Generative AI Models for Learning Flow Maps of Stochastic Dynamical Systems in Bounded Domains

2025-07-17

AI Summary

Modeling particle escape dynamics in stochastic differential equations (SDEs) over bounded domains remains challenging due to the difficulty of accurately capturing first-exit events under random forcing. Method: This paper proposes a hybrid data-driven framework integrating a conditional diffusion model with an escape prediction network. It jointly learns intra-domain stochastic evolution and boundary escape responses, leveraging training-free diffusion sampling, closed-form score functions, a binary-classification neural network, and binary cross-entropy loss to co-model non-escape transitions and escape decisions via probabilistic sampling. Contribution/Results: To our knowledge, this is the first method enabling accurate probabilistic modeling of first-exit events for SDEs on bounded domains. Theoretical analysis guarantees convergence. Experiments across multiscale benchmarks—including 1D analytically solvable SDEs, 2D advection–diffusion systems, and 3D magnetically confined fusion plasmas—demonstrate substantial improvements in reproducing boundary statistics and stochastic flow fields.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• Modeling stochastic dynamics and boundary interactions in bounded domains
• Capturing particle exit phenomena in stochastic differential equations
• Integrating diffusion models and neural networks for accurate SDE simulation

Innovation

• Hybrid data-driven approach combining diffusion model and neural network
• Neural network learns exit probabilities with convergence guarantees
• Training-free diffusion model generates non-exiting particle transitions