Linear array radiation pattern synthesis method and system based on hybrid optimization

Abstract

The invention relates to the technical field of array radiation pattern synthesis, in particular to a linear array radiation pattern synthesis method and system based on hybrid optimization. The method comprises the following steps: obtaining dipole array optimization parameter data; initializing an array parameter and an FGO population according to the acquired data; constructing a KAN proxy model and training the model by using an FGO population; performing hybrid iterative optimization on the KAN proxy model fused with the FGO; and performing array antenna pattern synthesis by using the optimized KAN proxy model. According to the probability hybrid evaluation strategy based on the KAN proxy model, through the whole FGO iteration process, low-potential candidate solutions are effectively screened out by dynamically allocating accurate electromagnetic simulation resources (such as high-proportion proxy evaluation in the initial stage and gradually-increased accurate verification in the later stage), and the calling frequency of a high-cost moment method is remarkably reduced.

Description

Technical Field

[0001] The present invention relates to the technical field of array radiation pattern synthesis, and in particular to a linear array radiation pattern synthesis method and system based on hybrid optimization. Background Art

[0002] With the rapid development of 5G / 6G communications, the Internet of Things, satellite communications, and radar detection, the demand for high-performance antennas, such as linear dipole array antennas (LAA) and elliptical dipole array antennas (CEAA), has surged. Antenna pattern synthesis, as a core design step, requires strict suppression of sidelobe levels while maintaining high mainlobe gain. However, this problem is inherently a high-dimensional nonlinear optimization problem, and traditional methods face three bottlenecks: (1) The analytical synthesis method is inefficient. Traditional pattern synthesis methods based on mathematical equations (such as Fourier transforms and Woodward sampling) struggle to simultaneously meet the stringent performance requirements of high gain and low sidelobes when dealing with complex radiation scenarios. Their calculations rely on gradient information and require repeated calls to electromagnetic simulations (such as the method of moments), resulting in lengthy computational times and low design efficiency, especially in high-dimensional parameter spaces.

[0003] (2) Inherent defects of metaheuristic algorithms, Although metaheuristic algorithms such as Fungal Growth Optimization (FGO), Seagull Algorithm (SOA), and Whale Algorithm (WOA) can circumvent the problem of non-differentiable objective functions, they still have significant shortcomings: Computational efficiency bottleneck: The algorithm requires numerous iterations to evaluate candidate solutions, and each evaluation requires invoking costly electromagnetic simulations, dramatically increasing the overall optimization time. High-dimensional performance degradation: As the number of antenna elements increases (the dimension increases), the algorithm's global search capability decreases sharply, making it prone to falling into local optimality and slowing convergence. Low utilization of historical data: The solution samples generated by the search process are not effectively mined, and the lack of random exploration guided by prior knowledge results in a large amount of ineffective calculations.

[0004] At the same time, deep learning agent models also have limitations. To accelerate optimization, existing research attempts to embed deep learning models such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs) as agent models into metaheuristic frameworks, but this faces new challenges: The curse of dimensionality: The parameter size of fully connected neural networks grows exponentially with the input dimension. High-dimensional optimization requires massive amounts of training data, which significantly increases the computational burden. Black box characteristics: The model decision process is unexplainable, making it difficult to coordinate with heuristic search strategies and unable to provide gradient information to guide the optimization direction; Static architecture defects: the fixed network structure cannot dynamically adapt to the evolution of solution space distribution in the optimization process, and needs to be frequently retrained to maintain accuracy, which weakens the acceleration effect. SUMMARY

[0005] In order to solve the problems of low computational efficiency, high-dimensional performance attenuation and difficulty of dynamic collaborative optimization of traditional proxy model in existing array antenna pattern optimization, the application provides a linear array radiation pattern synthesis method and system based on hybrid optimization.

[0006] In the first aspect, the application provides a linear array radiation pattern synthesis method based on hybrid optimization, which adopts the following technical scheme: A linear array radiation pattern synthesis method based on hybrid optimization, comprising: Obtaining dipole array optimization parameter data; Initializing array parameters and FGO populations according to the obtained data; Building a KAN proxy model and training the model using the FGO population; Hybrid iterative optimization of the KAN proxy model fused with FGO; Using the optimized KAN proxy model to synthesize array antenna patterns.

[0007] Further, the initialization of array parameters and FGO populations according to the obtained data comprises establishing a high-precision array antenna electromagnetic model using the full-wave moment method to solve the quantification problem of mutual coupling effects between array elements, wherein the surface of each antenna element is divided into unstructured triangular meshes, the entire array structure is discretized by MxN triangular patches, and the RWG vector basis function is used to describe the surface current distribution, and the basis function is defined on the common edge of adjacent triangular patches. is expressed as: , wherein is the length of the common edge, is the area of the adjacent triangular patch, is the coordinates of the corresponding vertex, and are adjacent triangular patches sharing the first n edge, and the direction convention is such that the basis function satisfies the normal current continuity on the common edge, avoiding the current discontinuity problem caused by traditional piecewise basis functions.

