by stochastic differential equations with non-Gaussian Lévy noise.
• Identifying the most likely transition paths between metastable states is crucial for understanding rare, high-impact events.
• Calculating these paths is challenging due to the inexpressible nature of the rate function in non-Gaussian Lévy noise systems.

Purpose of the Study:

• To develop a method for calculating the most likely transition paths in stochastic dynamical systems driven by non-Gaussian Lévy noise.
• To overcome the challenge of the rate function not being explicitly expressible by paths.
• To formulate an optimal control problem for identifying these critical transition paths.

Main Methods:

• Formulation of an optimal control problem to determine the most likely transition path.
• Development and application of a neural network method to solve the optimal control problem.
• Investigation of experimental cases for both Gaussian and non-Gaussian noise scenarios.

Main Results:

• Successfully formulated an optimal control problem to find the most likely transition path.
• Developed a neural network approach to efficiently solve this problem for systems with non-Gaussian Lévy noise.
• Demonstrated the method's efficacy through experiments involving both Gaussian and non-Gaussian noise.

Conclusions:

• The proposed neural network-based optimal control method provides an effective solution for calculating most likely transition paths in systems with non-Gaussian Lévy noise.
• This approach overcomes the limitations of explicit rate function calculations, offering a practical tool for analyzing rare events in complex systems.
• The findings have implications for various fields relying on stochastic modeling, including physics, chemistry, and finance.