Physics-Informed Functional Tucker Method with RKHS Factors for Sparse Spatiotemporal Reconstruction
Authors: Zhengyang Liu, Jianguo Huang, Junxi Hu, Yucheng Mi, Haojie Cai, Wenxuan Shen, Zhikun Zhang, Renfeng Peng, Guangtao Zhang, Zhiqiang Liu
arXiv: 2611.03717 [cs.CV] — Submitted 31 October 2026
Comments: Accepted at NeurIPS 2026
Subjects: Computer Vision and Pattern Recognition (cs.CV); Artificial Intelligence (cs.AI)
Abstract
Reconstructing physical dynamics from fixed sparse sensors and missing frames is a fundamental challenge in scientific research. Sparsity leaves large regions of the field unconstrained, making the inverse problem ill-posed: a model may fit the observed sensors while producing inaccurate off-sensor structures or unreliable predictions between support frames.
To address this, we propose PhysFTM—a physics-informed functional Tucker model that parameterizes the Tucker mode factors in reproducing kernel Hilbert spaces (RKHSs) and incorporates PDE constraints through finite-difference residuals evaluated at off-sensor collocation points. These residuals provide additional physical information in regions without direct observational supervision.
PhysFTM first reconstructs continuous spatial fields at support frames via an alternating minimization scheme, optimizing the Tucker core tensor and the RKHS-based factors separately. To enable missing-frame reconstruction, PhysFTM constructs a continuous trajectory in Tucker-core space, which is decoded by the learned spatial RKHS-FTM representation to recover the full spatiotemporal field.
Experiments on Allen–Cahn, Navier–Stokes, and three-dimensional Heat equations, as well as a three-dimensional gridded Copernicus ocean reanalysis, demonstrate accurate 2D/3D reconstruction under two missing-frame protocols. Code is available at github.com/liusirzzz/PhysFTM.
Key Contributions
- Physics-informed Tucker decomposition: Tucker mode factors are parameterized in RKHSs, enabling continuous functional representations of spatial fields.
- PDE-constrained learning at collocation points: Finite-difference residuals evaluated at off-sensor locations inject physical priors where no observations exist, mitigating the ill-posedness of sparse reconstruction.
- Alternating minimization scheme: The Tucker core tensor and RKHS-based factors are optimized separately to reconstruct continuous spatial fields at support frames.
- Core-space trajectory modeling: Missing frames are recovered by constructing a continuous trajectory in Tucker-core space and decoding it through the learned spatial RKHS-FTM representation.
- Broad empirical validation: Accurate 2D/3D reconstruction is demonstrated across Allen–Cahn, Navier–Stokes, and 3D Heat equation benchmarks, plus a 3D gridded Copernicus ocean reanalysis, under two missing-frame protocols.
Methodology Overview
PhysFTM tackles the ill-posed inverse problem of reconstructing physical fields from sparse fixed sensors and incomplete temporal frames. The approach combines two complementary sources of information:
- Observational supervision at sensor locations, captured through a functional Tucker decomposition whose mode factors live in RKHSs.
- Physical supervision through PDE residuals evaluated at collocation points, which constrain the solution in unobserved regions.
The alternating minimization scheme decouples the optimization of the core tensor from that of the RKHS factors, improving tractability. For temporal gaps, the model leverages the continuity of the Tucker-core space to interpolate missing frames and decode them into full spatiotemporal fields.
Availability
Code: https://github.com/liusirzzz/PhysFTM
Cite as: arXiv:2611.03717 [cs.CV]
via ArXiv CV