A metasurface array RCS self-healing compensation method based on deep unfolding network

Abstract

The present application relates to computational electromagnetics, intelligent metasurface design, deep learning and array self-healing technology, in particular to a metasurface array electromagnetic scattering self-healing compensation analysis method based on deep unfolding network. The present application focuses on the reliability and high invulnerability of programmable intelligent metasurface under complex electromagnetic environment and extreme physical conditions. Relying on the deep integration of artificial intelligence and full-wave vector electromagnetic physical layer, the present application proposes a dynamic self-healing control strategy for realizing the adaptive fast recovery and accurate reconstruction of radar cross section (RCS) characteristics under multi-angle oblique incidence conditions for active metasurface array containing uncertain random damage mode. The present application covers the physical constraint modeling of metasurface array, efficient numerical evaluation of electromagnetic scattering characteristics, and data-driven array failure fast compensation mechanism, which can be widely applied to new generation of electromagnetic stealth equipment, intelligent radar skin, invulnerability beamforming antenna and complex electromagnetic environment evaluation and other related frontier technical fields.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of microwave and antenna technology and computational electromagnetics, specifically involving a self-healing compensation analysis method for electromagnetic scattering of metasurface arrays based on deep unfolded networks. The core of this invention is the interdisciplinary integration of intelligent optimization algorithms and computational electromagnetics. By using intelligent optimization algorithms to repair and compensate for the electromagnetic scattering problem of metasurface arrays, it overcomes the shortcomings of existing traditional electromagnetic compensation methods, providing a superior approach for solving large-scale array problems. Background Technology

[0002] With the rapid development of modern advanced electromagnetic detection technologies such as phased array radar, multi-base station cooperative detection networks, broadband polarized radar, and over-the-horizon early warning systems, the sensitivity and anti-jamming capabilities of detection equipment have been unprecedentedly improved. Traditional passive static stealth technology mainly relies on electromagnetic scattering of a fixed shape and surface coating with radar-absorbing materials. This method is usually only effective for specific frequency bands and fixed detection directions, which not only greatly increases the weight and aerodynamic design burden of stealth platforms such as aircraft, but also makes it difficult to cope with the complex, ever-changing, multi-frequency, and multi-angle electromagnetic threats of the modern battlefield across the entire airspace. In order to achieve a high survivability rate for targets in high-frequency threat environments, the new generation of electromagnetic stealth technology is irreversibly evolving towards dynamic, intelligent, and adaptive directions.

[0003] Against this backdrop, programmable smart metasurface technology has emerged and rapidly become a key cutting-edge technology in computational electromagnetics and microwave engineering. This technology overcomes the bottleneck of limited electromagnetic properties of traditional natural materials by densely integrating active lumped devices such as varactor diodes, radio frequency microelectromechanical systems (MEMS), or graphene into planar or conformal array units at subwavelength scales, constructing artificial microstructures capable of strongly interacting with electromagnetic waves. Utilizing underlying digital control networks such as field-programmable gate arrays (FPGAs), smart metasurfaces can apply independent bias voltages to each array unit, thereby altering the equivalent capacitance, inductance, or resistance distribution within the unit and achieving real-time dynamic reconstruction of the amplitude, phase, and polarization state of incident electromagnetic waves. In radar cross-section reduction applications, by optimizing the electromagnetic response distribution in the array's two-dimensional space—for example, constructing a discrete phase coding pattern based on 180-degree phase cancellation or an amplitude contrast pattern with spatial gradients—smart metasurfaces can disrupt phase consistency in specific directions, scattering and deflecting strong electromagnetic energy originally concentrated in the target's specular reflection direction at large spatial angles, or converting it into heat energy for absorption, thereby achieving a significant reduction in the target's radar cross-section and dynamic stealth.

[0004] However, in harsh engineering applications and deployment environments, the electromagnetic reliability and damage resistance of metasurface arrays face extremely severe physical challenges. A typical large-aperture smart metasurface device usually contains tens of thousands of high-density arranged independent active units and a complex underlying multilayer feed network. During service, these fragile microwave RF devices are highly susceptible to damage from harsh external environmental stresses. For example, the intense thermal cycling caused by supersonic flight can lead to fatigue fracture of device solder joints, strong mechanical vibration can cause poor contact in the feed network, high-energy microwave directed energy weapons or electromagnetic pulse strikes can burn out semiconductor junctions, and direct physical damage such as shrapnel penetration during combat can cause complete physical destruction of local areas. Once the array suffers unpredictable random damage at uncertain locations, including failure of the phase modulation layer control voltage, constant resistive layer admittance, and dislocation or isolocation breakdown of the multilayer microwave dielectric structure, the carefully calculated and set global electromagnetic coding topology of the metasurface will be completely destroyed.