[0008] Further, the initialization of array parameters and FGO populations according to the obtained data further comprises establishing an array electromagnetic radiation model based on the EFIE equation, which is expressed as: , , , in, is the unit vector normal to the conductor surface, is the Green function, is the current distribution on the conductor surface, is the incident electric field, is the angular frequency, is the magnetic permeability, is the dielectric constant; EFIE is discretized into dimensional dense impedance matrix: , The final radiation pattern is given by the excitation current vector Export: , in is the wave vector, (θ, ) represents the spatial orientation of the dipole array.

[0009] Furthermore, the method of initializing array parameters and FGO population according to the acquired data also includes initializing the population of FGO for optimizing fungal growth by simulating the natural diffusion behavior of fungal hyphae. D Random generation in dimensional solution space Np candidate solutions, each individual X k =[ x k1 , x k2 ,..., x kD ] represents a position coordinate in the solution space, and its dimensional components x kd In the domain [ ad , bd ] obeys uniform distribution. During the search process, in order to prevent the constant growth rate from reducing the exploration accuracy, the fluctuation is increased, which is expressed as: , in, represents the exponential expression of hyphal growth rate, Indicates the The fitness value of a solution, is the current function evaluation.

[0010] Furthermore, the construction of the KAN agent model and the use of the FGO population to train the model include introducing the KAN model to model the radiation characteristics of the antenna array, and introducing multiple unary learnable activation functions. , first map the input variables separately, and then add them to another function , and finally form the output, denoted as: , wherein, denotes a bounded domain of a multivariate continuous function, i.e. ; and are univariate functions and and , , ; by defining the L1 norm of the activation function as the average value of its inputs, for a KAN layer with inputs and outputs, its L1 norm is defined as the sum of the L1 norms of all activation functions, the entropy of is defined as: .

[0011] Further, the KAN agent model is constructed and trained using the FGO population, including introducing two regularization terms in the total loss function, corresponding to the L1 norm of the activation function and the entropy value of its distribution, respectively, to encourage the model to tend to sparse and information-intensive structure, to realize the controllability of the activation function range, the concept of connection strength is introduced, and the local response range is adjusted by controlling the size of the function slope, wherein the total training target is the prediction loss of all KAN layers plus the L1 and entropy regularization of all KAN layers: , wherein and are relative amplitudes, used to control the overall regularization amplitude, denotes the number of network layers of KAN, after training using the sparsification penalty, the KAN will be sparsified on the node layer, for each node, its input and output are defined as: .

[0012] Further, the KAN agent model fused with FGO is mixed and iteratively optimized, including defining the optimization problem of the directional diagram synthesis of linear array and elliptical array as: under the constraint of given array antenna size, by optimizing the excitation phase, excitation amplitude or array element position, introducing KAN as an intelligent agent model, realizing the optimization of high gain and low sidelobe performance of the radiation directional diagram, the objective function is the probability The main lobe gain and side lobe level are predicted using KAN with probability The full-wave simulation is called, and the optimization problem is formulated as: , where, denote design variables, including excitation phase , excitation amplitude , element position , , used to balance the balance between the main lobe gain and the side lobe level , and denote the mixed evaluation function as the main lobe gain and side lobe level, respectively, denotes the impedance matrix, and denote the element surface current and incident field, respectively.

[0013] Further, the hybrid iterative optimization of the KAN agent model fused with FGO further includes determining the growth direction by subtracting two random solutions and in the current population, and calculating the new growth of the mycelium: , normalizing , the growth of the mycelium in the first state is represented as: , .

[0014] Further, the hybrid iterative optimization of the KAN agent model fused with FGO further includes using three random solutions to maintain the diversity of the optimization population, wherein a random factor is introduced to determine whether the mycelium grows in the current direction, and the growth speed of the mycelium is considered to help the mycelium accurately explore, which is represented as: , where, is a dimensional random number uniformly distributed in , and the balance probability between spore germination and lateral mycelium growth is set to 0.5.

[0015] In the second aspect, a linear array radiation pattern synthesis system based on hybrid optimization includes: A data acquisition module configured to acquire dipole array optimization parameter data; An initialization module configured to initialize array parameters and FGO population according to the acquired data; a training module configured to build a KAN agent model and train the model using a FGO population; an optimization module configured to perform hybrid iterative optimization on the KAN agent model fused with FGO; a synthesis module configured to perform array antenna pattern synthesis using the optimized KAN agent model.

[0016] In a third aspect, the present application provides a computer-readable storage medium having stored therein a plurality of instructions adapted to be loaded and executed by a processor of a terminal device to implement the hybrid optimization-based linear array radiation pattern synthesis method.

[0017] In a fourth aspect, the present application provides a terminal device comprising a processor and a computer-readable storage medium, the processor being configured to implement the instructions, and the computer-readable storage medium being configured to store a plurality of instructions adapted to be loaded and executed by the processor to implement the hybrid optimization-based linear array radiation pattern synthesis method.

