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arxiv: 2606.24851 · v3 · pith:SAT7BEFVnew · submitted 2026-06-23 · 💻 cs.LG

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

Jason Sulskis , Sathya Ravi This is my paper

Pith reviewed 2026-07-01 06:37 UTC · model grok-4.3

classification 💻 cs.LG
keywords neural operatorsFourier Neural OperatorHartley transformGreen's functionsspectral basesPDE operatorsmachine learning for PDEs
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The pith

The optimal spectral basis for neural operators is set by the symmetry of the PDE solution operator's Green's function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Hartley Neural Operator as a real-valued counterpart to the Fourier Neural Operator and tests both on a range of PDEs. It establishes that real Hartley bases perform better on self-adjoint elliptic operators whose Green's functions are real and symmetric, while complex Fourier bases are superior for time-dependent operators that involve phase information from oscillation or transport. This split is predicted by how well each basis aligns with the Green's function and is confirmed in benchmarks that hold training procedures fixed. A sympathetic reader would care because it replaces the search for a single best architecture with a rule for choosing the basis based on the operator's properties.

Core claim

Rather than one basis always winning, the best spectral basis matches the symmetry of the solution operator: self-adjoint elliptic operators with real symmetric Green's functions favor the real Hartley multiplier, while time-dependent operators with phase favor the complex Fourier multiplier, with the advantage growing as phase content increases and the heat equation as the borderline case.

What carries the argument

Green's function alignment, the property that determines whether a real diagonal multiplier (Hartley) or complex multiplier (Fourier) can exactly represent the operator's action in spectral space.

If this is right

  • For self-adjoint elliptic PDEs such as Poisson and biharmonic, the Hartley Neural Operator is favored.
  • For time-dependent PDEs such as wave, advection, Burgers, and Navier-Stokes, the Fourier Neural Operator is favored.
  • The performance difference is monotone in the operator's phase content.
  • The two operators are iso-parametric, differing only in the choice of real versus complex spectral basis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers of new neural operator variants could use Green's function symmetry as a design criterion when selecting transforms.
  • Extending the comparison to operators with mixed elliptic and hyperbolic character could test the boundaries of the predictive rule.
  • The theory suggests that for purely real-valued problems without phase, real bases avoid unnecessary conjugate symmetry overhead.

Load-bearing premise

The observed performance differences across PDE classes arise specifically from alignment between the Green's function and the spectral basis rather than from differences in retained modes, training, or other implementation details.

What would settle it

Training both operators on a new PDE class whose Green's function symmetry contradicts the elliptic-versus-time-dependent pattern and finding that the performance ordering does not follow the predicted rule would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.24851 by Jason Sulskis, Sathya Ravi.

Figure 1
Figure 1. Figure 1: Hartley Neural Operator architecture. Left: Network structure with input projection, spectral convolution blocks (3 for time-dependent, 4 for elliptic PDEs), residual 1×1 bypass, and output projection. Right: the Hartley spectral convolution keeps the low-frequency corners of the full real Hartley spectrum— four in two dimensions, eight octants in three—and applies one real weight matrix W(i) per corner, t… view at source ↗
Figure 2
Figure 2. Figure 2: HNO/FNO relative-L 2 ratio across PDEs (rows) and initial-condition/boundary combinations (columns). Green indicates HNO wins, red FNO; bold rows are the elliptic operators. The elliptic rows are green and the time-dependent rows red, with the FNO margin increasing from heat (phaseless) through wave to the transport-dominated operators. This figure reports the full relative-L 2 comparison. The results divi… view at source ↗
Figure 3
Figure 3. Figure 3: Spatial error advantage |FNO err| − |HNO err| per cell (green: HNO closer to truth; pink: FNO closer). Annotations give the HNO/FNO ratio and the fraction of the domain where HNO is better. Elliptic panels are diffusely green (a global basis match); time-dependent panels are pink with structured error that traces transport direction and vortices. 5 Discussion Our experiments support a single organizing pri… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces the Hartley Neural Operator (HNO) as the real-valued counterpart to the Fourier Neural Operator (FNO), replacing the complex FFT with the Discrete Hartley Transform while maintaining iso-parametric equivalence at equal width. It develops a Green's function alignment theory predicting that real symmetric Green's functions of self-adjoint elliptic operators (e.g., Poisson, biharmonic) favor HNO, while phase-carrying time-dependent operators (wave, advection, Burgers, Navier-Stokes) favor FNO, with the heat equation as a borderline case. Experiments across PDE classes, initial conditions, and boundaries report an elliptic-versus-time-dependent performance split that is monotone in operator phase content and matches the pre-derived theory, yielding the rule to match spectral basis to solution-operator symmetry.

