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：柔性智能超表面通过引入形态自由度，动态改变其三维形状以保障多径信号干涉相长，为无线通信提供了全新解决方案。然而，在柔性智能超表面系统中实现预期性能增益，关键在于在连续高维形变空间内获取精准的信道状态信息。为此，本文研究了柔性智能超表面辅助毫米波通信系统中的这一基础信道估计问题。首先，我们构建了基于模型的框架，将问题建模为函数逼近（采用插值与核函数法）或稀疏信号恢复问题（利用毫米波信道固有的角度稀疏性）。为进一步突破基于模型信道估计框架中显式假设的限制，我们提出基于傅里叶神经算子的深度学习框架。通过在傅里叶域参数化全局卷积算子，我们设计了高效的傅里叶神经算子架构，可学习从柔性智能超表面形状到信道响应的连续映射，且具备网格无关特性。更进一步，我们利用分层傅里叶神经算子架构，在空间分辨率层级上高效捕获多尺度特征。数值结果表明，所提分层傅里叶神经算子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，该算子能学习适应柔性智能超表面物理几何的各向异性空间滤波器，并在连续形变空间内精准重构非线性信道响应。