---
title: Hilbert Neural Operator (HNO)
url: https://www.emergentmind.com/topics/hilbert-neural-operator-hno
type: topic
---

# Hilbert Neural Operator (HNO)

Searching arXiv for the specified paper and closely related HNO usages to ground the article.
Searching for "2508.04882 Hilbert Neural Operator analytic signal domain"
Hilbert Neural Operator (HNO) is a neural-operator architecture for learning solution operators of partial differential equations by introducing an explicit amplitude–phase inductive bias from analytic signal theory into the spectral convolution pathway. In the formulation introduced in "Hilbert Neural Operator: Operator Learning in the Analytic Signal Domain," HNO learns mappings between infinite-dimensional function spaces, as in Fourier Neural Operator (FNO), but first applies the Hilbert transform so that instantaneous amplitude and phase become explicit features before the learnable spectral convolution is performed [2508.04882]. The architecture is motivated by physical systems in which causality, phase sensitivity, and non-stationarity are central, and it is positioned as a phase-aware variant of spectral neural operators rather than as a replacement for all Fourier-based approaches.

## 1. Definition and operator-learning setting

HNO is designed for operator learning: given an input function $u(x) \in U$ and a target function $f(x) \in F$, it aims to learn an operator
$$
G: U \to F, \qquad G(u) \approx f.
$$
In discretized form, training uses paired samples $\{(u_i,f_i)\}$, and the learned operator is evaluated on unseen discretizations, reflecting the standard neural-operator objective of discretization-invariant approximation [2508.04882].

The training criteria reported for HNO are the usual supervised function-space losses. The paper gives the $L^2$ mean-squared error
$$
L_{\mathrm{MSE}} = \frac{1}{N}\sum_{i=1}^{N}\|G_\theta(u_i)-f_i\|_2^2
$$
and the relative $L^2$ error
$$
L_{\mathrm{rel}} = \frac{1}{N}\sum_{i=1}^{N}\frac{\|G_\theta(u_i)-f_i\|_2}{\|f_i\|_2}.
$$
As with FNO, the use of global spectral convolutions and mode truncation is presented as a source of discretization-invariant approximations. The paper further argues that the analytic-signal representation provides a physics-inspired inductive bias that can stabilize learning in phase-sensitive and non-stationary regimes by making amplitude and phase explicit, while also noting that formal convergence theorems are not presented [2508.04882].

A common point of confusion is nomenclature. In the literature, the acronym HNO is not unique. "Real vs. Complex Spectral Bases for Neural Operators: The Role of Green's Function Alignment" uses HNO to mean **Hartley Neural Operator**, based on the Discrete Hartley Transform rather than the Hilbert transform [2606.24851]. "Parametric Value Approximation for General-sum Differential Games with State Constraints" uses HNO to mean **Hybrid Neural Operator** in a DeepONet-style, physics-informed setting [2503.06994]. Earlier work titled "An Operator Learning Approach via Function-valued Reproducing Kernel Hilbert Space for Differential Equations" also uses the phrase **Hilbert Neural Operator**, but in an RKHS and Hilbert–Schmidt integral-operator sense rather than the analytic-signal construction introduced in 2025 [2202.09488]. In current usage, therefore, "Hilbert Neural Operator" must be read in context.

## 2. Analytic signal theory and the Hilbert-transform inductive bias

The defining theoretical ingredient of this HNO is the analytic signal. For a one-dimensional signal $x(t)$, the Hilbert transform is
$$
H\{x\}(t)=\frac{1}{\pi}\,\mathrm{PV}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau.
$$
In the frequency domain,
$$
\mathcal{F}\{H\{x\}\}(\omega)=-i\,\mathrm{sgn}(\omega)\,X(\omega),
$$
where $X(\omega)=\mathcal{F}\{x\}(\omega)$. The associated analytic signal is
$$
x_a(t)=x(t)+iH\{x\}(t)=A(t)e^{i\phi(t)},
$$
with
$$
A(t)=\sqrt{x(t)^2+\big(H\{x\}(t)\big)^2}, \qquad
\phi(t)=\mathrm{atan2}(H\{x\}(t),x(t)),
$$
and instantaneous frequency
$$
\omega(t)=\frac{d\phi(t)}{dt}.
$$
These definitions are the basis for the claim that HNO exposes instantaneous amplitude and phase as explicit features for learning [2508.04882].

