Flexible Intelligent Metasurface

• Flexible intelligent metasurface (FIM) is a reconfigurable electromagnetic structure composed of low-cost radiating elements that physically morph to control signal propagation and sensing.
• FIM design leverages advanced optimization algorithms like block coordinate descent and meta-reinforcement learning to jointly tune phase shifts and geometry, achieving significant performance gains.
• Practical implementations demonstrate up to 32% spectral efficiency improvements and enhanced multi-target sensing by adapting element positions under precise mechanical and calibration constraints.

A flexible intelligent metasurface (FIM) is a reconfigurable electromagnetic structure composed of low-cost radiating elements that can independently modulate their positions and electromagnetic responses in real-time. Differing fundamentally from conventional rigid reconfigurable intelligent surfaces (RIS), an FIM physically morphs its 3D surface shape, thus offering an additional geometric degree of freedom for optimizing radio propagation, wireless sensing, and resource allocation. This surface morphing, typically achieved by actuation in the axis perpendicular to the metasurface plane, enables spatially variant electromagnetic (EM) field synthesis beyond phase-only or amplitude-only tuning. FIMs are seen as key enablers of next-generation wireless communications, integrated sensing, network energy efficiency, and adaptable physical-layer infrastructures.

1. Physical Structure and Control Mechanisms

A flexible intelligent metasurface consists of an array (e.g., uniform planar array, UPA) of EM elements whose positions in free space can be programmatically perturbed, usually within specified mechanical constraints. The typical configuration places elements at

$\mathbf{u}_n = [x_n,\; y_n,\; z_n]^T$

where $y_n$ is a controllable vertical displacement bounded within $[0, y_{\max}]$ or $|d_n| \leq d_{\max}$, while $x_n, z_n$ are fixed by fabrication. Each EM element can also tune its local reflection/transmission phase via varactor diodes, micro-electromechanical systems (MEMS), or equivalent technologies.

The physical limits on morphing speed, magnitude, and actuator resolution determine the FIM's reconfiguration capacity. Typical actuation times are on the order of milliseconds, compatible with wireless channel coherence times for current deployments. The element-wise phase/deformation calibration requires characterization of the element response versus displacement to support high-precision surface control (Hu et al., 8 Oct 2025).

2. System Modeling and Spatial Channel Representation

The fundamental system models incorporate the FIM's spatial reconfigurability in both the electromagnetic field response and system-level channel matrices. In communication and sensing applications:

• Spatial Channel Correlation: The spatial correlation matrix for the FIM-aided transmitter,

$$R_{\mathrm{FIM}}(\mathbf{y}) = \mathbb{E}\left\{ \mathbf{a}(\mathbf{y},\theta,\varphi)\,\mathbf{a}(\mathbf{y},\theta,\varphi)^\mathrm{H} \right\} \in \mathbb{C}^{N \times N}$$

depends on the locations $\mathbf{u}_n$ controlled by $\mathbf{y}$ (morphing vector). Under isotropic scattering, closed-form expressions using sinc functions parameterized by inter-element Euclidean distances arise. The morphing directly tunes $R_{\mathrm{FIM}}$ and all subsequent communication metrics (Kumar et al., 28 Dec 2025).

• Array Response and Steering Vectors: The FIM's per-target steering vector is,

$$\mathbf{a}(\theta_k, \varphi_k, \Delta \mathbf{d}) = \left[ \mathbf{a}_z(\theta_k) \otimes \mathbf{a}_x(\theta_k, \varphi_k) \right] \odot \mathbf{a}_y(\theta_k, \varphi_k, \Delta\mathbf{d})$$

where $\mathbf{a}_y$ encodes the geometric path-length modulation from morphing (Teng et al., 29 Jun 2025).

• Impact on Channel Gain: In multipath channels, the FIM enables aligning multiple paths by tuning the surface shape, thereby substantially increasing the effective end-to-end gain beyond traditional RIS by adapting the surface nonlinearly to the channel structure (Hu et al., 8 Oct 2025).

3. Optimization Algorithms for FIM Configuration

Joint optimization of the FIM surface shape and classical electromagnetic (EM) control variables is required. Distinct algorithmic paradigms have been advanced:

• Block Coordinate Descent (BCD): Problems such as multi-target wireless sensing maximize the cumulated probing power at target locations by alternating between (A) transmit covariance (solved as a semidefinite program for fixed shape) and (B) surface morphing (solved via projected gradient ascent with analytical derivatives of the response Gram matrix with respect to displacement) (Teng et al., 29 Jun 2025). Convergence to stationary points is guaranteed by the monotonicity of each coordinate update and boundedness of $P_c$.
• Projected Gradient Methods: For downlink systems under statistical channel state information (CSI), average sum spectral efficiency maximization over the morphing vector is performed via iterative projected gradient steps, using analytic gradients of the spectral efficiency with respect to each morphing coordinate. Step sizes are determined by Armijo backtracking, guaranteeing monotonic ascent (Kumar et al., 28 Dec 2025).
• Alternating Optimization (AO) with Swarm/Gradient Search: In coupled SISO/MISO settings, the FIM phase-shift and morphing variables are alternately optimized. For shape morphing, metaheuristics (particle swarm optimization, PSO) or multi-interval gradient descent (MIGD) address the local nonconvexity of the per-element scalar subproblems (Hu et al., 8 Oct 2025).
• Meta-Reinforcement Learning: Joint resource allocation, beamforming, FIM shaping, and RIS control pose nonconvex, high-dimensional, hybrid discrete-continuous problems. Actor-critic algorithms (Meta-SAC, Meta-TD3) with meta-learning enhancements have been applied to adapt FIM morphing, BS precoding, and RIS matrices, achieving superior energy efficiency and sum-rate compared to static or rigid setups (Farhadi et al., 6 Sep 2025, Eftekhari et al., 19 Sep 2025).