[0005] The consequences of such localized hardware damage are catastrophic. At the electromagnetic scattering level, the damaged unit cannot output the expected amplitude and phase response, causing the original spatial electromagnetic wave phase cancellation mechanism to completely fail. The severe mismatch in local phase and amplitude will induce strong electromagnetic parasitic interference and edge diffraction effects on the array aperture surface, thereby exciting abnormally strong scattering sidelobes at multiple spatial angles of non-mirror reflection. These abrupt and unexpected strong electromagnetic scattering regions are like searchlights in the night sky, easily detected by enemy high-sensitivity radar from multiple angles, instantly exposing targets with excellent stealth capabilities and seriously threatening the platform's survival.

[0006] For the recovery of electromagnetic properties of damaged intelligent metasurface arrays, i.e., self-healing compensation technology, existing traditional solutions mainly rely on heuristic global optimization strategies such as genetic algorithms, particle swarm optimization, or simulated annealing. The working mechanism of these methods is as follows: after detecting local damage to the array, the control voltage of the remaining healthy operating units is used as a high-dimensional optimization variable, with the lowest global radar cross section or the lowest sidelobe level at a specific angle as the objective function, and iterative optimization calculations are performed across multiple generations of the population. Although heuristic algorithms have the mathematical theoretical ability to converge to the global optimum, they suffer from significant computational complexity and time delay bottlenecks in practical computational electromagnetics engineering. This is because each fitness assessment of an individual in the population requires calling underlying algorithms for full-wave vector electromagnetic simulation, such as the method of moments or the finite-difference time-domain method, to solve integral or differential equations. For large-scale three-dimensional arrays containing hundreds or thousands of microwave units, a single full-wave calculation consumes massive amounts of memory and time, resulting in a complete self-healing compensation iteration often taking hours or even days. This massive consumption of computing resources and extreme computational latency are completely unable to meet the real-time dynamic response requirements of milliseconds to seconds in actual combat environments, making it difficult for traditional optimization self-healing methods to bridge the gap between theoretical simulation and engineering application.

[0007] In recent years, to overcome the computational limitations of heuristic algorithms, data-driven deep learning models have been introduced into the field of rapid response and feature prediction for electromagnetic arrays. By pre-generating massive amounts of damaged and compensated data pairs on supercomputers for offline training, deep networks can complete online forward inference in milliseconds. However, conventional deep neural network methods have revealed three core and fatal technical flaws when solving complex high-frequency electromagnetic scattering self-healing problems.

[0008] First, there is the problem of the lack of physical mechanisms and the uninterpretability of the model. Conventional deep convolutional networks or multilayer perceptrons can only construct a purely mathematical and statistical mapping from the damaged mask image to the compensation voltage value in a high-dimensional space. The network architecture is completely stripped of the electromagnetic field boundary conditions of Maxwell's equations and the strict physical constraints of microwave transmission line theory. This data-heavy, physics-light model performs reasonably well when dealing with the conventional damage distributions covered in the training dataset, but its prediction accuracy and generalization ability will drop precipitously when faced with complex, variable, large-scale, or topologically anomalous unseen damage patterns.

[0009] Secondly, there is the risk of outputting physically infeasible solutions. The electromagnetic response of microwave devices is limited by their physical material properties. For example, the junction capacitance adjustment range of a varactor diode has strict upper and lower boundary limits, and its corresponding equivalent impedance has a highly nonlinear relationship with the bias voltage. In the process of pursuing the minimization of the loss function, conventional deep learning models find it difficult to accurately apply and ensure that the output compensation bias parameters strictly fit the adjustment boundary of the actual physical device. This can easily generate invalid instructions that exceed the hardware impedance adjustment limit, leading to erroneous output states of the self-healing system or even deadlock of the underlying hardware control system.

[0010] Finally, the electromagnetic adaptability to multi-angle obliquely incident electromagnetic waves is extremely weak. In actual detection space, electromagnetic waves emitted by enemy detection radars often act on the surface of stealth targets in an oblique incidence manner. Oblique incidence of electromagnetic waves not only causes a sharp separation of spatial wave impedance between transverse electric waves and transverse magnetic waves under different polarizations, but also introduces an extremely complex spatial phase delay gradient on the array aperture surface. Conventional data-driven self-healing models usually rely only on static samples under normal perpendicular incidence conditions for training, and their internal weights cannot learn the dynamic coupling law between the incident angle of the electromagnetic wave and the wavefront of the array surface. Once the incident angle deflects, the spatial phase compensation distribution corresponding to the network output compensation parameters will be completely misaligned, resulting in severe distortion of the electromagnetic field pattern. This not only fails to suppress sidelobes but may also generate additional grating lobe effects.

