Real vs. Complex Spectral Bases for Neural Operators

Fourier Neural Operators (FNOs) are a class of models designed to learn solution operators for partial differential equations (PDEs) by performing global convolutions in the complex Fourier domain. However, for PDEs with real-valued solutions, the complex Fast Fourier Transform (FFT) introduces redundancy due to conjugate symmetry. This paper introduces the Hartley Neural Operator (HNO), which is a direct real-valued analogue to FNO, utilizing the Discrete Hartley Transform instead of FFT and learning real multipliers. The core argument is that the most effective spectral basis is determined by the properties of the specific operator being learned. For self-adjoint elliptic operators, which have real and symmetric Green's functions, the real Hartley multiplier is ideally suited, favoring HNO. Conversely, time-dependent operators, characterized by phase content from phenomena like oscillation or transport, are better represented by FNO due to its ability to handle complex arithmetic. The heat equation serves as a borderline case. Through extensive benchmarking across various PDE classes, initial conditions, and boundary conditions, the study confirms this predictive rule. The findings suggest that rather than a universally superior operator, the choice between real (Hartley) and complex (Fourier) bases should align with the symmetry and phase characteristics of the solution operator, providing a guiding principle for neural operator design.