DEEP LEARNING FOR HIGH-DIMENSIONAL SYSTEMS GOVERNING ANOMALOUS DIFFUSION*

Abstract

Anomalous diffusion in complex systems, such as intracellular transport and soft glassy materials, often emerges from complex dynamics between a tracer particle and multiple internal states of its heterogeneous environment. We develop a deep learning framework based on backward stochastic differential equations to solve high-dimensional anomalous Fokker-Planck equations with numerous internal states in an unbounded domain. The transition matrices of internal states can be singular or rectangular. Tailored algorithmic strategies are proposed for each case. The resulting scalable algorithms provide a way to probe the microscopic origins of anomalous diffusion in high-dimensional heterogeneous systems. Their effectiveness and accuracy are validated in several ways, depending on the number and dimension of the equations to be solved. From a modeling perspective, such systems are naturally described by high-dimensional coupled Langevin equations or their associated Fokker-Planck (FP) equations [12,13]. The internal states, which represent distinct dynamical modes or local environments, are governed by a transition matrix. A significant mathematical and computational challenge arises since this matrix can be singular or non-singular, reflecting different physical scenarios such as absorbing states or transitions between different environments. The high-dimensionality of the state space and the potentially singular transition matrix make the problem intractable for traditional numerical methods. To address these challenges, we propose a novel deep learning framework based on backward stochastic differential equations (BSDEs) to solve high-dimensional anomalous Fokker-Planck equations with numerous internal states in an unbounded domain. The framework is designed to handle both singular and non-singular transition matrices, providing a unified approach to a broad class of anomalous diffusion problems. The proposed algorithms are scalable and can effectively probe the microscopic origins of anomalous diffusion in high-dimensional heterogeneous systems. We validate the effectiveness and accuracy of our algorithms through extensive numerical experiments, demonstrating their capability to recover known solutions for intractable lower-dimensional systems and by exploring previously inaccessible regimes and comparing with the solutions obtained through other deep learning approaches in high dimensions. Ultimately, this scalable methodology not only provides a powerful computational tool but also serves to probe the microscopic origins of anomalous diffusion by enabling the direct interrogation of complex, high-dimensional coupled dynamics.