An Optimal Decision-Making Algorithm for Adjusting Aircraft Attitude Angles in Complex Electromagnetic Environments

1  Introduction

With the increasing complexity of the mission environment of modern aviation equipment, the electromagnetic environment faced by aircraft in the process of performing flight missions shows the characteristics of high intensity, long duration, and multi-source superposition. The widespread deployment of high-power radars, electromagnetic suppression systems, and a variety of fixed or semi-fixed radiation sources makes it inevitable that aircraft are exposed to a complex electromagnetic radiation environment for a long time in some mission areas. In such scenarios, it is difficult to effectively mitigate electromagnetic risks by relying solely on electromagnetic compatibility measures, such as electromagnetic shielding and flight-path avoidance. Airborne electronic systems may still be subjected to strong electromagnetic coupling effects, resulting in degraded system performance, abnormal signal processing, and even potential flight safety risks [1]. In recent years, the electromagnetic compatibility issues of aircraft onboard equipment in complex electromagnetic environments have received extensive attention and have become one of the important factors affecting the safe operation of aviation platforms [1]. In this context, digital twin technology provides a new technical approach for aircraft electromagnetic safety analysis and decision-making in complex electromagnetic environments. In recent years, electromagnetic digital twin has gradually become an important research direction for modeling complex electromagnetic environments and making real-time decisions. By constructing a virtual mapping of the electromagnetic environment, it is possible to achieve rapid prediction and online evaluation of electromagnetic responses [2]. By mapping the geometric structure, flight attitude, and electromagnetic characteristics of the real aircraft into a computable data model, the digital twin can evaluate the electromagnetic coupling characteristics of the aircraft in a specific electromagnetic environment in the virtual space in real time, which provides a quantitative basis for attitude adjustment and decision optimization. Especially in the mission scenario where electromagnetic radiation exists for a long time and is difficult to avoid through the flight path, optimizing and adjusting the attitude of the aircraft based on the digital twin model to actively reduce the coupled electric field strength at key locations inside the body has become a feasible idea to ensure the safe operation of the airborne electronic system. Fig. 1 shows the decision-making process of aircraft attitude adjustment in a complex electromagnetic environment: the ground interference source radiates electromagnetic waves outward, the aircraft is subjected to electromagnetic coupling during flight, and the on-board sensing unit obtains the coupled electric field intensity information inside the body in real time and transmits the data to the decision unit. The decision-making unit analyzes and evaluates the current electromagnetic environment based on the improved attitude angle optimization algorithm, calculates the optimal attitude adjustment scheme, and finally controls the aircraft to adjust the attitude angle, thereby reducing the influence of electromagnetic coupling interference.

Figure 1: Practical application scenarios of attitude angle adjustment decision algorithm.

Previous studies have shown that the electromagnetic coupling effect of aircraft is closely related to the frequency, incident direction, and polarization mode of external radiation sources [3], and the attitude angle of aircraft is significantly sensitive to the electromagnetic energy. At the same time, the change of attitude angle will cause changes in the distribution and scattering characteristics of the induced current on the body surface, thus affecting the propagation and coupling path of electromagnetic energy in the body structure, and eventually leading to significant differences in the intensity of the coupling electric field inside the body [4–6]. Therefore, under the framework of digital twin, the optimal suppression of electromagnetic coupling strength inside the airframe by actively adjusting the attitude angle of the aircraft has been considered as an active electromagnetic protection strategy with engineering application potential. Compared with passive protection methods relying on electromagnetic shielding, absorbing materials and filtering suppression [7,8], the active protection method based on attitude adjustment does not require additional structural modification, and can make full use of the aircraft’s own maneuverability and dynamically change the relative relationship between the aircraft and the external radiation field during the task execution, so as to improve the adaptability to the complex electromagnetic environment. However, there is a highly nonlinear relationship between the attitude angle and the coupled electric field strength inside the body, and it is affected by multiple factors such as frequency, incident direction, and body structure [9], In the digital twin-driven aviation decision-making framework, such issues are typically manifested as continuous optimization problems with complex physical constraints and multi-variable coupling characteristics [10]. It is difficult to solve the problem effectively by traditional analytical methods or empirical parameter adjustment strategies.

