Hartley Neural Operator: Real-Valued Spectral Methods

• Hartley Neural Operator is a real-valued neural operator that replaces the complex Fourier transform with the discrete Hartley transform for efficient spectral mixing.
• Its canonical architecture mirrors Fourier Neural Operators with spectral convolution blocks and real weight matrices, enabling iso-parametric comparisons while reducing redundancy.
• HNO finds applications in PDE solving and domain-specific tasks like 3D segmentation by leveraging parameter efficiency and improved resolution robustness.

Searching arXiv for the cited HNO papers and closely related work. Hartley Neural Operator (HNO) denotes a class of neural operators that replace the complex Fourier representation used by Fourier Neural Operators (FNOs) with the purely real Discrete Hartley Transform (DHT), so that spectral mixing is performed in a real-valued Hartley basis rather than in a complex Fourier basis (Sulskis et al., 23 Jun 2026). In current arXiv literature, HNO appears both as a formally defined real-valued mirror of FNO for PDE solution operators and as the basis of domain-specific variants, including compact 3D segmentation models and hybrid routed operators for non-stationary dynamics; related quantum work also describes a Hartley-basis latent operator paradigm that is best understood as a quantum analogue of an HNO-like model (Wong et al., 10 Jul 2025, Ma et al., 13 May 2026, Wu et al., 2024). Across these formulations, the recurring motivation is the same: many target fields are real-valued, and a Hartley basis preserves global spectral structure while avoiding complex arithmetic and aligning more directly with real symmetric kernels.

1. Mathematical foundation

The canonical Hartley kernel is the classical

$\mathrm{cas}(\theta)=\cos\theta+\sin\theta,$

and the DHT of a real signal $f[n]$ is written as

$$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$

A key identity used repeatedly in implementations is

$$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$

which makes Hartley processing accessible through FFT primitives while keeping the representation itself real-valued (Sulskis et al., 23 Jun 2026).

The defining architectural consequence is that HNO retains the global spectral structure of Fourier methods but stays fully real and avoids complex arithmetic. In the exact HNO formulation, the spectral layer is

$$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$

where $W$ is real-valued. The model therefore learns one real weight matrix per retained mode or corner, with no complex weight pair, no even/odd decomposition, and no explicit $k$/$-k$ coupling term in the learnable parameterization (Sulskis et al., 23 Jun 2026).

This real-valued formulation is motivated by the redundancy of complex Fourier representations for real data. For real signals, the Fourier spectrum obeys conjugate symmetry,

$\hat f(-k)=\overline{\hat f(k)},$

so half of the complex spectrum is determined by the other half. HNO removes that representation choice and substitutes a real spectral basis whose symmetry is different from the rFFT-style half-space used by FNO. In the comparison developed in the HNO paper, this leads to a basis-level distinction rather than a simple change in parameter count or width (Sulskis et al., 23 Jun 2026).

2. Canonical architecture and parameterization

The paper introducing HNO as the exact real-valued mirror of FNO specifies a matched macro-architecture: an input projection, several spectral convolution blocks, residual $1\times1$ bypass connections, GELU activations, and an output projection. It uses 3 spectral blocks for time-dependent PDEs and 4 spectral blocks for elliptic PDEs (Sulskis et al., 23 Jun 2026). The intent is to isolate the effect of spectral basis choice while holding the surrounding neural-operator scaffold fixed.

The spectral comparison is iso-parametric at equal width. FNO carries one complex multiplier per retained Fourier mode, whereas HNO carries one real multiplier per retained Hartley mode. Because the Hartley spectrum is not halved by conjugate symmetry, HNO retains twice as many frequency corners as FNO while using one real weight where FNO carries a complex pair; the paper therefore concludes that the two operators are iso-parametric at equal width and differ only in spectral basis. In 2D, the paper’s illustration keeps 2 corners for FNO and 4 corners for HNO; in 3D, it keeps 4 octants for FNO and 8 octants for HNO (Sulskis et al., 23 Jun 2026).

A second line of work reformulates this Hartley operator idea for extremely small 3D segmentation. HNOSeg-XS starts from the generic neural-operator update

$f[n]$0

then replaces Fourier-domain convolution with a Hartley-domain operator using shared parameters $f[n]$1 and a nonlinearity in frequency space: $f[n]$2 The HNO-XS block further compresses multiple frequency-domain updates so that the number of forward/inverse Hartley transform pairs is reduced from $f[n]$3 to just one pair per block, which the paper identifies as a major efficiency gain (Wong et al., 10 Jul 2025).