The paper’s motivation is signal-processing based. Many physical operators, including those relevant to waves, dispersive media, and causal filters, act on phase and amplitude in structured ways. Explicit phase information is therefore linked to sensitivity to group velocity, phase velocity, dispersion, and causality, while amplitude captures envelopes and modulations that are important in non-stationary dynamics. In the paper’s framing, applying learnable spectral kernels after the Hilbert transform biases the model toward phase-aware filtering because the Hilbert transform corresponds to a $\pm \pi/2$ phase shift depending on $\mathrm{sgn}(\omega)$ [2508.04882].

This perspective also clarifies what HNO is not. It is not introduced as a general remedy for non-periodic boundary conditions. The paper explicitly states that both FNO and the FFT-based HNO pathway inherit periodic extension assumptions when FFTs are applied on finite grids; HNO changes the feature space, not the spectral backend. Boundary artifacts can therefore still arise unless padding or windowing is used. The claimed advantage is a complementary phase–amplitude inductive bias, not elimination of FFT boundary issues [2508.04882].

For dimensions $d \ge 2$, the paper gives implementation guidance through the Riesz transform. For $\xi \in \mathbb{R}^d \setminus \{0\}$, the $j$-th Riesz transform has multiplier
$$
m_j(\xi)=-i\frac{\xi_j}{\|\xi\|},
$$
so that
$$
\mathcal{F}\{R_jx\}(\xi)=m_j(\xi)\hat X(\xi).
$$
A multidimensional analytic representation can then be formed from the original signal together with its Riesz transforms to obtain quadrature components. The paper notes, however, that this is implementation guidance rather than a reported experimental pipeline [2508.04882].

## 3. Architecture and layer mechanics

At the block level, HNO consists of a lift, repeated HNO layers, and a final projection. The input is first lifted by a linear map $P$ into a higher-dimensional latent feature $v_0(x)$. Each HNO layer then combines a local path and a global Hilbert spectral path. The local path applies a local linear operator $W$, such as a pointwise $1\times 1$ convolution, to capture short-range interactions. The global path proceeds through Hilbert transformation, Fourier transformation, mode-truncated spectral convolution, inverse Fourier transformation, and inverse Hilbert transformation [2508.04882].

The global Hilbert spectral path is given as follows:

1. Hilbert transform:
   $$
   \tilde v_k(x)=H\{v_k\}(x).
   $$
2. Fourier transform:
   $$
   \tilde V_k(\omega)=\mathcal{F}\{\tilde v_k\}(x).
   $$
3. Spectral convolution with a learnable spectral kernel $R_\phi$ on retained modes:
   $$
   \tilde V'_k(\omega)=R_\phi(\omega)\cdot \tilde V_k(\omega).
   $$
4. Inverse Fourier transform:
   $$
   \tilde v'_k(x)=\mathcal{F}^{-1}\{\tilde V'_k\}(x).
   $$
5. Inverse Hilbert transform:
   $$
   s_k(x)=H^{-1}\{\tilde v'_k\}(x)=-H\{\tilde v'_k\}(x).
   $$

The layer output is then
$$
v_{k+1}(x)=\sigma\big(Wv_k(x)+s_k(x)\big),
$$
and the paper summarizes the HNO layer by the equation
$$
v_{k+1}(x)=\sigma\!\left(Wv_k(x)+H^{-1}\!\left(\mathcal{F}^{-1}\!\left(R_\phi\cdot(\mathcal{F}(Hv_k))\right)\right)(x)\right).
$$
After a stack of such layers, a final projection $Q$ maps the latent representation to the output dimension [2508.04882].

The spectral kernel $R_\phi(\omega)$ is parameterized mode-wise and channel-wise, analogously to FNO. Because the Fourier domain is complex-valued, the weights can be complex, typically implemented through real and imaginary parts. Nonlinearities are applied after inverse Hilbert in the real domain, so the architecture does not require complex activations. The reported implementation uses standard autograd through FFT-based operators; gradients through complex weights may be realized via real–imaginary splits or Wirtinger calculus, but the paper states that standard autograd suffices in modern frameworks with FFT support [2508.04882].