[0018] To sum up, the present application has the following beneficial technical effects: The KAN-FGO hybrid optimizer built by deeply fusing the Kolmogorov-Arnold network (KAN) and the fungal growth optimization algorithm (FGO) achieves a double breakthrough in array antenna pattern design: at the level of computational efficiency, the probability hybrid evaluation strategy based on the KAN agent model is applied throughout the FGO iteration process, precise electromagnetic simulation resources are dynamically allocated (such as high proportion of agent evaluation in the early stage and gradually increasing accurate verification in the later stage), low-potential candidate solutions are effectively screened out, the number of calls to high-cost moment method is significantly reduced, the incremental learning mechanism of KAN is combined to online fuse historical solution data, the prediction accuracy of the model in the hotspot area of the current solution space is continuously improved, the double overhead of random search and numerical simulation is reduced, and the whole process convergence is greatly accelerated.

[0019] At the level of algorithm synergy, the explicit gradient analysis capability of KAN provides quantitative guidance for the direction of mycelium extension, enhances the directionality and effectiveness of local development, and the uncertainty estimation mechanism optimizes the global exploration path of spore germination, forming a closed-loop adaptive system of "agent evaluation→gradient guidance→data feedback→model update", breaking through the frequent retraining bottleneck and black box decision defects caused by the architectural rigidity of traditional static agent models, making the optimization process dynamically respond to the distribution evolution of the solution space, and improving the solution robustness of complex high-dimensional problems. BRIEF DESCRIPTION OF DRAWINGS

[0020] Figure 1 is a schematic diagram of the FGO-KAN optimization process of embodiment 1 of the present application; Figure 2is the radiation pattern obtained by phase optimization of Embodiment 1 of the present application; Figure 3 is the radiation pattern obtained by amplitude optimization of Embodiment 1 of the present application; Figure 4 is the radiation pattern obtained by element position optimization of Embodiment 1 of the present application. DETAILED DESCRIPTION

[0021] The present application is further described in detail below with reference to the accompanying drawings.

[0022] Embodiment 1 Referring to Figure 1 , the linear array radiation pattern synthesis method based on hybrid optimization of the present embodiment comprises: (1). Obtaining data, This step systematically collects the physical parameters, performance indicators and algorithm control quantities required for dipole array optimization, laying a data foundation for electromagnetic modeling and intelligent optimization. First, the array inherent parameters are obtained: including the geometric dimensions of the dipole unit (width 0.04 , height 0.46 , where represents the working wavelength), spatial arrangement (inter-element spacing 0.5 , array topology) and working frequency; secondly, the radiation performance target is defined: the mathematical expression of the target pattern function is clear; finally, the optimization algorithm parameters are configured: the FGO population size, the iteration termination criterion (maximum number of iterations T max) and the trade-off parameters at each stage are determined.

[0023] (2). Initializing array parameters and FGO population, This step first establishes an electromagnetic coupling model of the dipole array: based on the predefined array geometric parameters, an impedance matrix Z is generated using the method of moments, which quantitatively describes the electromagnetic coupling effect between elements and provides a physical basis for subsequent radiation pattern calculation. Secondly, the population initialization of Fungal Growth Optimization (FGO) is realized by simulating the natural diffusion behavior of fungal hyphae. Within the preset D dimensional solution space, a Np number of candidate solutions (referred to as "hypha individuals") are randomly generated, each individual X k [ x k1 , x k2 ,..., x kD ] represents a position coordinate in the solution space, and each dimension x kd of its components ad , bdThe internal administration is uniformly distributed. High-value initial solution set is provided for subsequent iteration, and convergence efficiency is significantly improved compared with traditional random initialization.

[0024] ① Array electromagnetic modeling, The application adopts a full-wave moment method (MoM) to establish a high-precision array antenna electromagnetic model, and focuses on solving the quantization problem of mutual coupling effect between array elements. First, the surface of each antenna element is divided into unstructured triangular meshes, and the entire array structure is discretized by M*N triangular facets (M is the number of elements, and N is the number of single-element reference division units). On this basis, the RWG (Rao-Wilton-Glisson) vector basis function is used to describe the surface current distribution: the basis function defined on the common edge of adjacent triangular facets Can be expressed as: , Wherein is the length of the common edge, is the area of the adjacent triangular facet, is the corresponding vertex coordinates. And are adjacent triangular facets sharing the first n Edge, the direction convention makes the basis function Satisfy the normal current continuity on the common edge, which fundamentally avoids the current discontinuity problem caused by the traditional segmented basis function.

[0025] Based on the electric field integral equation (EFIE), the array electromagnetic radiation model is established: , , , Wherein, Is the unit vector of the conductor surface normal vector, Is the Green function, which is used to describe the electromagnetic propagation characteristics of free space. Is the conductor surface current distribution. Is the incident electric field, Is the angular frequency, Is the magnetic permeability, Is the dielectric constant.