3. Green’s-function alignment and basis selection

The main theoretical contribution of the exact HNO paper is a Green’s-function alignment argument. For a self-adjoint elliptic operator $f[n]$4, the Green’s function $f[n]$5 is real and symmetric, and for translation-invariant cases satisfies $f[n]$6. The appendix result summarized in the paper states

$f[n]$7

Under this condition, the Hartley convolution theorem collapses from

$f[n]$8

to the simpler elementwise product

$f[n]$9

when the kernel is symmetric (Sulskis et al., 23 Jun 2026).

This is the regime in which HNO is argued to be structurally favored. For operators such as Poisson and biharmonic, the relevant spectra are real and even; the paper gives

$$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$0

for Poisson and

$$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$1

for biharmonic. In that setting, one real multiplier per mode directly matches the operator class, and the paper frames the benefit as a representation complexity reduction: FNO effectively carries $$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$2 real degrees of freedom for $$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$3 retained modes, while HNO carries $$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$4 real degrees of freedom (Sulskis et al., 23 Jun 2026).

The same analysis is used to delimit where HNO should not be expected to dominate. The appendix defines an operator as phaseless if its symbol $$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$5 is real and even, and phase-carrying otherwise. A real Hartley diagonal operator can represent exactly the phaseless case, but not operators with nonzero imaginary symbol. This produces the paper’s central predictive rule: the best spectral basis is a property of the operator, not a universal choice. A common misconception is therefore that HNO is a general replacement for FNO; the paper explicitly rejects that interpretation and instead argues for operator-dependent basis selection (Sulskis et al., 23 Jun 2026).

4. Empirical behavior on PDE benchmarks

The exact HNO benchmark suite compares HNO and FNO on Poisson, biharmonic, heat, wave, advection-diffusion, Burgers, and 2D Navier-Stokes, across three initial condition families—Gaussian random fields, eigenfunction expansions, and Gaussian bump superpositions—and under both periodic and homogeneous Dirichlet boundaries (Sulskis et al., 23 Jun 2026). Evaluation includes relative $$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$6 error and gradient error, with the reported formulas

$$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$7

The resulting pattern is a clean elliptic-versus-time-dependent split. HNO gets lower error on Poisson and biharmonic, with the largest gain on biharmonic. FNO wins on heat, wave, advection-diffusion, Burgers, and Navier-Stokes. The advantage for FNO grows with phase content: heat is the closest to a tie, wave shows a larger gap, and advection, Burgers, and Navier-Stokes show the largest FNO advantage (Sulskis et al., 23 Jun 2026).

The heat equation is treated as the border case because its propagator,

$$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$8

is real and even. The paper therefore describes heat as phaseless and reports that HNO can occasionally match or slightly outperform FNO on smooth inputs. By contrast, the wave equation, advection, Burgers, and Navier-Stokes are described as increasingly phase-rich or transport-dominated, which makes the complex Fourier basis increasingly advantageous (Sulskis et al., 23 Jun 2026).

The role of initial conditions is secondary in the reported experiments. Gaussian bumps are described as easiest, eigenfunction data as naturally FNO-friendly, and Gaussian random fields as the most informative broadband stress test. However, the paper says initial conditions mostly change how large the gap is, not which method wins. The empirical conclusion is therefore consistent with the Green’s-function theory: the operator dominates the basis decision (Sulskis et al., 23 Jun 2026).

5. Domain-specific extensions

HNOSeg-XS adapts the Hartley operator idea to 3D medical image segmentation. The paper replaces the Fourier transform by the Hartley transform, reformulates the problem in the frequency domain, and presents HNOSeg-XS as resolution robust, fast, memory efficient, and extremely parameter efficient. It reports fewer than 34.7k model parameters, overall best inference time $$H_k = \sum_{n=0}^{N-1} f[n]\;\mathrm{cas}\!\left(\frac{2\pi kn}{N}\right).$$9, and memory efficiency $$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$0 on BraTS'23, KiTS'23, and MVSeg'23 with a Tesla V100 GPU (Wong et al., 10 Jul 2025).

The model’s motivation is both mathematical and engineering-oriented. Complex-valued Fourier operations are described as “more complicated, less flexible, and require more memory,” whereas the Hartley transform maps real-valued signals to real-valued signals and supports ordinary deep learning layers and activation functions in the frequency domain. HNOSeg-XS also exploits parameter sharing across frequencies and adds nonlinearity directly in Hartley space to recover expressiveness lost by the shared real-valued form (Wong et al., 10 Jul 2025). In the reported resolution-robustness experiments, when training resolution is reduced, HNO-based models degrade much less sharply than CNN and transformer baselines. For example, on BraTS'23, reducing training resolution from $$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$1 to $$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$2 leads to Dice drops of 4.4% for HNOSeg and 3.5% for HNOSeg-XS, while nnFormer drops by 14.4%; on KiTS'23, reducing from $$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$3 to $$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$4 yields drops of 4.6% and 3.9% for HNOSeg and HNOSeg-XS, versus 23.2% for nnFormer (Wong et al., 10 Jul 2025).

