# MTL-FNO: A Lightweight Multi-Task Fourier Neural Operator for Sparse Field Reconstruction

arXiv: https://arxiv.org/abs/2605.26718 · HTML: https://mlsanity.com/paper/2605.26718/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 39 references that are in the map.
- First posted on arXiv 26 May 2026.

## Builds on (most cited first, 20 of 20)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Independent Component Alignment for Multi-Task Learning](https://mlsanity.com/paper/2305.19000/) (2023)
- [A Comprehensive Survey on Model Quantization for Deep Neural Networks in Image Classification](https://mlsanity.com/paper/2205.07877/) (2022)
- [Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning](https://mlsanity.com/paper/2101.00554/) (2021)
- [A Comprehensive Survey on Knowledge Distillation](https://mlsanity.com/paper/2503.12067/) (2025)
- [One Model is All You Need: Multi-Task Learning Enables Simultaneous Histology Image Segmentation and Classification](https://mlsanity.com/paper/2203.00077/) (2022)
- [Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data](https://mlsanity.com/paper/2109.12482/) (2021)
- [Data-Driven Aerospace Engineering: Reframing the Industry with Machine Learning](https://mlsanity.com/paper/2008.10740/) (2020)
- [MTL-LoRA: Low-Rank Adaptation for Multi-Task Learning](https://mlsanity.com/paper/2410.09437/) (2024)
- [Weakly-Supervised Deep Learning of Heat Transport via Physics Informed Loss](https://mlsanity.com/paper/1807.11374/) (2018)
- [PhySense: Sensor Placement Optimization for Accurate Physics Sensing](https://mlsanity.com/paper/2505.18190/) (2025)
- [MoRE: A Mixture of Low-Rank Experts for Adaptive Multi-Task Learning](https://mlsanity.com/paper/2505.22694/) (2025)
- [A Machine Learning Surrogate Modeling Benchmark for Temperature Field Reconstruction of Heat-Source Systems](https://mlsanity.com/paper/2108.08298/) (2021)
- [Customizable Combination of Parameter-Efficient Modules for Multi-Task Learning](https://mlsanity.com/paper/2312.03248/) (2023)
- [When Multi-Task Learning Meets Partial Supervision: A Computer Vision Review](https://mlsanity.com/paper/2307.14382/) (2023)
- [Low Rank Optimization for Efficient Deep Learning: Making A Balance between Compact Architecture and Fast Training](https://mlsanity.com/paper/2303.13635/) (2023)
- [Comparison of two artificial neural networks trained for the surrogate modeling of stress in materially heterogeneous elastoplastic solids](https://mlsanity.com/paper/2210.16994/) (2022)
- [Parameter-efficient Multi-Task and Multi-Domain Learning using Factorized Tensor Networks](https://mlsanity.com/paper/2310.06124/) (2023)

## Abstract (arXiv)

Efficient onboard multi-field sparse reconstruction is essential for the autonomous operation of aerospace vehicles. While existing deep learning models exhibit promise for single-field reconstruction, deploying multiple independent models leads to prohibitive model size growth and fails to exploit cross-field correlations, particularly under few-shot conditions. To address these challenges, we first propose a lightweight multi-task Fourier neural operator (MTL-FNO), an end-to-end joint training framework based on hard parameter sharing. In each layer, the parameters are divided into shared and task-specific components to capture common features across fields while preserving task-specific characteristics. Moreover, the task-specific fine-tuning parameters are implemented as low-rank terms, achieving substantial model compression. Second, to address the difficulty of co-optimizing shared and task-specific parameters along with their real and imaginary parts, we revisit the FNO's spectral weight from a polar-form perspective and devise a physically meaningful decoupled optimization scheme. Specifically, we apply polar decomposition to slice-wise disentangle the spectral weight into a unitary tensor encoding phase information and a positive semi-definite tensor characterizing amplitude. By decoupling the optimization of phase and amplitude, our method can effectively mitigate tasks conflict. Meanwhile, to preserve unitary geometric fidelity during training, the Cayley transform is introduced to reparameterize the unitary tensor, converting the constrained optimization problem to an unconstrained one. Finally, the effectiveness of the proposed method under few-shot conditions is validated on two representative engineering cases. Results show that MTL-FNO achieves accuracy comparable to or even surpassing that of standard FNO, while reducing total model size by 76% and 60%, respectively.