Channel Estimation for Flexible Intelligent Metasurfaces: From Model-Based Approaches to Neural Operators

TL;DR

This work tackles channel estimation for flexible intelligent metasurfaces (FIMs) whose deformation space is continuous and high‑dimensional. It first develops model‑based baselines (interpolation and sparsity) and then introduces a learning‑based framework using Fourier neural operators (FNO) and a hierarchical variant (H‑FNO) to learn a continuous operator that maps deformation shapes to channels, with mesh‑independence. Numerical results show that H‑FNO achieves superior NMSE and pilot efficiency, and exhibits interpretable, physically grounded features such as anisotropic Fourier filters and multi‑scale representations, enabling robust zero‑shot generalization across array sizes. The proposed approach offers a principled, scalable path to practical FIM deployments by leveraging operator learning and spectral priors to directly model the shape‑to‑channel mapping in realistic wireless environments.

Abstract

Flexible intelligent metasurfaces (FIMs) offer a new solution for wireless communications by introducing morphological degrees of freedom, dynamically morphing their three-dimensional shape to ensure multipath signals interfere constructively. However, realizing the desired performance gains in FIM systems is critically dependent on acquiring accurate channel state information across a continuous and high-dimensional deformation space. Therefore, this paper investigates this fundamental channel estimation problem for FIM assisted millimeter-wave communication systems. First, we develop model-based frameworks that structure the problem as either function approximation using interpolation and kernel methods or as a sparse signal recovery problem that leverages the inherent angular sparsity of millimeter-wave channels. To further advance the estimation capability beyond explicit assumptions in model-based channel estimation frameworks, we propose a deep learning-based framework using a Fourier neural operator (FNO). By parameterizing a global convolution operator in the Fourier domain, we design an efficient FNO architecture to learn the continuous operator that maps FIM shapes to channel responses with mesh-independent properties. Furthermore, we exploit a hierarchical FNO (H-FNO) architecture to efficiently capture the multi-scale features across a hierarchy of spatial resolutions. Numerical results demonstrate that the proposed H-FNO significantly outperforms the model-based benchmarks in estimation accuracy and pilot efficiency. In particular, the interpretability analysis show that the proposed H-FNO learns an anisotropic spatial filter adapted to the physical geometry of FIM and is capable of accurately reconstructing the non-linear channel response across the continuous deformation space.

Figures (14)

• Figure 1: FIM assisted multi-user communications.
• Figure 2: Frame protocol for FIM channel estimation.
• Figure 3: Fourier neural operator (FNO) architecture for FIM channel estimation.
• Figure 4: Hierarchical Fourier neural operator (H-FNO) architecture for multi-scale FIM channel estimation.
• Figure 5: NMSE performance against SNR for different channel estimation algorithms.