Compressibility of micromagnetic solutions in tensor train format

TL;DR

This paper demonstrates that tensor train representations can efficiently compress 3D micromagnetic data, significantly reducing computational complexity by exploiting spatial sparsity in large-scale magnetic simulations.

Contribution

The authors show that tensor train formats can overcome traditional micromagnetic simulation scaling issues by exploiting sparsity, enabling more efficient large-scale 3D magnetic modeling.

Findings

Tensor train compression scales approximately as L^{1.8} and (1/a)^{1.2}

TT representations preserve accuracy while reducing data size

Potential to develop new micromagnetic solvers beyond traditional methods

Abstract

For three-dimensional (3D) magnetic objects with linear size $L$ exceeding a few exchange lengths, the micromagnetic state exhibits pronounced informational sparsity: low-dimensional, high-gradient regions (e.g., domain walls) coexist with near-uniformly magnetized volumetric domains. Because standard micromagnetic simulation methods discretize the magnetization on near-uniform 3D grids with linear cell size $a$, they cannot take advantage of this sparsity. The computational problem scales as $\sim L^3$ and $\sim (1/a)^3$. In this Letter, we establish that direct tensor-train (TT) representations overcome these poor scalings by exploiting the spatial sparsity optimally, while preserving accuracy in a controlled way. Focusing on representative flux-closure configurations in soft-magnetic rectangular prisms, in the near-micrometer regime, we demonstrate that the parameter count of TT-compressed micromagnetic data scales approximately as $L^{1.8}$ and $(1/a)^{1.2}$. Hence the relative advantage over dense discretizations rapidly grows with the problem size and refinement level. These first results provide a strong motivation for future developments of micromagnetic solvers in TT format which could transcend the limitations of traditional simulators, with far reaching potential impacts on fundamental research and technology development.