[0011] Therefore, existing computational electromagnetics self-healing methods and conventional artificial intelligence algorithms cannot adequately solve the problem of efficient reconstruction of metasurface arrays under extreme damage conditions. Currently, this field urgently needs an innovative analysis and control framework that can inherit the ultra-high timeliness of forward inference from deep neural networks while deeply integrating microwave network equivalent analysis, full-wave vector electromagnetic calculation rules, and strict bias voltage regularization constraints into the algorithm model's underlying layer. How to overcome the theoretical limitations of traditional pure data-driven models and construct an intelligent analysis method with an electromagnetic physics forward feedback mechanism, capable of global spatial feature extraction based on radar cross-section residuals, and able to adaptively and robustly output high-precision self-healing compensation voltage under complex electromagnetic conditions with multi-angle oblique incidence has become the most critical and urgent technical bottleneck to be overcome in the intersection of next-generation high-reliability, high-damage-resistance intelligent stealth equipment development and computational electromagnetics. Summary of the Invention

[0012] The purpose of this invention is to address the technical challenges of low computational efficiency, lack of physical constraints, and poor adaptability to oblique incidence in traditional metasurface self-healing compensation methods. Specifically, this invention proposes a metasurface array RCS self-healing compensation method based on a deep unfolded network. This method not only hard-codes a strict electromagnetic full-wave vector physical layer within the network but also designs a context-aware mechanism based on RCS residuals and a multi-dimensional composite loss function. The detailed implementation steps of this method are as follows:

[0013] Step 1: For the metasurface array, establish its detailed microwave equivalent circuit and electromagnetic field coupling model. Divide the array into a two-dimensional mesh, with each cell containing a first layer (a varactor diode layer controlled by bias voltage V1) for phase modulation and a second layer (a graphene or resistive layer controlled by bias voltage V2) for amplitude modulation.

[0014] For a given operating frequency f and angular frequency ω = 2πf, combining the thin dielectric approximation and transmission line theory, the surface impedance Z of the top layer is... top The imaginary part is formed by the varactor diode and capacitor C. p Decide:

[0015]

[0016] Among them, Z top R represents the equivalent surface complex impedance of the top-level frequency-selective surface; FSS The equivalent series resistance of the top-layer metallic microstructure is represented by ω; the angular frequency of the incident electromagnetic wave is represented by L. FSS This represents the equivalent distributed inductance of the top-level structure. (C) p With control voltage V1 at [C min C max The dynamic and continuous changes between [ ].

[0017] Synthetic first-layer admittance (1 / Z) top Complex admittance Y, controlled by the second-layer voltage V2 g And the pure imaginary admittance Y of the underlying dielectric substrate sub The overall input impedance Z of the unit is obtained. in Under multi-angle oblique incidence conditions, the equivalent wave impedance of the TE wave, which varies with the spatial polar angle, is introduced to calculate the local reflection coefficient Γ of each element:

[0018]

[0019] Where Γ represents the local complex reflection coefficient of the array element; Z in This represents the overall equivalent input complex impedance of the array element; This represents the equivalent wave impedance of a transverse electric wave under oblique incidence conditions in free space. Here, η0 represents the intrinsic impedance of free space, and θ represents the polar angle of the scattering observation.

[0020] Furthermore, based on the array's physical arrangement and the obliquely incident wavefront, the far-field radar cross section (RCS) is calculated. First, the spatial phase delay factor ΔΦ for each element is calculated. i :

[0021] ΔΦ i =k0[x i (sinθcosφ-sinθ in cosφ in )+y i (sinθsinφ-sinθ in sinφ in )]

[0022] Where, ΔΦ i The i-th array element represents the spatial phase delay relative to the array's physical origin; k0 represents the free-space wavenumber; x i and y i θ and φ represent the x and y coordinates of the i-th array element in the physical Cartesian coordinate system, respectively; θ and φ represent the polar angle and azimuth angle of the scattered field in the spherical coordinate system, respectively; θ in and φ in These represent the polar angle and azimuth angle of the incident electromagnetic wave, respectively.

[0023] Based on this, the formula for calculating the far-field radar cross section (RCS) of the array is as follows:

[0024]

[0025] Where RCS represents the calculated logarithmic value of the radar cross section; λ represents the operating wavelength of the electromagnetic wave; A unit N represents the physical geometric area of ​​a single metasurface unit; unit Indicates the total number of cells contained in the metasurface array; cos 4 θ represents the low elevation angle approximate attenuation envelope weight of the microwave array element factor; Γ i Let represent the local complex reflection coefficient of the i-th unit; e represents the natural constant; ∈ represents the minimal positive bias constant to prevent the logarithmic function from exhibiting a singularity of negative infinity at zero. This full-wave physical layer model will be directly embedded as a differential operator into the computational graph of the deep neural network.

