Learning Function-to-Function Mappings: A Fourier Neural Operator for Next-Generation MIMO Systems

TL;DR

This work addresses the challenge of enabling physical-layer signal processing for next-generation MIMO systems by reframing wireless channels as function-to-function operators governed by Maxwell’s PDEs. It introduces Fourier Neural Operator (FNO), a mesh-free, operator-learning framework that efficiently learns global input-output mappings in the Fourier domain, aligning with the physics of EM propagation. Through case studies on holographic and flexible MIMO architectures, the paper shows FNO can serve as a generative, super-resolution, multi-modal, nonlinear inverse, functional autoencoder, and semantic codec, delivering accurate channel modeling and low-pilot-estimation overhead. The findings suggest FNO offers scalable, physics-consistent advantages for real-time, adaptive wireless systems and highlights open challenges in data, hardware constraints, closed-loop control, and acceleration for deployment.

Abstract

Next-generation multiple-input multiple-output (MIMO) systems, characterized by extremely large-scale arrays, holographic surfaces, three-dimensional architectures, and flexible antennas, are poised to deliver unprecedented data rates, spectral efficiency and stability. However, these advancements introduce significant challenges for physical layer signal processing, stemming from complex near-field propagation, continuous aperture modeling, sub-wavelength antenna coupling effects, and dynamic channel conditions. Conventional model-based and deep learning approaches often struggle with the immense computational complexity and model inaccuracies inherent in these new regimes. This article proposes a Fourier neural operator (FNO) as a powerful and promising tool to address these challenges. The FNO learns function-to-function mappings between infinite-dimensional function spaces, making them exceptionally well-suited for modeling complex physical systems governed by partial differential equations based on electromagnetic wave propagation. We first present the fundamental principles of FNO, demonstrating its mesh-free nature and function-to-function ability to efficiently capture global dependencies in the Fourier domain. Furthermore, we explore a range of applications of FNO in physical-layer signal processing for next-generation MIMO systems. Representative case studies on channel modeling and estimation for novel MIMO architectures demonstrate the superior performance of FNO compared to state-of-the-art methods. Finally, we discuss open challenges and outline future research directions, positioning FNO as a promising technology for enabling the enormous potential of next-generation MIMO systems.

Figures (6)

• Figure 1: Illustration of next-generation MIMO systems: (a) Extremely Large-Aperture Array (ELAA) MIMO, (b) Holographic MIMO, (c) 3D MIMO, and (d) Flexible MIMO.
• Figure 2: Design principle of Fourier neural operator. (a) Evolution between neural networks and neural operators, and (b) Pipeline of Fourier neural operator.
• Figure 3: Connection between the physical essence of wireless communication and FNO. First, framing wireless communication as a physical process governed by Maxwell's equations; second, abstracting this process into a mathematical problem of finding a solution operator; and third, introducing the FNO as a surrogate model to learn this operator directly from data.
• Figure 4: Applications of FNO in physical-layer signal processing of next-generation MIMO systems. FNO provides a potential foundational tool that reframes diverse physical-layer challenges as specific function-to-function mapping tasks, where each branch illustrates a distinct application, detailing the problem, the role of the FNO as a learned operator and the corresponding inputs and outputs of the mapping task.
• Figure 5: Channel modeling performance for holographic MIMO systems. The top row compares the magnitude of EM field across a 2D spatial plane, showing that the FNO prediction (center) closely matches the ground truth (left) with minimal absolute error (right). The bottom row analyzes the frequency spectrum, where the FNO prediction successfully replicates the key physical property of the ground truth by concentrating its energy within the wavenumber $\kappa_0$ boundary.