Data driven discovery of escape phenomena in stochastic systems

Stochastic dynamical systems, influenced by random disturbances, exhibit complex behaviors that are critical to understanding in various fields, such as physics, biology, and finance. The study of escape phenomena, where systems transition from stable states under the influence of noise, is essential for analyzing the dynamical behavior and learning the stochastic dynamics. This paper focuses on two key deterministic quantities, the mean exit time and the escape probability, which are widely used to analyze escape characteristics in stochastic dynamical systems. Traditional methods for computing escape problems, such as the finite difference method, finite element method, finite volume method, and Monte Carlo simulations, face challenges in high-dimensional systems and irregular domains. To address these limitations, we propose a comprehensive framework based on physics-informed neural networks. This framework is designed to solve both forward and inverse problems of escape phenomena in stochastic systems driven by Brownian motion. Our approach eliminates the need for mesh generation and naturally accommodates irregular domains, achieving a better balance between computational efficiency and accuracy. It not only overcomes the drawbacks of traditional numerical methods in solving mean exit time and escape probability but also enables learning stochastic dynamics from escape data. Through a series of numerical examples, we demonstrate the effectiveness and accuracy of the proposed method.