In recent years, Particle Swarm Optimization (PSO) and other intelligent optimization algorithms have been widely used in continuous variable optimization problems, and have achieved certain results in the field of flight control and parameter optimization [11,12]. However, when applied to attitude-angle optimization in complex electromagnetic environments, the standard PSO algorithm is prone to premature convergence and becoming trapped in local optima under multimodal objective functions and nonlinear constraints, thereby limiting its optimization stability and robustness [13,14]. At the same time, in the process of attitude angle search, full-wave electromagnetic simulation needs to be frequently invoked to evaluate the individual fitness of candidate solutions, and such simulation has high computational cost [15], which is difficult to meet the requirements of the optimization algorithm for the evaluation of high-frequency objective functions, and further restricts the feasibility of its engineering application.

Aiming at the above problems, this paper proposes an improved attitude angle optimization algorithm based on the particle Swarm optimization algorithm and genetic algorithm (AAO-PG). Based on the standard PSO framework, genetic crossover and mutation operators are incorporated to enhance population diversity and strengthen the global exploration capability. This hybrid strategy effectively alleviates premature convergence and improves the ability to escape local optima when solving multimodal optimization problems [16]. At the same time, in view of the electromagnetic performance constraints and optimization directionality requirements that need to be met in the process of attitude angle optimization, the electric field enhancement penalty term and zero improvement penalty term are constructed and integrated into the objective function. The constraint conditions are embedded into the objective function to realize the transformation of the constrained optimization problem into a tractable unconstrained form and improve the stability and convergence reliability of the algorithm [17–19]. In addition, in the study of aircraft electromagnetic coupling characteristics in a complex electromagnetic environment, if we rely on real physical tests to obtain data, it not only requires building a high-cost test environment, but also puts forward high requirements for test conditions, equipment capabilities, and safety, and it is difficult to carry out large-scale repeated experiments. Therefore, in this paper, a high-precision full-wave EM simulation method is used instead of the actual test to obtain EM response data with good physical consistency. However, in the process of attitude angle optimization, the full-wave electromagnetic simulation model needs to be invoked for each evaluation of candidate solutions, which is computationally expensive and difficult to meet the real-time evaluation requirements of the optimization algorithm in the high-frequency iteration process. To reduce the computational overhead, a machine learning-based surrogate model was developed within a digital twin framework to construct an electromagnetic coupling prediction model, thereby replacing computationally expensive real-time electromagnetic simulations during the optimization process. The Random Forest algorithm was adopted as the surrogate modeling method because of its strong nonlinear regression capability and robustness [20]. The model is trained based on FEKO simulation data, and the nonlinear mapping relationship between attitude parameters and the coupled electric field strength of monitoring points inside the body is established [21]. In the optimization stage, the prediction model is used to replace the real-time electromagnetic simulation to quickly evaluate the objective function, so as to significantly reduce the cost of single-function calculation and improve the overall optimization efficiency. Finally, extensive experiments including multi-scene simulations, Monte Carlo stability analysis, incident-field-strength validation, parameter sensitivity analysis, and search-range adaptability analysis are conducted to verify the effectiveness of the proposed method.

Therefore, the primary problem addressed in this paper is not digital twin modeling or electromagnetic response prediction themselves, but the optimal attitude-angle decision problem for aircraft operating in complex electromagnetic environments. Specifically, this work aims to determine an attitude-angle combination that minimizes the internal electromagnetic coupling interference while satisfying practical maneuvering constraints. The digital twin model, electromagnetic response prediction model, and stability analysis are introduced as supporting techniques for the formulation, acceleration, and validation of the attitude optimization process.

The main contributions of this paper can be summarized as follows.

Methodological contributions:

(1)   An improved attitude-angle optimization algorithm (AAO-PG) is proposed by integrating particle swarm optimization with genetic crossover and mutation mechanisms, together with an electric-field-based penalty strategy, to improve optimization performance under complex electromagnetic environments.

(2)   A scene-level electromagnetic response prediction model is developed to accelerate objective-function evaluation during attitude-angle optimization.

Engineering contributions:

(3)   A digital-twin-driven aircraft attitude decision framework is established by integrating electromagnetic simulation, response prediction, and intelligent optimization. Extensive simulation studies demonstrate its effectiveness, robustness, and engineering applicability in complex electromagnetic environments.

