Dynamical Tensor Train Approximation for Kinetic Equations
Geshuo Wang, Jingwei Hu
TL;DR
The paper tackles the computational barrier posed by high-dimensional kinetic equations by introducing a dynamical low-rank TT framework that discretizes velocity space with tensor trains while treating spatial coordinates as parameters. A projector-splitting integrator updates TT cores at each spatial grid point, enabling small TT-ranks and substantial reductions in memory and compute compared to full-tensor discretizations. The method is systematically applied to a range of kinetic models, including BGK, diffusion/Fokker-Planck, and VAFP, across both spatially homogeneous and inhomogeneous settings, with results demonstrating high accuracy and strong efficiency gains. The work lays a foundation for scalable kinetic simulations and discusses avenues for future enhancements, such as adaptive ranks and domain-decomposition strategies.
Abstract
The numerical solution of kinetic equations is challenging due to the high dimensionality of the underlying phase space. In this paper, we develop a dynamical low-rank method based on the projector-splitting integrator in tensor-train (TT) format. The key idea is to discretize the three-dimensional velocity variable using tensor trains while treating the spatial variable as a parameter, thereby exploiting the low-rank structure of the distribution function in velocity space. In contrast to the standard step-and-truncate approach, this method updates the tensor cores through a sweeping procedure, allowing the use of relatively small TT-ranks and leading to substantial reductions in memory usage and computational cost. We demonstrate the effectiveness of the proposed approach on several representative kinetic equations.