[0026] The EFIE is discretized into Dimensional dense impedance matrix by Galerkin weighted residual method: , The off-diagonal elements of the impedance matrix accurately characterize the electromagnetic coupling strength between array elements, and its calculation accuracy directly determines the reliability of the pattern synthesis. Compared with traditional methods (such as physical optics methods that ignore edge diffraction and equivalent circuit methods that simplify coupling paths), the present invention fully captures near-field coupling, edge scattering, and surface wave effects through the full-wave moment method, providing an electromagnetic response true value database for subsequent hybrid optimization. The final radiation pattern is determined by the excitation current vector Export: , in is the wave vector. (θ, ) represents the spatial orientation of the dipole array.

[0027] ② FGO population initialization, FGO is a new meta-heuristic algorithm proposed by Abdel-Basset et al. in 2025, which is inspired by the mycelial growth, branching and spore germination of fungi in nature. Spores produce different mycelial individuals at random locations suitable for growth. , these hyphae will explore the space to find a better nutritional environment. The number of hyphae in the entire search space is , the dimension of each hypha is , the initialization process of mycelium is shown in the formula: , in and In the optimization process, -dimensional upper and lower bound vectors. yes A uniformly distributed random value within . Operator representing the Hadamard product.

[0028] When certain conditions are met, hyphae will rapidly expand, searching for nutrient-rich environments within the search space. This phase is called the exploration phase. The growth rate and direction of hyphae are affected by various factors. For example, hyphae grow much faster in nutrient-rich environments than in nutrient-poor areas. Therefore, accounting for variations in hyphae growth rate and direction can effectively prevent the iteration process from falling into local optima. During the search process, a certain degree of fluctuation is added to prevent a constant growth rate from reducing exploration accuracy.

[0029] , in, An exponential expression of hyphal growth rate. Indicates the The fitness value of a solution, is the current function evaluation.

[0030] (3). Construct KAN surrogate model and train, To achieve fast prediction and optimization of electromagnetic radiation performance, constructing an efficient and interpretable surrogate model has become a research hotspot in recent years. Although traditional machine learning methods have advantages in accuracy and speed, they often rely on a large amount of training data and are difficult to fully express the complex nonlinear relationship between multiple variables. Kolmogorov-Arnold Network (KAN) is a modeling tool with transparent structure and solid theoretical foundation, which shows high approximation ability in limited sample scenarios. Its core idea is to use multiple layers of learnable unary functions to construct complex mapping relationships, thereby achieving efficient encoding of the input space of multiple variables. Compared with black-box deep networks, KAN is more sparse in structure, facilitating model compression and physical interpretation, and is particularly suitable for tasks in the electromagnetic field that require high generalization ability and physical consistency. Based on this, this paper introduces the KAN model to model the radiation characteristics of antenna arrays, thereby supporting subsequent research work such as pattern synthesis, parameter optimization, and dynamic control.

[0031] ① KAN architecture design, The theoretical basis of KAN is established on the Kolmogorov Superposition Theorem. This theorem states that any multivariate continuous function can be represented as the sum of several unary functions, i.e., any can be decomposed into multiple forms like , where is a unary activation function, is an output mapping function. KAN model is based on this theory, its structure introduces multiple unary learnable activation functions , which first maps the input variables respectively, then inputs them to another function through summation, and finally forms the output. Since all intermediate processes are composed of unary functions, its computational complexity is much lower than that of traditional neural networks, and it greatly enhances the interpretability and controllability of the model. The two summations in the above expression correspond to the weighted aggregation of input dimensions and the nonlinear combination of output dimensions, respectively, thereby realizing the mapping ability from low-dimensional functions to high-dimensional functions. Through this structure, KAN can flexibly model complex function relationships, and is particularly suitable for replacing traditional electromagnetic simulation models to achieve efficient approximation modeling and prediction.

[0032] , where, denotes a bounded domain of a multivariate continuous function, i.e., . and are unary functions and and , , .

[0033] By defining the L1 norm of an activation function as the average of its inputs. Then for a KAN layer with inputs and outputs, its L1 norm is defined as the sum of the L1 norms of all activation functions. The entropy of an activation function is defined as the formula: , , , ② Dynamic training and sparsification, The training target of KAN model not only minimizes the prediction error, but also needs to effectively control the complexity of the model to avoid overfitting and improve the generalization ability. Therefore, two regularization terms are introduced into the total loss function, corresponding to the L1 norm of the activation function and the entropy value of its distribution. These two indicators work together to encourage the model to tend to be sparse and information-intensive structure, thereby improving the efficiency and physical interpretability of the model. The introduction of L1 norm makes some redundant activation functions tend to zero, thereby realizing channel pruning and model compression; while the entropy regularization term punishes the distribution entropy of the activation function output, so that the activation function tends to have a concentrated response state, reflecting the importance ranking of input variables. This sparsification strategy cooperates with the dynamic training mechanism to retain more activation functions in the early training to fully learn the features, and gradually reduce the invalid or redundant activation functions in the later training, improving the simplicity and prediction stability of the model. In addition, to realize the controllability of the activation function range, the concept of connection strength is introduced, which adjusts the local response range by controlling the size of the function slope. The above mechanisms jointly construct a lightweight, efficient and self-adaptive KAN model with structure optimization capability, which is very suitable for modeling replacement and rapid optimization tasks in dynamic complex systems.

