TensorSpectra: Bridging the Spectral and Tensor Worlds for High-Dimensional PDEs
Abstract
High-dimensional partial differential equations (PDEs) remain challenging for spectral methods due to the exponential growth of multidimensional coefficient spaces. We propose TensorSpectra, a low-rank spectral framework that directly parameterizes spectral coefficient tensors in canonical polyadic (CP) form, reducing representation complexity from 𝒪(Nd) to 𝒪(dNr). TensorSpectra combines structure-preserving spectral calculus, de-aliased nonlinear interactions, operator-aware residual weighting, and progressive low-rank corrections to enable efficient PDEs solving in compressed coefficient space. We further establish a theoretical framework connecting spectral truncation, low-rank approximation, and optimization errors. Extensive experiments across five PDE families and 24 benchmark configurations demonstrate State-Of-The-Art accuracy among five representative baselines, with TensorSpectra consistently achieving the lowest relative L2 errors. More broadly, our findings offer a new perspective on high-dimensional PDEs solving, suggesting that exploiting intrinsic low-rank structure in spectral coefficient space provides a viable alternative to conventional physical-space approaches.
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BibTeX
@misc{hu2026tensorspectra,
title = {{TensorSpectra: Bridging the Spectral and Tensor Worlds for High-Dimensional PDEs}},
author = {Bingcheng Hu and Tianchi Yu and Zhaoxiang Li and Ivan Oseledets},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38087},
eprint = {2610.38087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38087},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - TensorSpectra: Bridging the Spectral and Tensor Worlds for High-Dimensional PDEs
AU - Bingcheng Hu
AU - Tianchi Yu
AU - Zhaoxiang Li
AU - Ivan Oseledets
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38087
UR - https://arxiv.org/abs/2610.38087
AB - High-dimensional partial differential equations (PDEs) remain challenging for spectral methods due to the exponential growth of multidimensional coefficient spaces. We propose TensorSpectra, a low-rank spectral framework that directly parameterizes spectral coefficient tensors in canonical polyadic (CP) form, reducing representation complexity from 𝒪(N^d) to 𝒪(dNr). TensorSpectra combines structure-preserving spectral calculus, de-aliased nonlinear interactions, operator-aware residual weighting, and progressive low-rank corrections to enable efficient PDEs solving in compressed coefficient space. We further establish a theoretical framework connecting spectral truncation, low-rank approximation, and optimization errors. Extensive experiments across five PDE families and 24 benchmark configurations demonstrate State-Of-The-Art accuracy among five representative baselines, with TensorSpectra consistently achieving the lowest relative L^2 errors. More broadly, our findings offer a new perspective on high-dimensional PDEs solving, suggesting that exploiting intrinsic low-rank structure in spectral coefficient space provides a viable alternative to conventional physical-space approaches.
ER -