The key distinction from FNO is architectural rather than merely motivational. FNO applies its spectral kernel directly to $\mathcal{F}\{v_k\}$, whereas HNO applies it to $\mathcal{F}(H v_k)$, i.e. to features shaped by the Hilbert transform that emphasize quadrature phase relationships. This is the central operational difference between the two models [2508.04882].

## 4. Training, implementation, and computational profile

The training objective is supervised $L^2$ regression, either in absolute or relative form:
$$
L(\theta)=\frac{1}{N}\sum_i \|G_\theta(u_i)-f_i\|_2^2,
$$
or the relative variant. The paper also lists several optional regularizers as design guidance rather than reported ingredients of the benchmark results: spectral sparsity penalties on the retained modes of $R_\phi$, smoothness penalties across modes, and phase-consistency penalties for time-causal tasks. It also states that physics-informed PDE residual penalties can be added, but these were not reported in the paper’s results [2508.04882].

The Hilbert transform is implemented with FFT multipliers. In one dimension, the procedure is:

- Compute $\hat X(\omega)=\mathrm{FFT}(x)$.
- Multiply by $-i\,\mathrm{sgn}(\omega)$:
  $$
  \hat H(\omega)=-i\,\mathrm{sgn}(\omega)\,\hat X(\omega).
  $$
- Apply inverse FFT to obtain $H\{x\}$.

This discrete implementation is applied per channel. For non-periodic data, padding or windowing is recommended to reduce edge artifacts. For nonuniform sampling, the paper advises resampling or the use of discrete Hilbert filters rather than FFT multipliers. For multidimensional signals, Riesz-transform multipliers are described as the natural extension, though again this is guidance rather than a reported experimental protocol [2508.04882].

A single HNO block can therefore be summarized operationally as: local pointwise linear mapping, Hilbert transform, FFT, low-mode truncation, complex spectral multiplication, inverse FFT, inverse Hilbert transform, then residual-style merging and nonlinearity. Batching is over independent samples, and uniform grids are emphasized as convenient for FFT-based operations. For Dirichlet or Neumann settings, the paper recommends padding or windowing and possible post-processing of boundary slices. For general geometries, HNO is said to share FNO’s FFT-pathway challenges, although domain deformation or mapping methods used in FNO variants may be combined with it [2508.04882].

The reported computational complexity remains spectral-operator scale: each transform costs $O(N\log N)$ per channel per batch, and the Hilbert pathway adds a small constant-factor overhead relative to FNO by introducing one extra Hilbert transform and its inverse per layer. Memory and latency are described as scaling similarly to FNO, with FFTs and spectral multiplications dominating the profile [2508.04882].

Implementation details are only partially specified. The paper describes typical, not exhaustively reported, hyperparameters as a small stack of HNO layers similar to FNO, such as $4$–$6$ layers, tens of retained modes per dimension, and hundreds of channels per layer, with Adam or a similar optimizer and FNO-like learning-rate, batch-size, and normalization practices. A public repository is listed at `https://github.com/sinapordanesh/XHNO`, and the paper notes that exact dependencies and license should be checked there [2508.04882].

## 5. Theoretical rationale, invariances, and limitations

The theoretical motivation is rooted in analytic signal theory rather than in new approximation theorems. The Hilbert transform produces a quadrature component, and together with the original signal forms the analytic signal $x_a=x+iH\{x\}=A e^{i\phi}$. The paper argues that physical systems that are causal and phase-sensitive often act as filters on phase and amplitude, making this feature domain a natural one for learning. In the frequency domain, since the Hilbert transform induces a sign-dependent quarter-cycle phase shift, learnable spectral kernels applied after the transform are interpreted as phase-aware filters [2508.04882].

The paper further attributes several inductive properties to the construction. For time shifts, the composition of spectral convolution with the Hilbert transform is described as commuting appropriately with shifts under FFT discretization, modulo boundary handling. For phase shifts, multiplication by $e^{i\psi}$ in the analytic representation corresponds to a phase offset, so the model can represent operators that impose phase delays. These arguments are qualitative and architectural rather than theorem-driven. The authors explicitly state that no new convergence theorems are proved [2508.04882].