[0034] The total training target is the prediction loss of all KAN layers plus the L1 and entropy regularization of all KAN layers: , where and are relative amplitudes, usually set to 1, to control the overall regularization amplitude, denotes the number of network layers of KAN. ​

[0035] To obtain a heavy KAN, set the connection And Activation function The transparency of So that small amplitude functions can be eliminated and important functions can be focused on, where Indicates Neurons in the layer , .

[0036] After training using the sparsity penalty, the KAN will be sparsified on the node layer. For each node (such as Layer Neurons), its input and output are defined as the formula, if both input and output scores are greater than the threshold parameter All unimportant neurons are pruned.

[0037] , During the training process, some activation functions may actually be symbolic (such as cos or log), so an interface is set to set them to the specified symbolic form Can set Activation to . Because the input and output of the symbolic formula may have shifts and scaling, the activation function cannot be simply set to the symbolic formula. Therefore, the pre-activation value And the post-activation value Are obtained from the sample, and the affine parameters Are fitted so that Fitting is performed by iterative grid search and prior regression of .

[0038] (4). FGO-KAN hybrid iterative optimization, Fusion of KAN gradient guidance and FGO mycelium behavior mechanism, realize efficient search and dynamic allocation of resources.

[0039] ① Probability mixed evaluation strategy, The pattern synthesis of array antenna is essentially a high-dimensional nonlinear optimization problem, and its complexity is derived from the nonlinear coupling characteristics of electromagnetic field equation and the complex mutual coupling effect between array elements. The full-wave moment method (MoM) is used as the core electromagnetic analysis tool in the application, which obtains the electromagnetic characteristics of the array by accurately solving the electric field integral equation (EFIE). The full-wave moment method can strictly consider the mutual coupling effect between unit dipoles, accurately capture the energy redistribution phenomenon caused by near-field coupling, and significantly improve the physical fidelity of pattern prediction.

[0040] The optimization problem of pattern synthesis for linear and elliptical arrays is defined as: under the constraint of given array antenna size, by optimizing excitation phase, excitation amplitude or element position, KAN is introduced as an intelligent agent model to realize the optimization of high gain and low sidelobe performance of radiation pattern. The objective function is constructed in the form of hybrid evaluation: with probability The main lobe gain and sidelobe level are predicted using KAN, with probability Full-wave simulation is called. The optimization problem can be formulated as: , where, represents the design variables, including excitation phase , excitation amplitude , element position . is used to balance the balance between the main lobe gain and the sidelobe level . and respectively represent the hybrid evaluation function as the main lobe gain and the sidelobe level. represents the impedance matrix, and respectively represent the element surface current and incident field.

[0041] ② Mycelium growth direction guidance, The growth direction of mycelium will be affected by the sudden change of environmental factors, so the growth direction of mycelium in the FGO optimization process is randomly generated. The growth direction is determined by the difference between the two solutions in the current population and , and the growth direction is positively correlated with the difference. According to the above formula, the new growth of mycelium can be calculated.

[0042] , When the nutrient environment exists, the growth rate of mycelium will be significantly improved, and the closer to the nutrient enrichment area, the faster the growth rate. Considering that the scale of fitness value is very high, resulting in the solution after updating exceeding the search range, therefore needs to be normalized. The growth of mycelium in the first state is as follows: , , The second state is aimed at the growth of mycelium away from the nutrient enrichment area, simulating the behavior of mycelium avoiding harmful substances, and the mathematical expression of this behavior is as follows: , ​The exploration phase in the growth behavior of the mycelium tip is that the mycelium located in the nutrition-rich area tries to explore other nutrition areas to achieve a better nutrition environment. The development phase is that the mycelium located in the nutrition-poor area will try to develop a new area to the nutrition-rich area. If is less than the exploration rate , the mycelium will enter the exploration phase, otherwise it will enter the development phase.

[0043] , , wherein is a very small number to prevent the occurrence of singular values. is a predetermined value for balancing the selection of the exploration and development phases. As the iteration proceeds, the mycelium will be in the development phase and grow towards the nutrition-rich area.

[0044] ③ Spore germination and branching mechanism, The lateral branching of the mycelium is the key to the fungus exploring different directions to avoid falling into a local optimal solution in the optimization process. In FGO, this behavior is embodied by two different modes. The first mode is to randomly select two solutions and from the current population to determine the direction, if it is positive, it means adding to the current mycelium and moving in the positive direction. Otherwise, it is subtracted from the current mycelium and moves in the negative direction.