[0026] Step 2: Define spatial mask matrices M1 (corresponding to the first phase modulation layer) and M2 (corresponding to the second amplitude modulation layer) that represent the healthy topological state of the array cells.

[0027]

[0028] in, V represents the complete fused bias voltage matrix of the k-th layer at the t-th compensation iteration; k_dmg M represents the actual damaged bias voltage matrix extracted after random physical damage; k This represents the health space mask matrix of the k-th layer (the corresponding element of the matrix is ​​1 when the cell is intact and 0 when the cell is damaged); This represents the candidate compensatory bias voltage matrix output by the neural network at step t for the damaged cell. This fusion formula ensures that the voltage state of all healthy cells remains hard-locked during the self-healing calculation process, with non-invasive compensatory intervention only performed on the damaged cells.

[0029] Step 3: Obtain the physical layer forward extrapolation prediction value (RCS) for the current damaged state or intermediate compensation state. pred Compared to the ideal radar cross section (RCS) target The residual matrix R between diff .

[0030] R aiff =RCS pred -RCS target

[0031] Among them, R diff RCS represents the radar cross section error residual matrix under the current state. pred RCS represents the predicted radar cross section matrix obtained by the forward calculation of the full-wave vector physical layer operator in step 1. target This represents the radar cross-section matrix of the baseline target under an undamaged, ideally encoded array. Core statistical indicators of this residual matrix (including residual mean, standard deviation, maximum range, root of square, positive and negative error energy integrals, pattern distortion skewness, and hotspot kurtosis) are calculated along the entire spatial scattering angle dimension, generating a fixed-dimensional contextual statistical feature vector C. ctx This completely decouples the spatial angular resolution required for network training from the forward inference resolution, thereby improving the model's ability to adapt across resolutions.

[0032] Step 4: The traditional high-dimensional global optimization numerical solution process is expanded into a multi-cascaded feedforward neural network over time. Each cascaded step includes a spatial convolutional coding module and a multi-head self-attention module to capture local impedance abrupt changes between array units and global electromagnetic coherence interference dependencies, generating a multi-channel spatial feature map F. spatial .

[0033] A Feature-Affine Linear Modulation (FiLM) layer is introduced, which uses the context feature vector generated in step 3 to modulate the spatial feature map with dynamic physical optimization direction:

[0034] F′spatial =F spatial ⊙γ(C ctx )+β(C ctx )

[0035] Among them, F′ spatial This represents the enhanced multichannel spatial feature map after context-modulated dynamic modulation; F spatial Represents the initial spatial feature map extracted by the convolution and self-attention modules; ⊙ represents the Hadamard product (element-wise multiplication operation); γ(C ctx ) represents the context feature vector C ctx The adaptive scaling factor matrix generated by the mapping controls the magnification ratio of specific spatial feature channels; β(C ctx ) represents the context feature vector C ctx The adaptive translation bias coefficient matrix generated by the mapping is equivalent to providing the specific gradient direction for physical residual compensation in the network; C ctx This represents the contextual statistical feature vector extracted in step 3. The modulated feature map is decoded to output the fine voltage correction for the current step. This process is repeated until a preset number of steps is reached, at which point the network-level initial high-precision compensation voltage is output.

[0036] Step 5: To ensure that the deep unfolded network does not produce non-physically overfitted solutions and to strictly suppress anomalous scattering sidelobes in the far-field radiation pattern, this invention constructs a composite loss function L that integrates multiple physical rules. total :

[0037] L total =L rcs +λ hs L hotspot +λ p L pea k+λ v L vreg +λ tv L tv +λ s L s

[0038] Among them, L total L represents the total loss in overall composite physics training; rcs L represents the fundamental mean square error loss between the predicted scattering map and the ideal target map; hotspot Indicates the penalty loss for both sides of the hotspot; L peak L represents the global maximum reflection peak alignment loss; vreg L represents the bias voltage regression regularization loss; tv L represents the total variational smoothing loss; s λ represents the transition smoothing loss based on mask boundaries; hs , λ p , λv , λ tv , λ s These represent the dynamic balance weight coefficients for each corresponding physical constraint.

