Real vs. Complex Spectral Bases for Neural Operators: Hartley Neural Operator and Green's Function Alignment

Forum topic · 小凯 · 2026-06-25

Summary

A new arXiv paper (2506.14669) by Jason Sulskis and Sathya Ravi introduces the Hartley Neural Operator (HNO), a real-valued mirror of the Fourier Neural Operator (FNO). HNO replaces the complex FFT with the purely real Discrete Hartley Transform and learns a single real multiplier per retained spectral mode. Since the real Hartley spectrum is not halved by conjugate symmetry, HNO retains twice as many frequency corners as FNO but with one real weight where FNO uses a complex pair, making the two operators iso-parametric at equal width. The paper's central thesis is that the optimal spectral basis is a property of the operator: self-adjoint elliptic operators (Poisson, biharmonic) with real, symmetric Green's functions are best served by real Hartley multipliers, while time-dependent operators carrying phase—waves, advection, Burgers, Navier-Stokes—favor the complex FNO basis. Experiments across PDE classes show a monotonic split by operator phase content, matching Green's function theory. The result is a predictive rule rather than a universal winner: match the spectral basis to the symmetries of the solution operator.

Paper Overview

Field: Machine Learning
Authors: Jason Sulskis, Sathya Ravi
Published: 2026-06-24
arXiv: 2506.14669

Introduction

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. This paper introduces the Hartley Neural Operator (HNO), the exact real-valued mirror of FNO.

Key Points

HNO design: replaces the FFT with the purely real Discrete Hartley Transform and learns a single real multiplier per retained spectral mode, with no complex arithmetic.
Iso-parametric comparison: 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. At equal width, the two operators have identical parameter counts and differ only in spectral basis.
Central thesis: the optimal basis is a property of the operator itself.
Self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that real Hartley multipliers can exactly diagonalize — HNO is favored here.
Time-dependent operators carry phase — oscillations in the wave equation, transport in advection, Burgers, and Navier-Stokes — which a real diagonal multiplier cannot represent. FNO is favored, increasingly so as phase content grows, making the phase-free heat equation a borderline case.
Empirical results: both operators trained under identical conditions and benchmarked across PDE classes, initial-condition families, and boundary conditions. The elliptic vs. time-dependent split is monotonic in operator phase content and matches the Green's function theory developed in the paper.

Conclusion

The contribution is not a universal winner but a predictive rule: match the spectral basis to the symmetries of the solution operator.

Original Abstract (excerpt)

> 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.

Full paper: arXiv:2506.14669