# Hilbert Neural Operator: Operator Learning in the Analytic Signal Domain

arXiv: https://arxiv.org/abs/2508.04882 · HTML: https://mlsanity.com/paper/2508.04882/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 11 references that are in the map.
- First posted on arXiv 6 Aug 2025.

## Builds on (most cited first, 5 of 5)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Real-time estimation of phase and amplitude with application to neural data](https://mlsanity.com/paper/2105.10404/) (2021)

## Built on by (most cited first, 2 of 2)

- [Extending Neural Operators: Robust Handling of Functions Beyond the Training Set](https://mlsanity.com/paper/2603.03621/) (2026)
- [Stochastic Adaptive Fourier Decomposition for Operator Learning](https://mlsanity.com/paper/2610.04241/) (2026)

## Abstract (arXiv)

Neural operators have emerged as a powerful, data-driven paradigm for learning solution operators of partial differential equations (PDEs). State-of-the-art architectures, such as the Fourier Neural Operator (FNO), have achieved remarkable success by performing convolutions in the frequency domain, making them highly effective for a wide range of problems. However, this method has some limitations, including the periodicity assumption of the Fourier transform. In addition, there are other methods of analysing a signal, beyond phase and amplitude perspective, and provide us with other useful information to learn an effective network. We introduce the Hilbert Neural Operator (HNO), a new neural operator architecture to address some advantages by incorporating a strong inductive bias from signal processing. HNO operates by first mapping the input signal to its analytic representation via the Hilbert transform, thereby making instantaneous amplitude and phase information explicit features for the learning process. The core learnable operation -- a spectral convolution -- is then applied to this Hilbert-transformed representation. We hypothesize that this architecture enables HNO to model operators more effectively for causal, phase-sensitive, and non-stationary systems. We formalize the HNO architecture and provide the theoretical motivation for its design, rooted in analytic signal theory.