[0039] To prevent the compensated radiation pattern from generating strong reflection sidelobes exceeding the standard threshold in any non-target direction, a two-sided hotspot penalty loss L is constructed. hotspot :

[0040]

[0041] Among them, L hotspot N represents the two-sided hotspot penalty loss value; angle This represents the total number of angle nodes evaluated in the discrete space of polar angle and azimuth angle; m represents the traversal index of the angle node; max() represents the nonlinear rectification maximum value function; This represents the residual value between the predicted RCS and the ideal RCS at the m-th angle node (corresponding to the aforementioned R). diff Matrix elements); R thr This represents the electromagnetic scattering margin threshold that the system can tolerate (e.g., 3.0 dB). This formula strictly limits the scattering characteristics at all angles from deviating significantly from the ideal envelope, effectively curbing the phenomenon of the network sacrificing global stealth performance for local performance gains.

[0042] To ensure the feasibility of the output compensation voltage in the hardware power supply network and avoid drastic non-physical jumps, a bias voltage regression regularization loss L is constructed. vreg :

[0043]

[0044] Among them, L vreg N represents the bias voltage regression regularization loss value; unit Indicates the total number of array cells; i represents the traversal index of the array physical cells; and These represent the candidate compensation voltages for the damaged cells in the first and second layers of the deep network output, respectively. and These represent the standard checkerboard preset voltages corresponding to the damaged unit under ideal, undamaged conditions; and represents the first and second layer health mask values ​​of the i-th unit, respectively; |·| represents the calculation of the absolute value norm.

[0045] Step 6: During the actual array deployment inference phase, the damaged array data is input into the trained deep unfolded network, and a high-quality baseline compensation voltage solution is rapidly obtained using a single forward propagation. Subsequently, the backpropagation computation graph of the underlying electromagnetic physical operators is activated, and a small number of lightweight gradient tracking iterations are performed using the adaptive moment estimation Adam optimizer to further eliminate minor numerical residuals. After compensation, the main lobe is located in the global field of view using the adaptive sidelobe level SLL algorithm, and radar RF indicators such as sidelobe level difference, 3dB main lobe bandwidth change rate, and global mean square error are extracted. Multi-dimensional self-healing visualization verification results are output. Attached image description:

[0046] The platform for this invention is as follows: the CPU is an Intel Xeon Gold 6226R, the graphics card is an NVIDIA RTX A6000, the memory is 256GB, the physical model material is a copper FSS frequency selective surface in the first layer, a graphene film in the second layer, and a copper base plate in the third layer. The neural network adopts the PyTorch architecture.

[0047] Figure 1 This is a flowchart illustrating the overall implementation of the present invention.

[0048] Figure 2 This is a physical model of the detailed material dimensions of the array unit in this invention.

[0049] Figure 3 This is a diagram of the 20×20 metasurface array unit used in this study.

[0050] Figure 4 An overview of the compensation network for the algorithm proposed in this invention.

[0051] Figure 5 The training loss curve in the experiment clearly shows that the model has fully converged after 350 epochs, reaching the ideal standard.

[0052] Figures 6(a) and (b) show the overall and main lobe magnification compensation effect of random sample sections under the first mode (first layer random damage, second layer no damage).

[0053] Figure 7 This is a diagram illustrating the overall 2D RCS compensation effect of the model. Detailed Implementation

[0054] To better understand the technical content of this invention, specific embodiments are described below in conjunction with the accompanying drawings.

[0055] Figure 1 It is the overall flowchart of the invention, such as Figure 1As shown, this invention first completes the initialization of global physical parameters and the generation of ideal electromagnetic codes, then establishes a full-wave vector physical layer model of the metasurface array, and further constructs random damage masks and multi-mode training samples. On this basis, the traditional electromagnetic optimization iterative process is expanded into a multi-cascaded deep neural network, and a context feature encoding module based on RCS residuals and a unit voltage correction module based on spatial self-attention are introduced into the network. In the training phase, multi-dimensional physical composite loss is used for deep supervision, and in the inference phase, the initial compensation voltage is obtained by forward propagation of the network, and the final compensation result is obtained by combining lightweight physical gradient refinement. Finally, the self-healing compensation effect is verified by radar cross section error, sidelobe level, main lobe width, and multi-dimensional visualization images.

[0056] In this invention, the following is employed: Figure 2 The multilayer programmable metasurface unit shown serves as the basic electromagnetic control unit. Its specific dimensional parameters are: L1 = 3.6 mm, L2 = 3.2 mm, L3 = 1.2 mm, L4 = 3.0 mm, W1 = 0.4 mm, W2 = 0.6 mm, W3 = 2.2 mm, W4 = 0.15 mm, W5 = 0.4 mm, T1 = 3535 μm, T2 = 50.235 μm, T3 = 1.5 mm. From top to bottom, the unit comprises a copper frequency-selective surface layer, a dielectric layer, a graphene or resistive amplitude modulation layer, a prepreg layer, an F4B dielectric substrate, and a bottom metal ground layer. Specifically, the first frequency-selective surface layer controls the equivalent capacitance of the varactor diode through a bias voltage V1 to achieve phase modulation; the second graphene or resistive film layer controls the surface admittance through a bias voltage V2 to achieve amplitude modulation.

