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

2026-06-23 · 17:29 UTC ·Jason Sulskis, Sathya Ravi ·1 min read

Brief

The paper proposes the Hartley Neural Operator (HNO), a real-valued analogue of Fourier Neural Operators that uses the Discrete Hartley Transform and a single real multiplier per spectral mode. The authors prove and empirically show HNO is preferred for self-adjoint elliptic operators (Poisson, biharmonic) whose Green's functions are real and symmetric, while FNO excels on time-dependent, phase-rich operators (wave, advection, Burgers, Navier–Stokes); benchmarks across PDE classes and conditions reveal a monotone performance split by operator phase content, yielding the practical rule: match spectral basis to operator symmetry.

Why it matters

The paper introduces the Hartley Neural Operator (HNO), which replaces the complex FFT in FNOs with the real Discrete Hartley Transform and learns a single real multiplier per retained spectral mode; HNO preserves twice as many frequency corners but uses one real weight where FNO uses a complex pair, making the two iso-parametric at equal width.

Key details

Theory and benchmarks (Poisson, biharmonic, wave, advection, Burgers, Navier–Stokes; varied initial-condition families and boundary conditions) show HNO outperforms FNO on self-adjoint elliptic operators with real symmetric Green's functions, while FNO wins on phase-rich time-dependent operators; the split is monotone in operator phase content and the heat equation is a borderline case. (Authors: Jason Sulskis, Sathya Ravi; arXiv 2026-06-23; submitted to the 62nd Allerton Conference.)

Abstract

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.

Comment: Submitted to/in consideration for the 62nd Allerton Conference on Communication, Control, and Computing