2  Electromagnetic Modeling and Attitude Angle Optimization Modeling

This paper focuses on the problem: under the given external electromagnetic environment conditions, by adjusting the attitude angle (pitch angle θ, roll angle γ, yaw angle ψ) of the aircraft, the intensity of the coupled electric field inside the fuselage can be reduced as much as possible. Therefore, it is necessary to establish a quantitative mapping model between the attitude angle and the electromagnetic coupling interference inside the body, and to construct a mathematical evaluation function that can be used for an intelligent optimization algorithm.

2.1 Aircraft Electromagnetic Simulation Model

In order to describe the influence of aircraft attitude change on the electromagnetic response inside the airframe, this paper first establishes the three-dimensional geometric and electromagnetic simulation model of the aircraft based on FEKO. Considering the computational efficiency and stability requirements of batch attitude angle scanning and large-scale data set construction, a relatively simple and representative helicopter is selected as the research object to ensure the physical rationality of the model and reduce the simulation complexity. The body shape is modeled according to the proportion of the real aircraft; the length of the body is about 11.7 m, the height is about 4 m, and the width is about 1.8 m. The outer skin material is uniformly set as the ideal conductor (PEC), so as to highlight the dominant influence of the flight attitude change of the aircraft equipment on the electromagnetic coupling strength.

The purpose of using a simplified PEC helicopter model in this study is to investigate the influence of attitude angle variations on electromagnetic coupling characteristics, rather than to reproduce the exact electromagnetic behavior of a specific aircraft platform. This model retains the key geometric scattering features of typical aircraft while maintaining the computational efficiency required for large-scale data generation and optimization analysis, thereby effectively supporting the research on attitude-dependent electromagnetic coupling mechanisms discussed here. It should be noted that real aircraft typically involve complex factors such as composite materials, apertures, antennas, and onboard electronic equipment, leading to electromagnetic response characteristics that differ from those of ideal conductor models. Consequently, the absolute values of the coupled electric fields obtained using the PEC model may deviate from those of actual aircraft. Nevertheless, the aircraft’s geometric configuration and its relative orientation with respect to incident electromagnetic waves remain the primary factors affecting attitude-related electromagnetic coupling effects. Therefore, this study focuses more on revealing the relationship between attitude angle changes and electromagnetic coupling responses, as well as their optimization mechanisms. Although the PEC simplification may affect the quantitative accuracy of optimization results, it does not alter the overall optimization trends or the conclusions regarding the effectiveness of the methods presented in this paper.

2.1.1 FEKO Simulation Condition Setting

In the electromagnetic environment modeling, the plane wave is used as the external excitation source, the incident field strength is set as 1 V/m, and the polarization mode is fixed as 0∘ polarization (parallel polarization) to ensure the comparability of electromagnetic response under different attitude conditions. Since the outer skin of the body is set as a PEC, which is automatically considered as an ideal conductor boundary by FEKO during the solution process, the outer surface of the model constitutes a closed PEC boundary without additional absorption or open domain boundaries.

In the subsequent attitude scanning process, the incident direction of the plane wave remains unchanged, and the relative incidence relationship is changed by the change of pitch, roll, and yaw angles, so as to simulate the different attitude conditions of the aircraft in the complex electromagnetic environment.

In order to meet the needs of batch simulation and automatic calculation, the airframe model is meshed by using the global Coarse (MeshSizeOption = Coarse) provided by FEKO, which can significantly reduce the computational cost of a single simulation on the premise of ensuring the basic accuracy of the airframe shape and electromagnetic scattering characteristics.

2.1.2 Setting of Sampling Points

Considering that the airborne electronic equipment is mainly concentrated in the nose and belly area of the aircraft, two monitoring points A and B (denoted as E1 and E2) are set along the longitudinal centerline inside the aircraft body, as shown in Fig. 2. The two points have the same height and are separated by about 2.5 m, which are used to describe the variation characteristics of the coupled electric field distribution inside the aircraft body under different attitude conditions.

Figure 2: Setting of model monitoring points.

2.2 Parameterized Description of Attitude Angle

The change of aircraft attitude angle will change the spatial relationship between the airframe and external radiation sources, and then affect the surface-induced current distribution and internal electromagnetic coupling path. In this paper, the standard attitude definition method of the helicopter model in FEKO is adopted, as shown in Fig. 3, and the body attitude change is described by three Euler angles:

(1)   Pitch (θ): the body rotates around the forward axis (X-U)

(2)   Roll (γ): the body rotates around the lateral axis (Y-V)

(3)   Yaw (ψ): the body rotates around the vertical axis (Z-N)

Figure 3: Aircraft attitude-angle coordinate system in FEKO.

