Hartley Neural Operator

Hartley Neural Operator: real-valued spectral basis for PDE operators with Green's function alignment theory

Researchers introduce the Hartley Neural Operator (HNO), a real-valued alternative to Fourier Neural Operators (FNO) that replaces the complex FFT with the Discrete Hartley Transform, yielding iso-parametric models with no complex arithmetic. The central finding is that the optimal spectral basis depends on the symmetry of the underlying PDE's Green's function: HNO outperforms FNO on self-adjoint elliptic operators (Poisson, biharmonic) whose Green's functions are real and symmetric, while FNO is favored for time-dependent operators with phase content (wave, advection, Navier-Stokes). Benchmarks across PDE classes confirm a monotone elliptic-versus-time-dependent split that matches the developed theory, yielding a predictive rule for basis selection rather than a universal winner.