[0045] The second mode is to determine the direction according to the current optimal solution and the randomly selected solution from the current population, and then generate the growth direction of the new mycelium branched from the current mycelium. When , the first mode has a greater impact on the new mycelium, and when , the second mode has a greater impact on the new mycelium.

[0046] , In terms of the growth rate of the mycelium, the growth rate of the lateral mycelium is almost the same as that of the parent mycelium. However, the growth rate of the lateral mycelium will change due to environmental changes, which is represented by the formula. If , the growth rate of the new mycelium and the parent mycelium is equivalent, otherwise the growth rate will be significantly reduced.

[0047] , Fungi reproduce by spores, assuming that the location of new spores is conducive to their germination and can precisely help explore and exploit the search space. The growth speed and direction of new mycelium formed after spore germination depend on the absolute difference between three random optimal solutions and the current solution. Here, three random solutions are used to maintain population diversity. Since the absolute difference is used, the growth direction of the spore is always in the positive direction of the current mycelium extension. Therefore, a random factor needs to be introduced to determine whether the mycelium grows in the current direction. In addition, the growth speed of the mycelium is related to nutrients, and a formula needs to be considered to help the mycelium explore more accurately. Finally, this behavior can be represented by a formula, where the balance probability between spore germination and lateral mycelium growth is set to 0.5 in this paper.

[0048]

[0049] (5). Dynamic proxy model collaborative update, Due to the continuous evolution of the solution space during optimization, static proxy models may have problems such as decreased prediction accuracy or response deviation. Therefore, this paper introduces a dynamic proxy modeling strategy, which makes the KAN model evolve collaboratively with the FGO algorithm during optimization. Through the "prediction-feedback-update" mechanism, the model's ability to adapt and local precision is continuously enhanced. This mechanism not only improves the efficiency of sample utilization, but also enhances the stability and global search ability of the model when solving complex nonlinear problems.

[0050] ① Uncertainty guides global exploration, KAN can calculate the uncertainty of the current prediction result based on the response distribution of the activation function and the output layer variance while predicting the radiation performance. This uncertainty information not only reflects the ability of KAN in a specific design area, but also provides a key reference for guiding optimization strategies. This paper uses this information to dynamically regulate the mycelium growth direction factor. When the solution is in a high uncertainty area, the spore population is guided to strengthen exploratory growth, thereby avoiding falling into a local optimum and enhancing the algorithm's global search ability. This search guidance mechanism based on prediction confidence provides an adaptive control strategy for the optimization process, effectively improving the solution efficiency on complex objective surfaces.

[0051] ② Incremental learning and model evolution, The new solution samples generated by FGO iteration, including design variables and their corresponding electromagnetic responses, are injected into the KAN training set. Through lightweight backpropagation incremental update of network parameters, the prediction accuracy of the model in the current solution space hotspot area is continuously improved, and the prediction deviation caused by the evolution of the static model is reduced. ​​​​​​

[0052] ③ Symbolic function fitting to ensure interpretability, For the specific activation function identified in the training, automatically fit its affine transformation parameters, convert the black box activation into an explicit mathematical expression, maintain the physical interpretability of the model decision-making process, and support the reliability of gradient guidance.

[0053] (6) Convergence judgment and result output, In the optimization iteration process, the system continuously monitors the convergence trend of the objective function value, the stability of the individual fitness distribution, and the satisfaction degree of the constraint condition, to judge whether the FGO-KAN algorithm meets the termination criteria. When the preset precision or the upper limit of the number of iterations is met, the current optimal solution is output, and the final electromagnetic performance verification stage is entered.

[0054] ① Full-wave verification and performance balance analysis, To test the effectiveness of the optimal parameter configuration obtained by the FGO algorithm in the actual electromagnetic environment, this paper uses the full-wave simulation platform based on the method of moments (such as CST, FEKO) to model and verify the optimization results. A number of optimization examples are modeled and verified, covering typical examples of different target main lobe gain and side lobe level, which verify the effectiveness and wide applicability of the optimization strategy.

[0055] ② Directional diagram synthesis and parameter solidification, For multiple optimization instances, the corresponding radiation directional diagrams are displayed, Figure 2 The performance of the radiation directional diagram is clearly marked, covering different array structures, different excitation schemes, and optimization target settings. Each directional diagram reflects the advantages of the FGO-KAN method in high gain and low side lobe radiation performance. The relevant directional diagrams and parameter configurations are listed in detail in the Figures 2-4 , providing clear references for subsequent experimental testing and engineering implementation.

[0056] Figure 2 The obtained is the excitation phase optimization considering LAA, which defines the range , containing 36 dipole elements, each element is excited by a V source. As shown in Table 3, RUN achieves the highest forward gain of 17.99 dB, which is 0.05 dB higher than the result of FGO-KAN. PLO achieves the lowest SLL of -1.04 dB, which is 0.16 dB lower than FGO-KAN. As shown in Table 1, FGO-KAN completes the calculation in 15.56 hours, which is significantly better than other optimization algorithms. Overall, FGO-KAN provides competitive radiation performance while significantly reducing the calculation time.

