Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures
Abstract
Parametrized PDEs with density-valued solutions are often difficult to approximate with classical linear reduced-order models, especially in transport-dominated regimes. We introduce an optimal-transport-based reduced-order modeling that represents each density by the Kantorovich potential transporting a fixed reference density to the target density, and then maps these potentials to transport signatures using a weighted Laplacian associated with the reference measure. This embeds the density-valued solution map in a Hilbert space while preserving control of the induced transport maps and Wasserstein error. We treat the signature map as a continuous matrix indexed by parameters and space, construct a low-rank skeleton decomposition using a maximal-volume criterion, and learn the parameter-to-coefficient map with a neural network for efficient non-intrusive online evaluation. The reconstructed solution is obtained by pushing forward the reference density, so mass and positivity preservation is built into the method. We prove a mean-squared Wasserstein error bound separating low-rank approximation, discretization, sampling, and learning errors, and demonstrate the method on density-valued continuity equation and Burgers equation, where transport signatures yield substantially lower-rank structure than the original density snapshots.
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BibTeX
@misc{yu2026structurepreserving,
title = {{Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures}},
author = {Jiajia Yu and Jingwei Hu and Fengyan Li and Shanyin Tong and Yunan Yang and Zhaiming Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.01696},
eprint = {2607.01696},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.01696},
}
RIS
TY - UNPB
TI - Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures
AU - Jiajia Yu
AU - Jingwei Hu
AU - Fengyan Li
AU - Shanyin Tong
AU - Yunan Yang
AU - Zhaiming Shen
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2607.01696
UR - https://arxiv.org/abs/2607.01696
AB - Parametrized PDEs with density-valued solutions are often difficult to approximate with classical linear reduced-order models, especially in transport-dominated regimes. We introduce an optimal-transport-based reduced-order modeling that represents each density by the Kantorovich potential transporting a fixed reference density to the target density, and then maps these potentials to transport signatures using a weighted Laplacian associated with the reference measure. This embeds the density-valued solution map in a Hilbert space while preserving control of the induced transport maps and Wasserstein error. We treat the signature map as a continuous matrix indexed by parameters and space, construct a low-rank skeleton decomposition using a maximal-volume criterion, and learn the parameter-to-coefficient map with a neural network for efficient non-intrusive online evaluation. The reconstructed solution is obtained by pushing forward the reference density, so mass and positivity preservation is built into the method. We prove a mean-squared Wasserstein error bound separating low-rank approximation, discretization, sampling, and learning errors, and demonstrate the method on density-valued continuity equation and Burgers equation, where transport signatures yield substantially lower-rank structure than the original density snapshots.
ER -