4. Applications in Wireless Communications and Sensing

The FIM paradigm provides notable enhancement in several domains:

• Wireless Multiuser MISO/SISO Communications: FIM-aided base stations dynamically reconfigure the antenna surface to maximize spectral efficiency, sum-rate, or energy efficiency. With strong channel spatial correlation, FIMs demonstrate up to 32% spectral efficiency gain over rigid antenna arrays, with diminishing returns as spatial correlation weakens or element spacing increases (Kumar et al., 28 Dec 2025). In MISO, joint beamforming and FIM shaping yields further gains, with the shape-morphing constituting the dominant improvement (Hu et al., 8 Oct 2025).
• Multi-Target Wireless Sensing: The addition of shape morphing aligns the mutual directions of the antenna array steering vectors for multiple targets, driving the Gram matrix toward quasi-rank-one structure and maximizing coherent power at all targets. This is unattainable with fixed or phase-only arrays (Teng et al., 29 Jun 2025).
• Integration with STAR-BD-RIS and NOMA: FIMs have been co-designed with simultaneously transmitting and reflecting beyond-diagonal RIS (STAR-BD-RIS) structures, and with non-orthogonal multiple access (NOMA), to enhance cellular and IoT networks. FIM allows direct 3D channel shaping at the source, which complements the more flexible propagation path control of STAR-BD-RIS. Meta-learning optimization strategies efficiently solve such highly-coupled, constraint-laden design problems (Farhadi et al., 6 Sep 2025, Eftekhari et al., 19 Sep 2025).

5. Complexity, Convergence, and Implementation Considerations

• Algorithmic Complexity: For BCD-based methods, each transmission-covariance update (SDP) incurs $O(N_t^{4.5}\log(1/\Upsilon))$ cost; shape updates incur $O(N_t^3)$ per iteration. Projected gradient methods for FIM-shaping in communications scale as $O(K N^3)$ per iteration for $K$ users and $N$ elements. Heuristic global search methods (e.g., PSO) for per-element optimization have complexity $O(Q T_{\mathrm{PSO}})$ for $Q$ particles and $T_{\mathrm{PSO}}$ iterations (Hu et al., 8 Oct 2025, Teng et al., 29 Jun 2025, Kumar et al., 28 Dec 2025).
• Convergence: The monotonic convergence of alternating or projected updates is asserted under differentiability and boundedness assumptions. Meta-learning RL approaches empirically reach higher energy efficiency or sum-rate than non-meta baselines, although more training episodes are generally required (Farhadi et al., 6 Sep 2025, Eftekhari et al., 19 Sep 2025).
• Physical Implementation: Real-world control requires element-wise calibration; surface-morphing times in the millisecond regime allow real-time adaptation at the scale of channel coherent intervals (Hu et al., 8 Oct 2025). The impact of actuator quantization, mechanical constraints, and phase-drift is mitigated in the optimization models via box constraints and robustness formulations.

6. Performance Gains and Quantitative Benchmarks

• Communications: For $d_E = \lambda/8$ element-spacing, FIM achieves a 32.3% gain in spectral efficiency over a rigid antenna array, with the advantage falling to 0.11% at $d_E = \lambda/2$ when spatial correlation is inherently weak (Kumar et al., 28 Dec 2025). In SISO/MISO channel gain, FIM outperforms RIS by over 3 dB for typical configurations and realizes substantial gains in rich multipath (Hu et al., 8 Oct 2025).
• Sensing: FIM-based designs yield significant enhancement in cumulative probing power and target SNR for multi-target scenarios, allowing adaptive focusing not possible with rigid arrays (Teng et al., 29 Jun 2025).
• Joint Systems: For a FIM + STAR-BD-RIS hybrid architecture, meta-TD3 learning achieved notable sum-rate improvements (>15%) against conventional FIM-only and RIS-only baselines under strict SINR and power constraints (Eftekhari et al., 19 Sep 2025).

7. Research Directions and Future Trends

Emerging topics related to FIM include:

• Extensions to hardware-constrained morphing (speed, curvature limits), robustness to actuation errors, and actuator power minimization.
• Hybridization with distributed RIS/SURF architectures for joint coverage and fine-grained propagation control.
• Scalable meta-learning and model-free optimization strategies addressing the curse of dimensionality for large FIMs and complex system topologies.
• Integration into joint radar-communication systems where continuous FIM reconfiguration may provide unprecedented coexistence and performance capabilities.

The field is rapidly advancing, with ongoing work on hardware prototyping, robust morphing algorithms, and the development of theoretical performance bounds unique to the physically flexible design space (Hu et al., 8 Oct 2025, Teng et al., 29 Jun 2025, Kumar et al., 28 Dec 2025, Farhadi et al., 6 Sep 2025, Eftekhari et al., 19 Sep 2025).