[0057] Table 1. Calculation time of each comparison algorithm FGO-KAN FGO AO FATA PLO NRBO RUN Phase 15.56 21.69 37.37 37.62 38.58 39.96 56.49 Amplitude 18.13 25.50 41.26 42.54 43.32 53.55 64.36 Position 11.05 16.80 42.71 47.16 48.68 59.31 95.74 CEAA 16.01 13.93 69.64 24.26 90.96 36.52 70.95 Table 2. Radiation performance after phase optimization FGO-KAN FGO AO FATA PLO NRBO RUN Main gain 17.94 17.84 17.77 17.80 17.84 17.79 17.99 SLL -0.88 -0.55 -1.02 -0.04 -1.04 1.24 -0.25 Figure 3 The optimization of excitation amplitudes for a 40-element LAA is obtained, with parameter ranges set as Each dipole is independently excited by a V-voltage gap source. As shown in Table 3, AO achieves the highest main gain of 18.36 dB, 0.21 dB higher than FGO-KAN, but the SLL is limited to -12.92 dB. In contrast, both FGO-KAN and RUN achieve the lowest SLL of -21.74 dB. Overall, FGO-KAN, FGO, and RUN achieve a better trade-off between high gain and low SLL. As shown in Table 1, FGO-KAN completes optimization in 18.13 hours, significantly faster than other methods, and remains the most efficient and balanced solution.

[0058] Table 3. Radiation performance after amplitude optimization FGO-KAN FGO AO FATA PLO NRBO RUN Main gain 18.15 18.11 18.36 17.61 18.01 17.67 18.11 SLL -21.74 -21.69 -12.92 -6.28 -18.93 -2.41 -21.74 Table 4. Radiation performance after element position optimization FGO-KAN FGO AO FATA PLO NRBO RUN Main gain 17.60 17.60 17.43 17.49 17.57 17.39 17.59 SLL -2.37 -2.32 -1.23 -2.27 -2.35 -0.89 -2.24 Figure 4 The optimization of element positions for a 32-element LAA is obtained, with each element driven by a V-feed voltage. The variable is defined on the interval , where represents the initial configuration. As shown in Table 4, FGO-KAN achieves the highest forward gain of 17.60 dB and the lowest side-lobe loss of -2.37 dB. In addition, FGO-KAN completes optimization in 11.05 hours, demonstrating its excellent computational efficiency while meeting the requirements of high gain and low side-lobe loss.

[0059] Embodiment 2 The embodiment provides a linear array radiation pattern synthesis system based on hybrid optimization, comprising: The data acquisition module is configured to: A computer-readable storage medium, wherein a plurality of instructions are stored, the instructions are suitable for being loaded and executed by a processor of a terminal device, and the instructions are suitable for being loaded and executed by a processor of a terminal device.

[0060] A terminal device comprises a processor and a computer readable storage medium, the processor is used to realize instructions; the computer readable storage medium is used to store a plurality of instructions, the instructions are suitable for being loaded by the processor and executing the linear array radiation pattern synthesis method based on hybrid optimization.

[0061] The above are preferred embodiments of the present application, not limited to the protection scope of the present application, therefore: any equivalent changes made according to the structure, shape, principle of the present application should be covered within the protection scope of the present application.

Claims

1. A linear array radiation pattern synthesis method based on hybrid optimization, characterized in that: include: Obtaining dipole array optimization parameter data; Initialize array parameters and FGO population based on the acquired data; Build a KAN agent model and train it using the FGO population; Perform hybrid iterative optimization on the KAN proxy model integrated with FGO; The optimized KAN proxy model is used to synthesize the array antenna pattern.

2. A linear array radiation pattern synthesis method based on hybrid optimization according to claim 1, characterized in that: The array parameters and FGO population are initialized according to the acquired data, including establishing a high-precision array antenna electromagnetic model using the full-wave moment method to solve the quantification problem of the mutual coupling effect between array elements, wherein each antenna array element surface is divided into an unstructured triangular mesh, the entire array structure is discretized by M×N triangular facets, and the surface current distribution is described using RWG vector basis functions, and the basis functions defined on the common edges of adjacent triangular facets are Expressed as: ， in is the length of the common side, is the area of ​​adjacent triangles, is the corresponding vertex coordinate, and To share n The adjacent triangles of the edge are oriented in such a way that the basis function The normal current continuity is satisfied on the common edge, avoiding the current discontinuity problem caused by the traditional piecewise basis function.

