MTL-FNO: A Lightweight Multi-Task Fourier Neural Operator for Sparse Field Reconstruction
Authors: Siyu Ye, Shihang Li, Zhiqiang Gong, Benrong Zhang, Weien Zhou, Yiyong Huang, Wen Yao
Organizations: Defense Innovation Institute, Academy of Military Science, Beijing, 100071, China · Intelligent Game and Decision Laboratory, Beijing, 100071, China
Abstract
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.
Sparse representations are often expected to make models smaller and also reduce inference cost. For Fourier Neural Operators (FNOs), these objectives are not equivalent or do not always align: removing parts of the learned operator can leave the underlying transforms and dense computations unchanged, while changing the grid on which the model is evaluated can introduce overhead of its own. We therefore distinguish sparsity in the representation, in the stored parameters, in the theoretical operation count, and in measured runtime, and present an empirical study of several routes toward sparse FNOs that tests each transition between them separately. Coarsening the execution grid reduces the theoretical cost without reducing measured latency, and adding a correction term recovers accuracy at the cost of making the model slower. Even an 83% parameter reduction remains slower than the dense baseline under ordinary execution. These results motivate a stricter definition of useful sparsity: the deployed operator must preserve solution accuracy and map its reduced support to a genuinely cheaper execution path.
Neural fields (NFs) map continuous coordinates to signals such as color or density, but fast high-quality reconstruction from sparse observations remains difficult. Classical Neural Tangent Kernel (NTK) regression gives closed-form fits, yet it is fundamentally linear and cannot accumulate reusable task priors. We develop three algorithms that address these gaps. NTK-KIP learns a distilled support set of coordinates (and optional labels) so that a finite NTK can inpaint large missing regions from little observed data, yielding a compact non-linear representation instead of a raw kernel solve. MetaQuill meta-learns a shared initialization for an INR so that new scenes can be adapted by updating only a small task-specific weight offset, which provides true feature learning and a reusable prior. Finally, MetaQuill-KIP fuses both ideas: it seeds the task with a KIP-style non-linear warm start, then refines only that small offset around the meta-learned initialization. MetaQuill-KIP achieves high-PSNR reconstructions and semantically plausible inpainting under very sparse observations, while requiring only lightweight per-instance adaptation, whereas diffusion-style baselines typically depend on large pretrained generative priors and costly per-image tuning. This shows that NTK-driven neural fields can be made both non-linear and meta-learnable, narrowing the gap between analytic kernels and practical few-shot reconstruction.
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.