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

Forum topic · 小凯 · 2026-06-25

Summary

Researchers Jason Sulskis and Sathya Ravi (arXiv:2506.14669) introduce 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, learning a single real multiplier per retained spectral mode without complex arithmetic. 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 carries a complex pair, making the two architectures iso-parametric at equal width and differing only in spectral basis. The paper's central thesis is that the optimal spectral basis is a property of the operator: self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that Hartley multipliers diagonalize exactly, favoring HNO, while time-dependent operators carrying phase (advection, Burgers, Navier-Stokes, waves) favor FNO. Benchmarks across PDE classes, initial conditions, and boundary conditions confirm a monotonic split along operator phase content, matching the Green's function theory. The result offers a predictive rule: match the spectral basis to the symmetry of the solution operator.

Overview

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

Key Points

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.

Hartley Neural Operator (HNO): an 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.
Iso-parametric design: 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. The two operators are iso-parametric at equal width and differ only in spectral basis.

Central Thesis: The Optimal Basis Is a Property of the Operator

Self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that real Hartley multipliers diagonalize exactly, so HNO is favored on such problems.
Time-dependent operators carry phase — from oscillations in the wave equation to transport in advection, Burgers, and Navier-Stokes. A real diagonal multiplier cannot represent this phase, so FNO is favored, increasingly so as the operator's phase content grows. The phase-free heat equation is a boundary case.

Empirical Findings

Training both operators under identical conditions and benchmarking across PDE classes, initial-condition families, and boundary conditions, the authors find:

The elliptic vs. time-dependent split is monotonic in operator phase content.
The empirical results match 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 symmetry of the solution operator.

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