发表机构
Yonsei University（延世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对FNO高频学习受限问题，提出CAFE+FNO，通过CAFE+的傅里叶-切比雪夫特征乘法组合生成核，在五个PDE基准上优于现有变体，且参数不随模式数增加。
AI 中文摘要
傅里叶神经算子（FNO）通过傅里叶空间中的核参数化学习偏微分方程（PDE）的解算子，但频率截断可能限制高频变化的学习。AM-FNO和SirenFNO使用共享网络从谱坐标生成所有网格模式的核，这使得坐标编码和生成器设计变得重要。最近关于隐式神经表示（INR）的工作提出通过显式特征组合构建频率交互，而不是依赖后续的MLP隐式形成这些交互。基于这一方法，我们提出了CAFE+FNO，它将内容感知频率编码+（CAFE+）融入傅里叶核生成中。CAFE+通过并行仿射分支和Hadamard积组合傅里叶-切比雪夫特征，形成两个特征族内部及跨特征族的交互。一个核MLP将每个归一化谱坐标的表示映射为复通道混合矩阵。每一层在所有存储模式之间共享其生成器，使得在固定架构下可训练参数的数量与模式数量无关。我们在五个PDE基准上将CAFE+FNO与现有FNO变体进行比较，并对基配置、乘法组合和带宽可学习性进行消融研究。代码和实验配置可在该https URL获取。
英文摘要
The Fourier Neural Operator (FNO) learns solution operators of partial differential equations (PDEs) through Fourier-space kernel parameterization, but frequency truncation can limit the learning of high-frequency variations. AM-FNO and SirenFNO generate kernels for all grid modes from spectral coordinates using shared networks, making coordinate encoding and generator design important. Recent work on implicit neural representations (INRs) has proposed constructing frequency interactions through explicit feature composition rather than relying on subsequent MLPs to form them implicitly. Building on this approach, we propose CAFE+FNO, which incorporates Content-Aware Frequency Encoding+ (CAFE+) into Fourier kernel generation. CAFE+ combines Fourier--Chebyshev features through parallel affine branches and a Hadamard product, forming interactions within and across the two feature families. A kernel MLP maps the resulting representation of each normalized spectral coordinate to a complex channel-mixing matrix. Each layer shares its generator across all stored modes, making the number of trainable parameters independent of the number of modes for a fixed architecture. We compare CAFE+FNO with existing FNO variants on five PDE benchmarks and conduct ablation studies on basis configuration, multiplicative composition, and bandwidth learnability. Code and experimental configurations are available at https://github.com/fabsk101/CAFEPlusFNO.git.
