Data driven discovery of escape phenomena in stochastic systems

Area of Science:

• Dynamical Systems and Chaos Theory
• Computational Physics
• Stochastic Processes

Background:

• Stochastic dynamical systems are crucial in physics, biology, and finance, but their complex behaviors, especially escape phenomena, are challenging to analyze.
• Traditional numerical methods (finite difference, finite element, finite volume, Monte Carlo) struggle with high-dimensional systems and irregular domains when calculating escape characteristics like mean exit time and escape probability.

Purpose of the Study:

• To develop a mesh-free framework using physics-informed neural networks (PINNs) for analyzing escape phenomena in stochastic systems driven by Brownian motion.
• To address limitations of traditional methods in high-dimensional and irregular domains for computing mean exit time and escape probability.
• To enable learning stochastic dynamics from escape data through solving both forward and inverse problems.

Main Methods:

• A comprehensive framework based on physics-informed neural networks (PINNs) is proposed.
• The PINN approach is applied to solve forward and inverse problems related to escape phenomena in stochastic systems.
• The method avoids mesh generation, naturally handling irregular domains and offering improved computational efficiency and accuracy.

Main Results:

• The proposed PINN framework effectively computes mean exit time and escape probability in stochastic systems.
• It demonstrates superior performance compared to traditional numerical methods, particularly in complex scenarios.
• Numerical examples confirm the accuracy and effectiveness of the PINN-based approach for analyzing escape characteristics and learning stochastic dynamics.

Conclusions:

• Physics-informed neural networks provide a powerful and versatile tool for studying escape phenomena in stochastic dynamical systems.
• This mesh-free approach offers a significant advancement over traditional numerical techniques, enhancing both computational efficiency and accuracy.
• The framework's ability to learn stochastic dynamics from escape data opens new avenues for research in complex systems analysis.