Gappy probabilistic manifold decomposition for nonlinear field reconstruction
Abstract
This paper proposes gappy probabilistic manifold decomposition (Gappy PMD), a nonlinear method for reconstructing high-dimensional fields from extremely sparse measurements. Gappy PMD reconstructs the field on the nonlinear manifold learned by probabilistic manifold decomposition (PMD). We further propose a differentiable point selection method for reduced-order model (ROM)-based field reconstruction (DPS). Using differentiable meshless interpolation within the ROM-based reconstruction framework, DPS makes the full-field reconstruction error differentiable with respect to the sampling locations and directly optimizes these locations. In addition, a theoretical error analysis for Gappy PMD is also given. It splits the squared reconstruction error into two orthogonal parts: one normal to the reconstruction manifold and the other induced by sparse sampling and observation noise. Under a stability condition on the sampling operator, this error vanishes with the PMD approximation error and the noise. The Gappy PMD is evaluated on three numerical test cases: flow past a cylinder, lid-driven cavity flow, and backward-facing step flow. For the same reduced dimension and sampling points, Gappy PMD attains mean relative L2 errors one to two orders of magnitude below Gappy POD. Optimizing the sampling points with DPS further improves reconstruction accuracy and robustness.
Related
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BibTeX
@misc{feng2026gappy,
title = {{Gappy probabilistic manifold decomposition for nonlinear field reconstruction}},
author = {Qihan Feng and Jiaming Guo and Jiarun Meng and Dunhui Xiao},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29014},
eprint = {2608.29014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29014},
}
RIS
TY - UNPB
TI - Gappy probabilistic manifold decomposition for nonlinear field reconstruction
AU - Qihan Feng
AU - Jiaming Guo
AU - Jiarun Meng
AU - Dunhui Xiao
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29014
UR - https://arxiv.org/abs/2608.29014
AB - This paper proposes gappy probabilistic manifold decomposition (Gappy PMD), a nonlinear method for reconstructing high-dimensional fields from extremely sparse measurements. Gappy PMD reconstructs the field on the nonlinear manifold learned by probabilistic manifold decomposition (PMD). We further propose a differentiable point selection method for reduced-order model (ROM)-based field reconstruction (DPS). Using differentiable meshless interpolation within the ROM-based reconstruction framework, DPS makes the full-field reconstruction error differentiable with respect to the sampling locations and directly optimizes these locations. In addition, a theoretical error analysis for Gappy PMD is also given. It splits the squared reconstruction error into two orthogonal parts: one normal to the reconstruction manifold and the other induced by sparse sampling and observation noise. Under a stability condition on the sampling operator, this error vanishes with the PMD approximation error and the noise. The Gappy PMD is evaluated on three numerical test cases: flow past a cylinder, lid-driven cavity flow, and backward-facing step flow. For the same reduced dimension and sampling points, Gappy PMD attains mean relative L^2 errors one to two orders of magnitude below Gappy POD. Optimizing the sampling points with DPS further improves reconstruction accuracy and robustness.
ER -