Hilbert Neural Operator: Operator Learning in the Analytic Signal Domain

Abstract: 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 \textbf{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.