[0057] like Figure 3 As shown, this invention employs a 20×20 two-dimensional metasurface array, comprising 400 array elements with a period of 10 mm. The operating frequency is set to 10 GHz, corresponding to a free-space wavelength of 30 mm. The main physical parameters are: operating frequency f = 10 GHz, speed of light c = 3 × 10⁻⁶. 8 m / s, free space wave impedance η0 = 377Ω, relative permittivity ε of F4B dielectric r =2.65, frequency selection surface equivalent series resistance is R FSS =8Ω, frequency selection surface equivalent inductance is L FSS =2.0nH, the varactor diode capacitance adjustment range is 0.08pF to 2.0pF.

[0058] In this invention, to improve the phase cancellation effect between two coding states, the substrate thickness is first scanned within a range of 1.0 mm to 4.0 mm. The reflection phase difference and reflection amplitude of the two candidate coding states at different thicknesses are calculated, and the substrate thickness that makes the phase difference between the two states closest to 180° and has a high reflection amplitude is selected as the preferred thickness. Subsequently, the first layer bias voltage V1 and the second layer bias voltage V2 are jointly scanned to obtain two ideal phase coding states V. 1，bit0、 V 1，bit1 And the corresponding optimal second-layer bias voltage V 2，opt During the scanning process, the local reflection amplitudes of both coded states are constrained simultaneously to prevent phase coding from degenerating into a simple absorption state. For any two candidate states, the phase cancellation residual is defined as:

[0059]

[0060] Where Γ0 and Γ1 represent the complex reflection coefficients corresponding to the two coding states, and Δφ represents their phase difference. The optimal state minimizes the residual r, thereby improving the theoretical RCS reduction capability.

[0061] After completing the unit encoding calibration, an ideal two-dimensional encoding pattern in the form of a checkerboard is constructed using 2×2 superunits as the basic period. That is, for the unit in the i-th row and j-th column of the array, a value is assigned according to the following rules:

[0062]

[0063] When bit=0, the first layer bias voltage is V. 1，bit0 When bit=1, the first layer bias voltage is V. 1，bit1 The second-layer bias voltage is uniformly selected from the optimal value V. 2,opt Thus, the two-layer bias matrix V under the ideal lossless state is obtained. 1，ideal and V 2，ideal .

[0064] To simulate uncertain damage to metasurface array elements in a real-world service environment, this invention defines health masks M1 and M2 for the first phase-modulated layer and the second amplitude-modulated layer, respectively. When an array element's corresponding layer is intact, its mask value is 1; when it is damaged, the mask value is 0. Training samples with damage ratios of 20%, 30%, and 35% are randomly generated, and a learning-based approach is used to gradually increase the damage intensity. During the random damage generation process, each damaged element can exhibit complete or partial failure; in the case of complete failure, the bias voltage is set to 0, and in the case of partial failure, the bias voltage fluctuates randomly between 0 and 1.

[0065] To verify the robustness of the present invention under different damage topologies, this embodiment further constructs four typical damage modes: First-layer isolated damage mode: only the first layer V1 is damaged at a random location, while the second layer remains healthy; Second-layer isolated damage mode: only the second layer V2 is damaged at a random location, while the first layer remains healthy; Two-layer ectopic damage mode: both the first and second layers are damaged, but the damage locations do not overlap; Two-layer co-located damage mode: the first and second layers are damaged simultaneously at the same location, which is the most severe damage case.

[0066] Through the aforementioned multi-mode sample construction method, the network can learn the electromagnetic response variation patterns under different physical damage mechanisms, thereby improving the generalization ability of the compensation model to unknown damage modes.