U-HNO extends the Hartley/Fourier neural operator line in a different direction. It is a U-shaped hybrid neural operator whose central mechanism is Sparse-Point Adaptive Routing (SPAR): at each spatial location and each resolution level, a hard per-pixel mask selects whether a global Fourier branch or a local multi-scale Gaussian branch should dominate. At block level,

$$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$5

and routing is defined by

$$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$6

followed by a residual $$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$7 projection and GELU (Ma et al., 13 May 2026).

The training objective combines pointwise supervision, a finite-difference $$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$8 gradient term, and a cross-branch consistency regularizer. Across 1D Burgers, Kuramoto–Sivashinsky, KdV, 2D advection, Allen–Cahn, Navier–Stokes, Darcy flow, and 3D transonic compressible Navier–Stokes from PDEBench, the paper reports state-of-the-art rollout accuracy on the majority of tasks in both relative $$\mathcal{H}\{f\}(k)=\mathrm{Re}\{\mathcal{F}\{f\}(k)\}-\mathrm{Im}\{\mathcal{F}\{f\}(k)\},$$9 and $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$0, with especially large gains on sharp localized features. On Burgers it reports $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$1 relative $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$2 and $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$3 relative $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$4; on 3D-CFD it reports $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$5 relative $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$6 and $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$7 relative $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$8 (Ma et al., 13 May 2026). Despite the name, this model is not a pure Hartley spectral operator in the exact-HNO sense: its global branch is explicitly a Fourier-style global branch with a learned complex-valued Fourier mixing tensor. The paper also emphasizes that SPAR is not compute-saving, since both branches are evaluated everywhere, and reports a roughly $$y=\mathcal{H}^{-1}\!\left[W\cdot\mathcal{H}[x]\right], \qquad W \in \mathbb{R}^{d_v \times d_v \times k_\mathrm{max}},$$9 inference-latency gap on NS-2D relative to FNO (Ma et al., 13 May 2026).

6. Quantum analogue and related directions

A related 2024 quantum paper develops what is best understood as a quantum analogue of a Hartley Neural Operator or Hartley-feature neural model. Instead of representing functions in a Fourier feature space with complex amplitudes, it uses a real-valued Hartley basis built from the Hartley kernel and introduces both a differentiable Hartley feature map and a quantum Hartley transform (QHT) circuit (Wu et al., 2024). The Hartley basis state is written as

$W$0

and the QHT is defined as a basis-change circuit between computational and Hartley bases.

The paper’s motivation parallels the classical HNO motivation. It argues that the DHT is real-to-real and self-inverse, unlike the Fourier transform, whose quantum encoding naturally produces complex amplitudes and Hermitian-symmetric redundancy for real data. The authors state that real-valued Hartley states are more natural for many physics- and probability-based tasks, reduce parameter count, and act as an inductive bias and natural regularization. The corresponding QHT satisfies the involution property

$W$1

matching the classical fact that the DHT is its own inverse (Wu et al., 2024).

This quantum Hartley framework is used for regression, stochastic differential equations, and generative modeling. For function fitting, the paper combines the Hartley feature map with a real-amplitude variational ansatz called HERA. For stochastic differential equations, it treats the Hartley model as a differentiable ansatz in latent space for Fokker–Planck-derived densities, specifically Ornstein–Uhlenbeck and geometric Brownian motion. For generative modeling, sampling is performed by reversing the trained circuit and applying the inverse QHT, with extended registers used for fine sampling and separate Hartley feature maps plus a correlation circuit used for bivariate modeling (Wu et al., 2024). The paper’s own summary is explicit: if one thinks of an HNO as a neural operator working in a Hartley basis, this work provides the quantum version of that idea.

Taken together, these papers place HNO within a broader real-spectral program. In its exact PDE-operator form, HNO is a real-valued mirror of FNO whose suitability depends on Green’s-function symmetry and operator phase content. In compact segmentation models, Hartley processing is used to reduce parameter count and improve resolution robustness. In hybrid routed models, Hartley/Fourier operator ideas are combined with local branches and spatially adaptive gating to address non-stationary dynamics. And in quantum models, the same Hartley-basis logic is reinterpreted as a real-amplitude latent operator framework (Sulskis et al., 23 Jun 2026, Wong et al., 10 Jul 2025, Ma et al., 13 May 2026, Wu et al., 2024).