Compressibility of micromagnetic solutions in tensor train format
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866917451154849792 |
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| author | Valet, Thierry Vukadinovic, Nicolas |
| author_facet | Valet, Thierry Vukadinovic, Nicolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_28008 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Compressibility of micromagnetic solutions in tensor train format Valet, Thierry Vukadinovic, Nicolas Mesoscale and Nanoscale Physics Computational Physics 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. |
| title | Compressibility of micromagnetic solutions in tensor train format |
| topic | Mesoscale and Nanoscale Physics Computational Physics |
| url | https://arxiv.org/abs/2604.28008 |