Generative AI Models for Learning Flow Maps of Stochastic Dynamical Systems in Bounded Domains
This work addresses computational challenges in simulating stochastic dynamical systems with boundary interactions, which is incremental as it builds on existing machine learning methods but adapts them for bounded domains.
The authors tackled the challenge of simulating stochastic differential equations in bounded domains by developing a hybrid data-driven approach that combines a conditional diffusion model with an exit prediction neural network, achieving accurate modeling of interior dynamics and boundary exit phenomena across one-, two-, and three-dimensional test cases.
Simulating stochastic differential equations (SDEs) in bounded domains, presents significant computational challenges due to particle exit phenomena, which requires accurate modeling of interior stochastic dynamics and boundary interactions. Despite the success of machine learning-based methods in learning SDEs, existing learning methods are not applicable to SDEs in bounded domains because they cannot accurately capture the particle exit dynamics. We present a unified hybrid data-driven approach that combines a conditional diffusion model with an exit prediction neural network to capture both interior stochastic dynamics and boundary exit phenomena. Our ML model consists of two major components: a neural network that learns exit probabilities using binary cross-entropy loss with rigorous convergence guarantees, and a training-free diffusion model that generates state transitions for non-exiting particles using closed-form score functions. The two components are integrated through a probabilistic sampling algorithm that determines particle exit at each time step and generates appropriate state transitions. The performance of the proposed approach is demonstrated via three test cases: a one-dimensional simplified problem for theoretical verification, a two-dimensional advection-diffusion problem in a bounded domain, and a three-dimensional problem of interest to magnetically confined fusion plasmas.