Real vs. Complex Spectral Bases for Neural Operators: The Role of Green’s Function Alignment

This paper investigates whether the Fourier transform is the optimal spectral basis for neural operators. While Fourier Neural Operators (FNO) have become a standard for learning partial differential equations (PDEs), the authors hypothesize that the complex-valued nature of the Fourier basis introduces unnecessary redundancy and potential mismatch for real-valued physical systems. They propose the Hartley Neural Operator (HNO) as a real-valued alternative and seek to determine if spectral basis selection should be tailored to the underlying symmetry of the PDE.

The authors introduce the Hartley Neural Operator (HNO), which replaces the Fast Fourier Transform (FFT) with the Discrete Hartley Transform (DHT). Because the Hartley spectrum is purely real and lacks the conjugate symmetry of the Fourier spectrum, the HNO uses a single real multiplier per spectral mode. The authors ensure a fair comparison by making the HNO and FNO iso-parametric—meaning they have the same number of trainable parameters and are trained using identical hyperparameters, optimizers, and schedules. They benchmark these models across a diverse suite of PDEs, including elliptic (Poisson, biharmonic), parabolic (heat), hyperbolic (wave), and nonlinear (Burgers, Navier-Stokes) equations, testing across various initial conditions and boundary types.

The study reveals a clear, predictive split: the choice of spectral basis should be determined by the symmetry of the solution operator. For self-adjoint elliptic PDEs, which possess real and symmetric Green's functions, the HNO is significantly more effective because its real-valued multipliers can diagonalize the operator exactly. Conversely, for time-dependent PDEs that involve phase-carrying phenomena like transport or oscillation, the FNO is favored. The authors demonstrate that this performance gap is monotone with respect to the operator's phase content, with the heat equation (which lacks phase) acting as the borderline case between the two methods.