这篇论文告诉你，求解偏微分方程时该用复傅里叶还是实哈特利基——没有万能赢家，得看算子有无相位。
该论文提出Hartley Neural Operator (HNO)，作为Fourier Neural Operator (FNO)的纯实数镜像，用实离散Hartley变换替代复FFT。HNO在每个保留谱模式上学习单个实权重，无复数运算。实验表明，对于自伴椭圆偏微分方程（如泊松、双调和方程），HNO表现更优，因为其实对称Green函数可被实数对角化；对于含相位的时间依赖方程（如波动、对流、Burgers、Navier-Stokes），FNO更优，且优势随相位含量增加而增强。研究给出了基于算子对称性选择谱基的预测规则。
Real vs. Complex Spectral Bases for Neural Operators: The Role of Green's Function Alignment
Fourier Neural Operators (FNO) learn solution operators of partial differential equations by parameterizing global convolutions in the complex Fourier domain. For real-valued PDE solutions, the complex FFT carries representational redundancy through conjugate symmetry. We introduce the Hartley Neural Operator (HNO), the exact real-valued mirror of FNO: it replaces the FFT with the purely real Discrete Hartley Transform and learns a single real multiplier per retained spectral mode, with no complex arithmetic. Because the real Hartley spectrum is not halved by conjugate symmetry, HNO retains twice as many frequency corners as FNO but one real weight where FNO carries a complex pair, so the two operators are iso-parametric at equal width and differ only in spectral basis. Our central thesis is that the best basis is a property of the operator. Self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that the real Hartley multiplier diagonalizes exactly, and HNO is favored there. Time-dependent operators carry phase, from oscillation in the wave equation to transport in advection, Burgers, and Navier-Stokes, which a real diagonal multiplier cannot represent, so FNO is favored there, and increasingly so with the operator's phase content, leaving the phaseless heat equation as the borderline case. Training both operators identically and benchmarking across PDE classes, initial-condition families, and boundary conditions, we find an elliptic-versus-time-dependent split that is monotone in operator phase content and matches the Green's-function theory we develop. Rather than a universal winner, our findings give a predictive rule: match the spectral basis to the symmetry of the solution operator.