The change of attitude angle will affect the angle of attack of the nose, the lateral irradiation area of the fuselage, and the local scattering characteristics of the outer skin at the same time, which will cause the nonlinear change of the coupled electric field distribution in multiple sections. Combined with typical aircraft maneuvering conditions and flight safety constraints, the attitude-angle search range is set to θ∈[−10∘,10∘], γ∈[−10∘,10∘], and ψ∈[−10∘,10∘]. This range covers the moderate attitude variations commonly encountered during practical flight missions while avoiding extreme attitudes that are less representative of normal operational scenarios. At the same time, it provides sufficient attitude diversity for optimization analysis and dataset construction.

2.3 Attitude Angle Evaluation Function

The feasible search domains for the three attitude angles are determined jointly by flight safety constraints and operational feasibility. They are defined as follows: Pitch angle θ∈J=[−10∘,10∘], Roll angle γ∈G=[−10∘,10∘], Yaw angle ψ∈Y=[−10∘,10∘]. The feasible search space is the Cartesian product Ω=J∙G∙Y of these three angles. This space is a three-dimensional continuous, bounded, closed region that covers the main attitude change intervals that may occur in typical helicopter flight tasks. On this feasible domain, the attitude angle optimization problem studied in this paper can be strictly formulated as a nonlinear programming problem with boundary constraints. This problem has non-convex and multi-modal characteristics—the objective function and attitude variables form a highly nonlinear mapping through electromagnetic coupling physical processes, and there are multiple local extrema, which cannot be effectively solved using analytical gradient methods. Therefore, this paper adopts an improved intelligent optimization algorithm for global search.

This paper models the aforementioned optimization problem as a multi-objective optimization problem that incorporates physical constraints, and multiple electromagnetic performance indicators are mapped into a single objective function by the weighted quantization method, which can be solved by the subsequent intelligent optimization algorithm. In order to quantitatively evaluate the comprehensive influence of different attitude adjustments on the electromagnetic coupling effect inside the body, and consider the maneuver cost caused by attitude adjustment, this paper constructed the following benchmark attitude evaluation function:

f(θ,γ,ψ)=−∑i=12widEi,eff+λssafe_theta(1)

where dEi,eff represents the effective coupled electric field intensity change at the ith monitoring point. w1 and w2 are the weights of two monitoring points, which can be set according to the importance of cabin segments, and (w1 + w2) is 1. The weighting coefficients are set to w1 = w2 = 0.5, reflecting that the two monitoring points are considered equally important in evaluating the electromagnetic coupling characteristics inside the aircraft. The λs is the angle penalty weight, which is set to 0.5 to strike a balance between reducing electromagnetic coupling and operational costs, preventing excessive attitude adjustments, and maintaining the optimization capability.

In the above evaluation function, the change in the coupled electric field strength of the monitoring point before and after the attitude adjustment is first defined. Let the coupled electric field intensities of the two monitoring points under the reference attitude (θ, γ, ψ) = (0∘,0∘,0∘) be E10 and E20, respectively, and the coupled electric field intensities after attitude angle adjustment are E1 and E2.

The attenuation of coupled electric field strength is defined as follows:

dE1,eff=E10−E1,dE2,eff=E20−E2(2)

In order to describe the maneuvering cost caused by attitude adjustment, the attitude angle adjustment is introduced as follows:

Δθ=|θ|+|γ|+|ψ|(3)

Considering that the attitude angle adjustment may be close to zero, a smooth lower bound function is further defined to avoid division by zero or numerical instability in subsequent calculations:

safe_theta=max(Δθ,α)(4)

where α represents the minimum controllable attitude adjustment of the aircraft, which is set to 0.5∘. This indicates the minimum meaningful attitude adjustment considered during the optimization process. This setting can prevent numerical instability when the total attitude change approaches zero, while maintaining sensitivity to small attitude changes, allowing the optimization algorithm to search stably in the vicinity of small angles and avoiding the occurrence of safe_theta being 0.

