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.

翻译：柔性智能超表面（FIM）通过引入形态自由度，动态改变其三维形状以确保多径信号相干叠加，为无线通信提供了新的解决方案。然而，在FIM系统中实现预期的性能增益，关键在于获取连续且高维形变空间上的精确信道状态信息。因此，本文研究了面向FIM辅助毫米波通信系统的这一基本信道估计问题。首先，我们发展了基于模型的框架，将该问题构造为基于插值与核方法的函数逼近问题，或利用毫米波信道固有角度稀疏性的稀疏信号恢复问题。为了进一步突破基于模型的信道估计框架中显式假设的局限，我们提出了一种基于傅里叶神经算子（FNO）的深度学习框架。通过在傅里叶域参数化全局卷积算子，我们设计了高效的FNO架构来学习将FIM形状映射至信道响应的连续算子，该算子具有网格无关特性。此外，我们利用分层傅里叶神经算子（H-FNO）架构，在空间分辨率层级结构中高效捕获多尺度特征。数值结果表明，所提出的H-FNO在估计精度和导频效率上显著优于基于模型的基准方法。特别地，可解释性分析表明，H-FNO学习了适应FIM物理几何的各向异性空间滤波器，能够在连续形变空间上精确重建非线性信道响应。