Neural Operator-Grounded Continuous Tensor Function Representation and Its Applications
Authors: Ruoyang Su, Xi-Le Zhao, Sheng Liu, Wei-Hao Wu, Yisi Luo, Michael K. Ng
Organizations: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China · School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China · Department of Mathematics, Hong Kong Baptist University, Hong Kong, China
Recently, continuous tensor functions have attracted increasing attention, because they can unifiedly represent data both on mesh grids and beyond mesh grids. However, since mode-n product is essentially discrete and linear, the potential of current continuous tensor function representations is still locked. To break this bottleneck, we suggest neural operator-grounded mode-n operators as a continuous and nonlinear alternative of discrete and linear mode-n product. Instead of mapping the discrete core tensor to the discrete target tensor, proposed mode-n operator directly maps the continuous core tensor function to the continuous target tensor function, which provides a genuine continuous representation of real-world data and can ameliorate discretization artifacts. Empowering with continuous and nonlinear mode-n operators, we propose a neural operator-grounded continuous tensor function representation (abbreviated as NO-CTR), which can more faithfully represent complex real-world data compared with classic discrete tensor representations and continuous tensor function representations. Theoretically, we also prove that any continuous tensor function can be approximated by NO-CTR. To examine the capability of NO-CTR, we suggest an NO-CTR-based multi-dimensional data completion model. Extensive experiments across various data on regular mesh grids (multi-spectral images and color videos), on mesh girds with different resolutions (Sentinel-2 images) and beyond mesh grids (point clouds) demonstrate the superiority of NO-CTR.
Figures & tables
Fig. 1: Discrete and linear mode- n operator and proposed continuous and nonlinear mode- n operator. The basic unit performed upon by discrete and linear mode- n operator is the mode- n fiber vectors of the discrete core tensor G , which are then mapped to the mode- n fiber vectors of the discrete target tensor X . The basic unit performed upon by continuous and nonlinear mode- n operator is the mode- n univariate fiber functions of the continuous core tensor function G , which are then mapped to the mode- n univariate fiber functions of the continuous target tensor function X .
Fig. 2: Visual and quantitative recovery results with discrete and linear mode- n operators (LRTFR [ 34 ] ), continuous and nonlinear mode- n operators (proposed NO-CTR) on the MSI Toy (top) and point cloud Frog (bottom) when the sampling rates are 10% .
Fig. 3: Flow chart of NO-CTR and the corresponding multi-dimensional data completion model. We first cleverly leverage neural operators to suggest continuous and nonlinear mode- n operators. Specifically, the basic unit performed upon by continuous and nonlinear mode- n operator is the mode- n univariate fiber functions of the continuous core tensor function G , which are then mapped to the mode- n univariate fiber functions of the continuous target tensor function X . Empowering with continuous and nonlinear mode- n operators, we propose a neural operator-grounded continuous tensor function representation (NO-CTR), which can faithfully represent continuous target tensor function X as a continuous core tensor function G and a series of continuous and nonlinear mode- n operators {F⟨n⟩}n=1N . To examine the capability of NO-CTR, we suggest an NO-CTR-based multi-dimensional data completion model.
TABLE I: Quantitative Recovery Results of MSI Completion
Fig. 4: Visual recovery results of MSI completion at a sampling rate of 10% .
TABLE II: Quantitative Recovery Results of Color Video Completion
Fig. 5: Visual recovery results of color video completion at a sampling rate of 10% .
TABLE III: Quantitative Recovery Results of Sentinel-2 Image Completion
Fig. 6: Visual recovery results of Sentinel-2 image completion at a sampling rate of 10% .
TABLE IV: Quantitative Recovery Results of Point Cloud Completion
Fig. 7: Visual recovery results of point cloud completion at a sampling rate of 10% .