Through the above definition, the evaluation function comprehensively considers the two factors of “coupling electric field strength weakening gain” and “attitude adjustment cost”. Because the negative sign is introduced in front of the evaluation function, the optimization problem is transformed into the minimization process of f, so as to achieve “as small as possible attitude adjustment to obtain as large as possible coupling electric field strength weakening effect”, which provides a clear optimization goal for the search direction and convergence process of the subsequent improved attitude angle optimization algorithm.

3  Improved Attitude Angle Optimization (AAO-PG) Algorithm

3.1 Algorithm Motivation and Comparative Analysis

The Particle Swarm Optimization (PSO) algorithm is a stochastic optimization method based on swarm intelligence. The basic idea of PSO comes from the foraging behavior of birds. The algorithm finds the optimal solution by simulating the cooperative movement of particles in the search space. Each particle represents a candidate solution of the problem and has two attributes: position and velocity. During the search process, the particle constantly adjusts its movement direction according to its own experience and the group’s experience. In the PSO algorithm, firstly, the position and velocity of the particle swarm are randomly initialized, and the fitness value of each particle is calculated. Then, the particles move in the search space and are guided themselves to the optimal direction by individual experience (Pbest) and group experience (Gbest). As the iteration goes on, the particle swarm gradually converges to the optimal solution region. When the maximum number of iterations is reached, or the convergence condition is satisfied, the algorithm terminates and outputs the current optimal solution. The pose evaluation model constituted by Eq. (1) is a typical optimization problem, which can be solved by the PSO algorithm.

In the complex electromagnetic environment, due to the highly nonlinear relationship between the electromagnetic response and the attitude angle, the objective function often has multiple local optima. The basic particle swarm optimization algorithm is prone to particle aggregation and the decline of population diversity in the later iterations, which leads to the premature convergence of the search process to the local optimal solution. At the same time, the basic evaluation function is difficult to effectively distinguish between “effective attitude adjustment” and “ineffective maneuver” in the case of no significant improvement of the electric field or local numerical fluctuations, which may lead to the stagnation of the optimization process in the pseudo-optimal region with limited electromagnetic benefits. In order to solve the above problems, this paper introduces an improvement strategy based on the basic PSO algorithm framework and enhances the evaluation function. On the one hand, by periodically introducing the selection, crossover, and mutation operations of the genetic algorithm in the search process of particle swarm optimization, the diversity of the population is restored, the ability of the algorithm to jump out of local optimum is enhanced, and the global search performance is improved while maintaining the fast convergence characteristics of particle swarm. On the other hand, based on the original evaluation model of electric field weakening and attitude cost, a penalty mechanism based on the variation trend of electric field is introduced, which is helpful to suppress the interference of electric field strengthening and zero improved attitude on the optimization process, so as to effectively avoid the premature convergence of the particle swarm near the invalid solution and improve the search stability and reliability of the algorithm in complex electromagnetic environment.

Various metaheuristic optimization algorithms have been widely applied to engineering optimization problems, including Genetic Algorithm (GA), Differential Evolution (DE), Grey Wolf Optimizer (GWO), Whale Optimization Algorithm (WOA), and Particle Swarm Optimization (PSO). GA possesses strong global exploration capability through crossover and mutation operations, but its convergence speed is often relatively slow. DE generally performs well in continuous optimization problems but may require a large number of objective-function evaluations. More recent swarm-intelligence algorithms such as GWO and WOA have demonstrated competitive optimization performance in some benchmark problems; however, their convergence behavior and parameter sensitivity may vary significantly across different application scenarios. For the attitude-angle optimization problem investigated in this paper, computational efficiency is an important consideration because a large number of candidate solutions must be evaluated during the iterative optimization process. Compared with many other metaheuristic algorithms, PSO has the advantages of a simple mathematical structure, a small number of control parameters, fast convergence, and relatively low computational complexity. These characteristics make PSO particularly suitable as the basic optimization framework for the proposed digital-twin-based attitude decision system. However, the objective function considered in this study is highly nonlinear and multimodal. Under such conditions, the population diversity of the basic PSO algorithm tends to decrease during the later stages of the search process, making the algorithm susceptible to premature convergence and local-optimum stagnation. To address this limitation, the crossover and mutation mechanisms of GA are introduced into the PSO framework. The crossover operation promotes information exchange among high-quality candidate solutions, while the mutation operation increases population diversity and enhances the capability of escaping local optima. Consequently, the proposed AAO-PG algorithm combines the fast convergence characteristics of PSO with the strong exploration capability of GA, achieving a better balance between exploitation and exploration. This mechanism is further validated through the average convergence analysis presented in Section 5.2. From an engineering perspective, another advantage of the proposed AAO-PG algorithm is its compatibility with surrogate-model-based optimization frameworks. Since the optimization process relies on repeated evaluations of the random-forest prediction model rather than computationally expensive full-wave electromagnetic simulations, the relatively low computational overhead of PSO-GA further contributes to the practical applicability of the proposed attitude decision framework. Nevertheless, the proposed method still has some limitations. Similar to most population-based metaheuristic algorithms, AAO-PG cannot theoretically guarantee convergence to the global optimum. In addition, its optimization performance may be influenced by the selection of algorithm parameters such as crossover probability, mutation probability, and penalty coefficients.