3. The linear array radiation pattern synthesis method based on hybrid optimization according to claim 2, characterized in that: The method of initializing array parameters and FGO population according to the acquired data also includes establishing an array electromagnetic radiation model based on the electric field integral EFIE equation, which is expressed as: ， ， ， in, is the unit vector normal to the conductor surface, is the Green function, is the current distribution on the conductor surface, is the incident electric field, is the angular frequency, is the magnetic permeability, is the dielectric constant; EFIE is discretized into dimensional dense impedance matrix: ， The final radiation pattern is determined by the excitation current vector Export: ， in is the wave vector, (θ, ) represents the spatial orientation of the dipole array.

4. The linear array radiation pattern synthesis method based on hybrid optimization according to claim 3, characterized in that: The method of initializing array parameters and FGO population according to the acquired data also includes initializing the population of FGO for fungal growth optimization by simulating the natural diffusion behavior of fungal hyphae. D Random generation in dimensional solution space Np candidate solutions, each individual X k =[ x k1 , x k2 ,..., x kD ] represents a position coordinate in the solution space, and its dimensional components x kd In the domain [ ad , bd ] obeys uniform distribution. During the search process, in order to prevent the constant growth rate from reducing the exploration accuracy, the fluctuation is increased, which is expressed as: , in, represents the exponential expression of hyphal growth rate, Indicates the The fitness value of a solution, is the current function evaluation.

5. The method for synthesizing linear array radiation patterns based on hybrid optimization according to claim 4, characterized in that: The KAN agent model is constructed and trained using the FGO population, including introducing the KAN model to model the radiation characteristics of the antenna array, and introducing multiple unary learnable activation functions. , first map the input variables separately, and then input them into another function by adding them together , and finally form the output, expressed as: ， in, In the formula, we can express a bounded domain of a multivariate continuous function, that is, ; and are all unary functions and and , , ; By defining the activation function The L1 norm of The average value of the inputs, for input and KAN layer with outputs , its L1 norm is defined as the sum of the L1 norms of all activation functions, The entropy of is defined as: .

6. The linear array radiation pattern synthesis method based on hybrid optimization according to claim 5, characterized in that: The KAN proxy model is constructed and trained using the FGO population, including the introduction of two regularization terms in the total loss function, corresponding to the L1 norm of the activation function and the entropy of its distribution, to encourage the model to tend to a sparse and information-concentrated structure. In order to achieve controllability of the range of the activation function, the connection strength is introduced. The concept of , by controlling the size of the function slope to adjust the local response range, where the total training goal is is the prediction loss of all KAN layers Add L1 and entropy regularization to all KAN layers: ， in and is the relative amplitude, Used to control the overall regularization amplitude, Represents the number of network layers of KAN. After training with sparsification penalty, KAN will be sparsified on the node layer. For each node, its input and output definitions are expressed as: .

7. The linear array radiation pattern synthesis method based on hybrid optimization according to claim 6, characterized in that: The hybrid iterative optimization of the KAN agent model fused with FGO is described, including defining the optimization problem of the combined radiation pattern of linear array and elliptical array as follows: under the constraint of a given array antenna size, by optimizing the excitation phase, excitation amplitude or array element position, KAN is introduced as an intelligent agent model to achieve the optimization of the high gain and low sidelobe performance of the radiation pattern, and the objective function is based on probability. Use KAN to predict the mainlobe gain and sidelobe level with probability Calling full-wave simulation, the optimization problem is formulated as: ， in, represents the design variables, including the excitation phase , excitation amplitude , array element position , , used to balance the main lobe gain and sidelobe levels The balance between and are respectively expressed as the mixed evaluation functions of the main lobe gain and side lobe level, represents the impedance matrix, and are represented by the array element surface current and the incident field respectively.

8. The method for synthesizing linear array radiation patterns based on hybrid optimization according to claim 7, characterized in that: The hybrid iterative optimization of the KAN agent model fused with FGO also includes two random solutions in the current population. and Make a difference to determine the growth direction and calculate the new growth of hyphae : ， Will Normalization is performed, and the first state The growth of hyphae is expressed as: ， 。 9. The linear array radiation pattern synthesis method based on hybrid optimization according to claim 8, characterized in that: The hybrid iterative optimization of the KAN agent model fused with FGO also includes adopting three random solutions to maintain the diversity of the optimized population, wherein a random factor is introduced To determine whether the mycelium grows in the current direction, at the same time, considering that the growth rate of mycelium is related to nutrients to help mycelium explore accurately, it is expressed as: ， in, yes Evenly distributed within The probability of equilibrium between spore germination and lateral hyphal growth is set to 0.

5.

10. A linear array radiation pattern synthesis system based on hybrid optimization, characterized in that: include: A data acquisition module is configured to acquire dipole array optimization parameter data; The initialization module is configured to initialize array parameters and FGO population according to the acquired data; The training module is configured to build a KAN agent model and train the model using the FGO population; The optimization module is configured to perform hybrid iterative optimization on the KAN proxy model integrated with FGO; The synthesis module is configured to use the optimized KAN proxy model to perform array antenna pattern synthesis.

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