发表机构
School of Mathematics, Southeast University（东南大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对分数阶偏微分方程重复求解昂贵的问题，提出分数阶傅里叶神经算子，注入解析 Mittag-Leffler 传播子处理线性部分，仅训练残差网络，实现分辨率鲁棒且误差最低。
AI 中文摘要
分数阶偏微分方程将依赖于记忆的时间导数与非局部分数阶拉普拉斯算子耦合在一起，在变化的分数阶、初始数据或扩散系数场下重复求解这些方程的计算成本很高。神经算子提供了一种快速的替代方案，但标准架构必须从数据中重新学习主要的线性分数阶演化过程。我们提出了分数阶傅里叶神经算子（FrFNO），一种分辨率鲁棒的条件算子。它将冻结线性部分的解析 Mittag-Leffler 响应作为一次预计算、与分辨率无关的传播子表注入，并仅针对由变系数和非线性平流引起的残差训练一个谱卷积网络，同时连续地以两个分数阶为条件。核心理论结果是，在零样本超分辨率下，注入的传播子填补了标准傅里叶神经算子置零的带外模式，将带外全场尾部替换为更小的残差尾部（在弱扰动下为 K 无关的常数因子缩减）。残差和完整解具有相同的 Sobolev 阶数，因此优势在于振幅缩减而非更陡的尾部。网格细化将误差驱动到残差控制的下限。在非线性二维时空分数阶 Burgers 问题上，FrFNO 在训练分辨率和零样本超分辨率下，在六个基线（FNO、PINO、PDNO、CNO、DeepONet、U-Net）中实现了最低的相对 L^2 误差，产生了最小的谱相位误差，并在整数阶极限和长积分窗口下仍然是最优的。相同的构造可扩展到耦合分数阶系统，如分数阶 Allen-Cahn 和 Navier-Stokes 方程。代码、训练脚本和参考输出可在该 https URL 获取。
英文摘要
Fractional partial differential equations couple a memory-dependent time derivative with a nonlocal fractional Laplacian, and their repeated solution under varying fractional orders, initial data, or diffusivity fields is computationally expensive. Neural operators offer a fast surrogate, but standard architectures must relearn the dominant linear fractional evolution from data. We propose the \emph{Fractional Fourier Neural Operator} (FrFNO), a resolution-robust conditional operator. It injects the analytic Mittag--Leffler response of the frozen linear part as a once-precomputed, resolution-independent propagator table, and trains a spectral convolutional network only on the residual induced by variable coefficients and nonlinear advection, conditioned continuously on both fractional orders. The central theoretical result is that under zero-shot super-resolution the injected propagator fills the out-of-band modes that a standard Fourier neural operator sets to zero, replacing the out-of-band full-field tail by the smaller residual tail (a $K$-independent constant-factor reduction under weak perturbation). The residual and the full solution carry the same Sobolev order, so the advantage is amplitude reduction rather than a steeper tail. Mesh refinement drives the error to a residual-controlled floor. On a nonlinear two-dimensional space--time fractional Burgers problem FrFNO achieves the lowest relative $L^2$ error among six baselines (FNO, PINO, PDNO, CNO, DeepONet, U-Net) at the training resolution and under zero-shot super-resolution, yields the smallest spectral phase error, and remains the best at the integer-order limit and for long integration windows. The same construction extends to coupled fractional systems such as fractional Allen-Cahn and Navier--Stokes equations. Code, training scripts, and reference outputs are available at https://github.com/Derek2021Pang/FrFNO.
Comments22 pages, 4 figures