[0067] Figure 4 This paper illustrates the deep unfolded self-healing compensation network structure employed in this invention. This embodiment unfolds the traditional iterative solution process based on physical optimization into four cascaded compensation steps. Each step includes physical layer forward computation, RCS residual extraction, spatial feature encoding, residual context modulation, and voltage correction output. In the k-th unfolding step, the network first calculates the predicted RCS based on the current voltage state. Subsequently, the predicted RCS is subtracted from the ideal RCS to obtain the residual. This residual directly reflects the pattern error between the current compensation state and the ideal electromagnetic scattering state, serving as a crucial physical feedback signal for subsequent voltage correction. Each single-step correction network uses a six-channel two-dimensional tensor as its spatial input. The six channels are: current first-layer voltage V1; current second-layer voltage V2; first-layer health mask M1; second-layer health mask M2; damage deviation of the first-layer relative to the ideal encoding; and damage deviation of the second-layer relative to the ideal encoding. The above six-channel input first undergoes two-dimensional convolution, normalization, and GELU activation function extraction to extract local spatial features, and then the expressive power is enhanced through a residual convolution module. To capture the coherent interference between distant array units, this embodiment introduces a spatial self-attention module after the convolutional features. This module flattens the 20×20 array features into 400 spatial position sequences and introduces learnable position encoding, calculating the global coupling relationship between different units through multi-head self-attention. Simultaneously, this invention sets up an RCS residual context encoding module. This module does not directly rely on a fixed angular resolution, but extracts fixed-dimensional statistical features from the residual radiation map, including residual mean, standard deviation, maximum value, minimum value, root mean square, positive residual energy, negative residual energy, skewness, and kurtosis. These nine statistical quantities are mapped into a context vector through a multilayer perceptron. Subsequently, a feature-affine linear modulation layer is used to dynamically modulate the spatial feature map. Through this structure, the network can dynamically adjust the importance of different spatial feature channels according to the current RCS residual state, thereby achieving physical feedback-based self-healing compensation by "adjusting array voltage based on radiation map error".

[0068] Finally, the modulated feature map is decoded to output the voltage correction values ​​ΔV1 and ΔV2 for the first and second layers. To ensure that the voltage output meets the hardware-feasible range, a hyperbolic tangent function is used to limit the single-step correction amplitude, and the voltage is clipped to between 0 and 1 after each update. Each step only updates damaged cells, while healthy cells always maintain their original actual voltage.

[0069] During the training phase, this embodiment employs a composite loss function composed of multiple physical constraints to ensure that the network output not only numerically approximates the ideal RCS but also meets multiple requirements such as radar pattern, hardware bias, and spatial smoothing. The overall loss function includes the basic RCS mean square error loss, bilateral hotspot penalty loss, peak alignment loss, voltage regularization loss, total variational smoothing loss, and mask boundary smoothing loss. Furthermore, since this invention adopts a deep unfolding structure, intermediate RCS prediction results can be obtained at each unfolding step. Therefore, this embodiment simultaneously applies auxiliary supervision loss to each intermediate step and assigns greater weight to subsequent steps, enabling the network to form a progressive optimization process from coarse compensation to fine compensation.

[0070] In this embodiment, the number of training samples is 20,000, the number of validation samples is 4,000, the batch size is 400, and the maximum number of training rounds is 500. The optimizer uses AdamW, combined with a warm-up learning rate and cosine annealing learning rate scheduling strategy. During training, different early stopping patience values ​​are set according to the incident angle to avoid overfitting and save training time.

[0071] like Figure 5 As shown, both training loss and validation loss decrease rapidly with the number of iterations and are significantly lower than the uncompensated baseline loss. After about 350 training cycles, the model basically enters a stable convergence state, indicating that the proposed physically embedded deep unfolded network can effectively learn the electromagnetic compensation law of randomly damaged metasurface arrays.

[0072] After offline training is completed, during the actual deployment and inference phase, the damaged array's V1, V2, M1, and M2 are input into the trained deep unfolded network. After one forward propagation, the complete compensation voltage V is output. 1，comp and V 2，comp .

[0073] To further eliminate the small residuals between network prediction and refined physics calculation, this embodiment performs a small amount of lightweight physics gradient refinement based on the network output. Specifically, the network output voltage is mapped to the logit space to ensure that the voltage remains within the hardware-feasible range of 0 to 1 during the optimization process; then, the Adam optimizer is used to perform gradient tracking on the damaged cell voltage for about 30 steps. During the refinement process, healthy cells remain locked, and only the candidate compensation voltages at the damaged locations are optimized.

[0074] The refinement objective function includes RCS mean square error, peak error, hotspot penalty, and voltage regularization term. Since the deep unfolded network already provides a high-quality initial solution, the number of iterations required for the refinement process is far fewer than that of traditional heuristic global optimization algorithms, thus achieving a balance between real-time performance and compensation accuracy.

[0075] Finally, this invention generates candidate test samples for each of the four damage modes, automatically selects random samples for display, and outputs the corresponding one-dimensional RCS compensation map, as well as the corresponding main lobe magnification map and two-dimensional RCS compensation map, as shown below. Figure 5 As shown in Figures 6 and 7, the results demonstrate that this invention is applicable to first-layer damage, second-layer damage, double-layer ectopic damage, and double-layer isotopic damage scenarios, exhibiting strong damage mode generalization ability.