Sampling Rate
10%
20%
MSI
Mode- n Operators
PSNR
SSIM
PSNR
SSIM
Flowers
No Operator
30.565
0.898
34.367
0.948
Discrete & Linear
39.175
0.984
43.119
0.984
Continuous & Nonlinear
42.591
0.994
49.717
0.999
Toy
No Operator
26.190
0.835
29.087
0.895
Discrete & Linear
38.907
0.982
41.395
0.991
TABLE V: Quantitative Recovery Results of MSI Completion With Different Mode- n Operators
Sampling Rate
10%
20%
MSI
Architecture
PSNR
SSIM
PSNR
SSIM
Flowers
No Operator
30.565
0.898
34.367
0.948
FNO
40.940
0.992
47.446
0.997
DeepONet
42.591
0.994
49.717
0.999
Toy
No Operator
26.190
0.835
29.087
0.895
FNO
41.167
0.994
46.076
0.998
TABLE VI: Quantitative Recovery Results of MSI Completion With Different Neural Operators
Sampling Rate
10%
20%
MSI
Core
PSNR
SSIM
PSNR
SSIM
Flowers
LRTFR
39.586
0.983
44.232
0.992
PE-MLP
42.086
0.993
47.326
0.998
SIREN
42.591
0.994
49.717
0.999
Toy
LRTFR
37.781
0.980
41.948
0.991
PE-MLP
40.831
0.993
45.765
0.998
TABLE VII: Quantitative Recovery Results of MSI Completion With Different Representation Methods for the Continuous Core Tensor Function
Sampling Rate
10%
20%
MSI
Method
PSNR
SSIM
Parameters
PSNR
SSIM
Parameters
Flowers
SIREN
38.681
0.977
16 711
46.041
0.995
47 671
LRTFR
36.879
0.965
130 191
43.249
0.991
331 023
NO-CTR
42.591
0.994
507 341
49.717
0.999
472 721
Toy
SIREN
37.259
0.974
16 711
45.306
0.995
47 671
LRTFR
37.661
0.975
130 191
43.099
0.991
595 231
TABLE VIII: Quantitative Recovery Results and Number of Parameters of MSI Completion With Different Continuous Tensor Function Representations
Fig. 8: Quantitative recovery results w.r.t. different numbers of sensors on the MSIs Flowers (left) and Toy (right) at a sampling rate of 10% .
Fig. 9: Quantitative recovery results w.r.t. different numbers of branch networks on the MSIs Flowers (left) and Toy (right) at a sampling rate of 10% .
Tensor Ring (TR) decomposition is a powerful tool for high-order data modeling, but is inherently restricted to discrete forms defined on fixed meshgrids. In this work, we propose a TR functional decomposition for both meshgrid and non-meshgrid data, where factors are parameterized by Implicit Neural Representations (INRs). However, optimizing this continuous framework to capture fine-scale details is intrinsically difficult. Through a frequency-domain analysis, we demonstrate that the spectral structure of TR factors determines the frequency composition of the reconstructed tensor and limits the high-frequency modeling capacity. To mitigate this, we propose a reparameterized TR functional decomposition, in which each TR factor is a structured combination of a learnable latent tensor and a fixed basis. This reparameterization is theoretically shown to improve the training dynamics of TR factor learning. We further derive a principled initialization scheme for the fixed basis and prove the Lipschitz continuity of our proposed model. Extensive experiments on image inpainting, denoising, super-resolution, and point cloud recovery demonstrate that our method achieves consistently superior performance over existing approaches. Code is available at https://github.com/YangyangXu2002/RepTRFD.
Yangyang Xu, Junbo Ke, You-Wei Wen +1
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University · Department of Statistics and Data Science, Southern University of Science and Technology
This paper presents NOIR, a framework that reframes core medical imaging tasks as operator learning between continuous function spaces, challenging the prevailing paradigm of discrete grid-based deep learning. Instead of operating on fixed pixel or voxel grids, NOIR embeds discrete medical signals into shared Implicit Neural Representations and learns a Neural Operator that maps between their latent modulations, enabling resolution-independent function-to-function transformations. We evaluate NOIR across multiple 2D and 3D downstream tasks, including segmentation, shape completion, image-to-image translation, and image synthesis, on several public datasets such as Shenzhen, OASIS-4, SkullBreak, fastMRI, as well as an in-house clinical dataset. It achieves competitive performance at native resolution while demonstrating strong robustness to unseen discretizations, and empirically satisfies key theoretical properties of neural operators. The project page is available here: https://github.com/Sidaty1/NOIR-io.
Sidaty El Hadramy, Nazim Haouchine, Michael Wehrli +1
Department of Biomedical Engineering, University of Basel, Allschwil, Switzerland · Harvard Medical School, Brigham and Women’s Hospital, Boston, United States
Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces. We propose a novel reframing of their use. By introducing an auxiliary base-space, any finite-dimensional function can be viewed as an operator acting by composition on functions of the base-space. Through a range of benchmarks on analytic functions of increasing complexity and dimensionality, we demonstrate that NOs can match or outperform standard multilayer perceptrons and Kolmogorov--Arnold Networks in accuracy while requiring significantly fewer parameters and training time. As a real-world application, we apply a two-dimensional Tensorized Fourier Neural Operator (TFNO) to the nuclear chart, learning a correction to state-of-the-art nuclear mass models as a partially observed residual field. A TFNO ensemble reaches a held-out root-mean-square error of 198.2 keV, placing it among the best recent neural-network approaches while retaining high parameter efficiency and short training times. More broadly, these results introduce NOs as a scalable framework for finite-dimensional function interpolation, from analytic benchmarks to structured scientific data.
Department of Physics, CCTP and ITCP, University of Crete, 71303, Greece · Theoretical Physics Department, CERN, Geneva, Switzerland · Physik-Institut, Universität Zürich, 8057 Zürich, Switzerland