3.2 Evaluation Function and Algorithm Design

The overall application process of the improved attitude angle optimization algorithm in the attitude angle optimization problem in a complex electromagnetic environment is shown in Fig. 4, and the specific steps are as follows:

(1)   Firstly, the reference attitude (θ, γ, ψ) = (0∘,0∘,0∘) is selected, and the reference coupling electric field strength of the monitoring point inside the body is obtained by electromagnetic simulation: E10, E20.

(2)   Initialize the particle swarm position xi = (xi1, xi2, …, xid) and the speed vi = (vi1, vi2, …, vid), where i = (1, 2, …, n), where n is the population size and d is the dimension of the problem to be solved. Where the position of each particle represents a candidate solution in the solution space, and the velocity represents the search direction and step size of the particle in the solution space.

(3)   Fitness evaluation of generation 0. The following two penalty terms are introduced based on Eq. (1). Electric field enhancement penalty term:

Pinc=β(|dE1,eff|+|dE2,eff|+λssafe_theta),if (dE1,eff<0) or (dE2,eff<0)(5)

Figure 4: Flowchart of the improved attitude angle optimization algorithm.

In Eq. (5), β > 0 is the penalty weight coefficient, is set to 100, and when the electric field shows an enhancement trend, the penalty term is activated.

Electric field zero improvement penalty term:

Pzero=λexp⁡(−Δθ),if dE1,eff<ε and dE2,eff<ε(6)

In Eq. (6), λ is the penalty coefficient, set to 1000,and ε is the small threshold, set to 10−3 V/m. When the electric field variation is below this threshold, it can be regarded as insignificant within the overall variation range and is not considered an effective improvement. When there is almost no improvement in the electric field, the penalty term is activated.

According to the benchmark evaluation function Eq. (1) and penalty term Eqs. (5) and (6), the final fitness evaluation function is as follows.

ffinal(ϑ,γ,ψ)=f(ϑ,γ,ψ)+Pinc+Pzero(7)

The two penalty terms (Eqs. (5) and (6)) essentially incorporate the electromagnetic protection performance constraints—that during the optimization process, there should be no increase in electric field and no ineffective maneuvers that do not bring any actual improvement—by embedding them into the objective function using the penalty function method. These two constraints stem from the physical requirements of electromagnetic protection rather than the geometric limitations of the search domain. They are difficult to directly incorporate into the position-velocity update mechanism of PSO, so they are transformed into penalty components of the objective function, converting the original constrained optimization problem into an unconstrained form of ffinal = f+Pinc+Pzero(Eq. (7)). The parameter settings adopted in the proposed optimization framework are as follows: w1 = w2 = 0.5, λs = 0.5, α = 0.5∘, ε = 10−3 V/m, β = 100, λ = 1000. These parameters are empirically selected through preliminary numerical experiments and can provide stable and reliable optimization performance in different electromagnetic scenarios.

(4)   Enter the main iteration loop part. Firstly, adaptively adjust the parameters of the improved attitude angle optimization algorithm according to the iteration progress:

w(k)=wstart+(wend−wstart)kK(8)

c1(k)=c1,start+(c1,end−c1,start)kK(9)

c2(k)=c2,start+(c2,end−c2,start)kK(10)

where k = (1, 2, …, K) represents the iteration stage of the current particle swarm search, K is the maximum iteration number, kK represents the running progress proportion of the algorithm, which is used to realize the linear dynamic adjustment of the inertia weight. wstart is the initial inertia factor, wend is the termination inertia factor; c1,start is the initial individual learning factor, c1,end is the termination individual learning factor. c2,start is the initial population learning factor, and c2,end is the termination population learning factor c1.

