发表机构
University of Sydney; University of Macau; Macau University of Science and Technology（悉尼大学; 澳门大学; 澳门科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对傅里叶神经算子固定基与硬截断的局限，提出SAFDNO，利用神经极点预测器构建自适应TM系统，在九个PDE基准上六项规则网格问题最优，并提升超分辨率性能。
AI 中文摘要
傅里叶神经算子（FNOs）为求解偏微分方程（PDEs）提供了一种高效范式。然而，FNOs依赖固定的傅里叶基和硬频率截断，这固有地限制了其对非周期、局部化和细尺度解结构的建模能力。我们提出了随机自适应傅里叶分解神经算子（SAFDNO），这是一种谱神经算子，用基于随机自适应傅里叶分解（SAFD）理论推导出的自适应Takenaka-Malmquist（TM）正交系统替代预定义的傅里叶模式。SAFDNO不执行经典SAFD中昂贵的贪婪极点搜索，而是通过神经极点预测器摊销随机极点选择，并直接从潜在特征构建自适应TM系统。所得算子对Hardy空间中解析分支的系数执行学习滤波，相对于输入自适应TM系统，保留了谱算子的全局感受野，同时提供了比使用固定谱基更灵活的表征。在九个PDE基准问题上，SAFDNO在强神经算子基线中所有六个规则网格问题上取得了最佳性能，尤其在Darcy、Burgers和Navier-Stokes问题上表现突出，这些场景中固定傅里叶模式在建模局部振荡、尖锐过渡和多尺度结构时往往效果较差。SAFDNO还展现出更强的零样本超分辨率性能，并在更细离散化部署时性能退化更少。这些结果表明，输入自适应TM系统为神经算子学习提供了固定谱表征的有前景替代方案。
英文摘要
Fourier Neural Operators (FNOs) offer an efficient paradigm for solving partial differential equations (PDEs). However, FNOs rely on a fixed Fourier basis and hard frequency truncation, which inherently limit their ability to model non-periodic, localized, and fine-scale solution structures. We propose the Stochastic Adaptive Fourier Decomposition Neural Operator (SAFDNO), a spectral neural operator that replaces predefined Fourier modes with an adaptive Takenaka-Malmquist (TM) orthonormal system derived from the theory of Stochastic Adaptive Fourier Decomposition (SAFD). Instead of performing an expensive greedy pole search in classical SAFD, SAFDNO amortizes stochastic pole selection through a neural pole predictor and constructs the adaptive TM system directly from latent features. The resulting operator performs learned filtering on coefficients of analytic branches in the Hardy space with respect to the input-adaptive TM system, preserving the global receptive field of spectral operators while providing a more flexible representation than using fixed spectral bases. Across nine PDE benchmark problems, SAFDNO achieves the best performance on all six regular-grid problems among strong neural operator baselines, with especially notable gains on Darcy, Burgers, and Navier-Stokes, where fixed Fourier modes are often less effective at modeling localized oscillations, sharp transitions, and multiscale structures. SAFDNO also exhibits stronger zero-shot super-resolution performance and shows less performance degradation when deployed on finer discretizations. These results suggest that the input-adaptive TM system provides a promising alternative to fixed spectral representations for neural operator learning.