This absence of new theory is central to the model’s current status. HNO is justified by signal-processing theory and empirical evidence, not by a formal approximation or stability theorem specific to the Hilbert-transformed architecture. The paper accordingly frames the method as hypothesis-driven: HNO is hypothesized to model operators more effectively for causal, phase-sensitive, and non-stationary systems. A plausible implication is that the architecture should be assessed primarily on tasks where analytic-signal representations are standard, rather than on generic PDE benchmarks alone.

Several limitations are stated directly. FFT-based implementations inherit periodic boundary assumptions and can produce edge artifacts without padding or windowing. Hilbert transforms can be sensitive to noise and finite-length effects, and instantaneous features may be unstable near boundaries. Complex-valued spectral weights introduce numerical considerations. Scaling to high-dimensional irregular domains shares FNO’s difficulties, suggesting the possible need for domain deformations or mesh-agnostic operators [2508.04882].

These limitations also identify a common misconception: the Hilbert transform does not by itself confer nonperiodic boundary handling, geometric flexibility, or robustness to all phase-related phenomena. It changes the inductive bias; it does not remove the standard constraints of FFT-based neural operators.

## 6. Empirical results, interpretation, and relation to neighboring operator families

The reported experiments cover Burgers’ equation in 1D and 2D, Darcy flow in 2D, incompressible Navier–Stokes in 2D, and the Lorenz-63 system as a 1D time series, with the Lorenz dataset taken from Laplace Neural Operator [2508.04882]. This set includes nonlinear PDEs, a linear elliptic problem, and chaotic ODE dynamics. The paper notes that causality and non-stationarity are present in the time-dependent problems.

The preliminary test errors reported are as follows:

| Task | HNO | Comparator(s) |
|---|---:|---:|
| Burgers-1D | $5.72\times 10^{-4}$ | FNO: $6.57\times 10^{-4}$ |
| Lorenz63-1D | $5.06\times 10^{-3}$ | FNO: $3.94\times 10^{-3}$; LNO: $2.03\times 10^{-2}$ |
| Burgers-2D | $6.03\times 10^{-5}$ | FNO: $4.11\times 10^{-5}$; LNO: $1.18\times 10^{-4}$ |
| DarcyFlow-2D | $3.83\times 10^{-2}$ | FNO: $1.64\times 10^{-2}$ |
| NavierStokes-2D | $1.90\times 10^{-2}$ | FNO: $6.84\times 10^{-3}$ |

The paper’s own interpretation is cautious. FNO outperforms HNO on most benchmarks, while HNO shows comparable performance overall and outperforms FNO on 1D Burgers. The results are explicitly described as preliminary, and the paper suggests that HNO may be better suited to tailored problems such as acoustic wave PDEs, where analytic signals are standard tools [2508.04882].

This empirical pattern matters for interpretation. The architecture is not presented as a uniformly stronger successor to FNO. Rather, it is a specialized spectral neural operator whose value depends on the operator class. That perspective resonates with the broader 2026 analysis of spectral-basis choice in neural operators, which argues that basis selection should match operator symmetry and phase content, although that work’s HNO denotes the Hartley Neural Operator rather than the Hilbert Neural Operator [2606.24851]. The overlap is terminological, but the comparison highlights a shared design principle: spectral inductive bias should be aligned with operator structure.

Ablation studies were not reported. In particular, the paper does not provide experiments removing the Hilbert transform, isolating amplitude-only or phase-only features, or varying spectral mode counts systematically. Future work is suggested in precisely these directions. The lack of ablations leaves the contribution theoretically motivated and empirically suggestive, but not yet fully dissected [2508.04882].

The paper also situates HNO among possible future hybrids. It mentions multidimensional analytic representations via Riesz transforms, physics-informed losses that regularize phase behavior, adaptive spectral filters, and hybrid architectures combining HNO with FNO, WNO, or LNO, as well as tailored benchmarks in acoustic and electromagnetic wave PDEs [2508.04882]. This suggests a research program in which Hilbert-domain operator learning is treated as a specialized module for phase-aware systems rather than as a universal neural-operator template.

Source: https://www.emergentmind.com/topics/hilbert-neural-operator-hno