The inertia weight decreases linearly from wstart=0.9 to wstart=0.4, enabling the algorithm to conduct global exploration with a larger step size in the early stage of the search and focus on local development with a smaller step size in the later stage. The individual learning factor decreases linearly from c1,start=2.5 to c1,end=0.6, while the group learning factor increases linearly from c2,start=0.6 to c2,end=2.5. The alternating changes of c1 and c2 achieve a smooth transition of the search behavior from “individual experience-driven” to “group consensus-driven”: In the early iterations, a larger c1 prompts particles to fully explore their own neighborhood and maintain population diversity, while a larger c2 guides the particle swarm to converge towards the global optimal neighborhood and conduct fine search in the later stage.

(5)   Then update the position and velocity of the particles. According to the individual optimal solution, the global optimal solution, and the current particle position and velocity information in the population, update the velocity and position of the k + 1 generation particles according to the following formula.

The speed is:

vij(k+1)=w(k)vij(k)+c1(k)r1(Pbesti−xij(k))+c2(k)r2(Gbest−xij(k))(11)

The location is:

xij(k+1)=xij(k)+vij(k+1)(12)

In Eqs. (11) and (12), i = (1, 2, …, n), j = (1, 2, …, d). r1 and r2 are random numbers between [0, 1]. Pbesti is the individual optimal solution of the current ith particle. Gbest is the global optimal solution of the current population. vij(k) and xij(k) denote the velocity and position of the jth dimensional variable of the ith particle at the kth generation, respectively.

(6)   Determine whether the genetic condition is satisfied (generational inheritance); if triggered, the genetic operation will be carried out. Firstly, the first few particles with better fitness are selected as the parent set. Then, two particles xp1 and xp2 are randomly selected from the parent set, and the children are generated by linear crossover:

xchild=αxp1+(1−α)xp2(13)

where α is a random number between [0, 1].

To further enhance the search randomness and prevent the offspring particles from being completely limited to the parent distribution, a small probability mutation operation is applied to the offspring particles generated by crossover:

xchildj+1=xchildj+δj(14)

where δj is the variation range, xchildj denotes the offspring particle generated by the jth crossover.

(7)   The fitness of the new generation of particles is evaluated, and the individual optimal value and the global optimal value are updated.

(8)   Determine the termination condition. If the termination condition is satisfied (the maximum number of iterations is reached or the fitness value converges), the algorithm terminates. Otherwise, return to process 4) Continue the iteration.

(9)   After the termination of the algorithm, the global optimal attitude angle combination (ϑ∗, γ∗, ψ∗), the corresponding internal coupling electric field strength, and the objective function value are output as the optimal attitude angle decision result in a complex electromagnetic environment.

For clarity and reproducibility, the complete implementation procedure of the proposed AAO-PG optimization framework is formally summarized in Algorithm 1.

3.3 Analysis of Algorithm Complexity

The computational cost of the AAO-PG algorithm consists of two parts: particle swarm iteration and periodic genetic operation. The fitness evaluation relies on a random forest prediction model. Let the population size be n and the maximum number of iterations be K. The random forest contains several decision trees. The main part of the PSO: Each particle in each generation executes velocity update (Eq. (11)) and position update (Eq. (12)) successively, each requiring constant time. Then, the random forest is called for fitness evaluation. A single random forest prediction requires traversing all decision trees. The average depth from the root node to the leaf node of each tree is proportional to log S (S is the number of training samples), so the single fitness evaluation cost is O(log S). The total cost of the PSO main part for all K generations and n particles is O(K⋅n⋅log⁡S). The genetic operation part: The genetic operation is not executed every generation but is triggered periodically (set to trigger once every G generations). When triggered, selection (sorted by fitness, cost O(n⋅log⁡n)), linear crossover (Eq. (13), assuming the number of particles participating in crossover is nc, cost O(nc)) and mutation (Eq. (14), applying perturbation with a small probability in each dimension, cost O(nc)) are executed successively. The total cost of a single trigger is O(n⋅log⁡n+nc). In all K generations, it is triggered K/G times, and the additional cost of the genetic operation is O((K/G)⋅(n⋅log⁡n+nc)). Combining the two parts, the total time complexity of AAO-PG is O(K⋅n⋅log⁡S+(K/G)⋅(n⋅log⁡n+nc)). Compared with the standard PSO (only containing the first term), AAO-PG only adds a low-frequency genetic operation cost proportional to 1/G. Both have the same time complexity. Additionally, it should be noted that if the random forest proxy model is not used and FEKO full-wave simulation is directly called, the computational cost of each fitness evaluation will increase sharply from O(log S) to the time scale of a single full-wave simulation (usually tens of seconds to several minutes), which is the fundamental reason for introducing the random forest proxy model in Section 4 of the paper to compress the fitness evaluation cost.