Claims

1. A metasurface array RCS self-healing compensation method based on deep unfolded networks. The purpose of this invention is to solve the technical problems of low computational efficiency, lack of physical constraints, and poor adaptability to oblique incidence in traditional metasurface self-healing compensation methods. This method not only hard-codes a strict electromagnetic full-wave vector physical layer within the network, but also designs a context-aware mechanism based on RCS residuals and a multi-dimensional composite loss function. It includes the following steps: Step 1: For metasurface arrays, establish a detailed microwave equivalent circuit and electromagnetic field coupling model; Step 2: Define spatial mask matrices M1 (corresponding to the first phase modulation layer) and M2 (corresponding to the second amplitude modulation layer) to represent the healthy topological state of the array cells; Step 3: Obtain the physical layer forward extrapolation prediction value (RCS) of the current damaged state or intermediate compensation state. pred Compared to the ideal radar cross section (RCS) target The residual matrix R between diff ; Step 4: Expand the traditional high-dimensional global optimization numerical solution process into a multi-stage feedforward neural network over time steps. Step 5: To ensure that the deep unfolded network does not produce non-physical overfitting solutions and to strictly suppress anomalous scattering sidelobes in the far-field radiation pattern, this invention constructs a composite loss function L that integrates multiple physical rules. total ; Step 6: In the actual array deployment inference stage, the damaged array data is input into the trained deep unfolded network. The adaptive sidelobe level SLL algorithm is used to locate the main lobe in the global field of view. Radar radio frequency indicators such as sidelobe level difference, 3dB main lobe bandwidth change rate and global mean square error are extracted, and multi-dimensional self-healing visualization verification results are output.

2. The method for RCS self-healing compensation of metasurface arrays based on deep unfolded networks according to claim 1, characterized in that: In Step 1, a microwave equivalent circuit and electromagnetic field coupling model of the metasurface array were constructed. By independently controlling the phase and amplitude of each element through dual bias voltages, the overall input impedance and local reflection coefficient of the element were calculated. Based on this, the RCS of the array was derived by combining spatial phase delay. Finally, this full-wave physical model was directly embedded as a differential operator into the computational graph of the deep neural network.

3. The method for self-healing RCS compensation of metasurface arrays based on deep unfolded networks according to claim 1, characterized in that: In Step 2, a spatial mask matrix representing the health state of each layer of the array is defined, and this matrix is ​​used to fuse the actual damaged voltage with the compensatory voltage generated by the neural network. This mechanism ensures that the voltage state of healthy cells is strictly locked during the self-healing iterative calculation, and precise, non-invasive compensatory intervention is performed only on damaged cells.

4. The method for self-healing RCS compensation of metasurface arrays based on deep unfolded networks according to claim 1, characterized in that: In Step 3, the residual matrix between the current predicted RCS and the ideal target RCS is calculated, and its core statistical indicators in the full spatial dimension are extracted to generate a fixed-dimensional context feature vector.

5. The method for RCS self-healing compensation of metasurface arrays based on deep unfolded networks according to claim 1, characterized in that: In Step 4, the traditional global optimization process is expanded into a multi-level neural network, and the local impedance mutation and global electromagnetic coherence features of the array are extracted using convolution and self-attention mechanisms. Subsequently A feature-based linear modulation layer is introduced, and the spatial feature map is dynamically modulated using the context feature vector of Step 3 to guide the physical optimization direction. Finally, the decoder iteratively outputs a fine voltage correction amount to generate a network-level high-precision initial compensation voltage.

6. The method for RCS self-healing compensation of metasurface arrays based on deep unfolded networks according to claim 1, characterized in that: In Step 5, a composite loss function integrating multiple physical rules is constructed to prevent the network from generating non-physical overfitting solutions and to strictly suppress anomalous scattering sidelobes. This function not only includes the basic error but also introduces a two-sided hotspot penalty loss to limit the full-domain scattering error to not exceeding the tolerance threshold, and a bias voltage regression regularization loss to constrain the compensation voltage to be close to the ideal state.

7. The method for self-healing RCS compensation of metasurface arrays based on deep unfolded networks according to claim 1, characterized in that: In Step 6, a high-quality reference compensation voltage solution is first obtained rapidly through a single forward propagation of the deep unfolded network. Subsequently, the backpropagation of the underlying physical operators is activated, and the Adam optimizer is used for minor fine-tuning iterations to eliminate small residuals. Finally, core radar RF indicators such as sidelobe level and 3dB bandwidth are extracted, and multi-dimensional visualization verification results of the self-healing effect are output.

Images