Significance. If the central claim holds under rigorous controls, the work supplies a predictive, theory-driven rule for spectral-basis selection in neural operators instead of universal preference or post-hoc tuning. The explicit iso-parametric construction, the Green's-function derivation, and the cross-class experimental confirmation are strengths that could inform architecture choices for elliptic versus evolutionary PDEs.

major comments (2)
  1. [§4] §4 (Experimental Setup) and the iso-parametric claim in the abstract: the statement that HNO and FNO differ only in spectral basis requires explicit confirmation that retained modes match in physical frequency content. Because HNO retains twice the frequency corners, the manuscript must document the precise wavenumber cutoffs, boundary handling, and highest retained wavenumber for each operator; without this, the reported elliptic/time-dependent split could arise from unequal effective resolution or optimization landscape rather than Green's-function alignment.
  2. [§3] §3 (Green's Function Alignment Theory): the claim that a real diagonal Hartley multiplier exactly diagonalizes the real symmetric Green's function for elliptic operators needs a concrete verification step (e.g., explicit matrix representation or low-dimensional example for the Poisson operator) to show that the alignment is not merely formal but accounts for the discrete transform truncation used in the neural operator.
minor comments (2)
  1. Figure captions and axis labels should explicitly annotate which PDEs belong to the elliptic versus time-dependent categories to make the monotone phase-content trend immediately visible.
  2. Notation for the number of retained modes (e.g., k_max) should be unified between the FNO and HNO descriptions to avoid any ambiguity about parameter counting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the iso-parametric construction and the Green's function theory. We address each major comment below and will revise the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: [§4] §4 (Experimental Setup) and the iso-parametric claim in the abstract: the statement that HNO and FNO differ only in spectral basis requires explicit confirmation that retained modes match in physical frequency content. Because HNO retains twice the frequency corners, the manuscript must document the precise wavenumber cutoffs, boundary handling, and highest retained wavenumber for each operator; without this, the reported elliptic/time-dependent split could arise from unequal effective resolution or optimization landscape rather than Green's-function alignment.

    Authors: We agree that explicit documentation is required to substantiate that the operators differ only in spectral basis. In the revised manuscript we will add to §4 a table and text specifying, for every PDE: grid resolution, number of retained modes, highest wavenumber k_max (matched across FNO and HNO so that the same physical frequencies are covered), and boundary handling (periodic in all cases). This will confirm identical effective resolution and isolate the performance differences to the choice of real versus complex basis. revision: yes

  2. Referee: [§3] §3 (Green's Function Alignment Theory): the claim that a real diagonal Hartley multiplier exactly diagonalizes the real symmetric Green's function for elliptic operators needs a concrete verification step (e.g., explicit matrix representation or low-dimensional example for the Poisson operator) to show that the alignment is not merely formal but accounts for the discrete transform truncation used in the neural operator.

    Authors: We agree that a concrete low-dimensional verification would strengthen the claim and address truncation concerns. In the revision we will insert into §3 an explicit 1D Poisson example: the discrete Green's function matrix on a periodic grid, its DHT representation, and the resulting real diagonal multiplier for the retained modes. This will demonstrate that the alignment holds under the same mode truncation employed by the neural operator. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theory grounded in standard Green's function properties

full rationale

The paper states its central thesis directly from known properties of self-adjoint elliptic operators (real symmetric Green's functions diagonalized by real multipliers) versus phase-carrying time-dependent operators. Experiments are framed as confirmation of this pre-existing theory across PDE classes rather than the origin of any fitted prediction. The iso-parametric design (HNO vs FNO differing only in basis) is presented as an explicit construction choice, not derived from data. No self-citations, self-definitional loops, or renamings of known results appear in the abstract or described structure. The derivation chain remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on domain assumptions about Green's function properties for different operator classes and introduces the HNO as a new method variant; no free parameters or invented physical entities are described.

axioms (2)
  • domain assumption Self-adjoint elliptic operators have real, symmetric Green's functions that the real Hartley multiplier diagonalizes exactly
    Stated as the basis for favoring HNO on elliptic problems.
  • domain assumption Time-dependent operators carry phase from oscillation or transport that a real diagonal multiplier cannot represent
    Used to explain why FNO is favored for wave, advection, Burgers, and Navier-Stokes.
invented entities (1)
  • Hartley Neural Operator (HNO) no independent evidence
    purpose: Real-valued neural operator using the Discrete Hartley Transform instead of complex FFT
    New method introduced to serve as the exact real mirror of FNO.

pith-pipeline@v0.9.1-grok · 5822 in / 1353 out tokens · 38950 ms · 2026-07-01T06:37:33.597045+00:00 · methodology

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