4  Random Forest Prediction Model

4.1 Modeling Background and Problem Description

In the study of aircraft attitude angle optimization in a complex electromagnetic environment, the attitude-electromagnetic coupling interference relationship model should be established based on real physical test data. However, the electromagnetic experiment of the whole aircraft has the characteristics of high cost, long cycle, complex scene construction, and high safety risk, and it is difficult to carry out large-scale repeated experiments under the combination conditions of multiple frequency bands, multiple incidence angles, and multiple attitudes. Therefore, in this paper, a high-precision full-wave electromagnetic simulation method is used instead of real physical experiments to obtain physically consistent electromagnetic coupling interference data. Based on this, this paper relies on the FEKO platform to construct the aircraft electromagnetic simulation model, generate the internal electric field distribution data under different attitude angles, and provide basic samples for subsequent modeling and optimization.

In the previous Section 3, the improved attitude angle optimization framework is constructed, and the physical constraints are integrated into the unified objective function through the penalty function form to realize the global search and constrained optimization of the aircraft attitude angle in the complex electromagnetic environment. However, in the actual optimization process, each particle position update needs to call the electromagnetic simulation model to evaluate the candidate attitude, calculate the coupled electric field strength of the corresponding internal monitoring point of the body, and obtain the objective function value. Since the attitude angle optimization belongs to the continuous high-dimensional search problem, the improved attitude angle optimization algorithm usually needs a large number of function evaluations in the iterative process. When the objective function depends on the FEKO electromagnetic simulation results based on the full-wave method, the single simulation calculation is time-consuming, which makes it difficult to meet the real-time evaluation requirements of the optimization algorithm in high-frequency iterations, and seriously restricts the optimization efficiency and engineering application feasibility. Therefore, reducing the computational overhead of a single attitude assessment while maintaining the physical constraints and the prediction accuracy to meet the engineering requirements has become a key issue for fast attitude angle decision.

In order to solve the above bottleneck problem, this paper introduces the machine learning method to construct a fast prediction model between the attitude angle parameters and the coupled electric field strength of the monitoring point inside the body. In the offline stage, the model is trained based on a large number of FEKO simulation data to learn the nonlinear mapping relationship between the attitude angle and the coupled electric field strength of the monitoring point inside the body. In the optimization phase, the real-time electromagnetic simulation is replaced with a quick calculation of the objective function value. Through this “simulation-prediction” method, the overall computational efficiency of the optimization algorithm can be greatly improved under the premise of ensuring that the prediction error is controllable, and the efficient solution of the attitude angle optimization problem in a complex electromagnetic environment can be realized.

4.2 Dataset Construction and Feature Design

The dataset used in this paper was generated based on the FEKO electromagnetic simulation platform. The Lua interface of FEKO is driven by a Python script to perform batch automatic simulation calculation of the helicopter model under different combinations of electromagnetic environmental parameters and attitude angles, and the coupled electric field intensity values of the monitoring points inside the body under the corresponding attitude conditions are recorded, so as to construct the sample set of attitude angle-electromagnetic response. In terms of setting electromagnetic environment parameters, this paper focuses on two factors: operating frequency and plane wave incident direction.

(1)   In order to cover the typical electromagnetic environment characteristics under complex electromagnetic environment conditions and take into account the differences in the electromagnetic response mechanism of the airframe under different frequency bands, this paper sets the research frequency range as 200 MHz–2 GHz. This frequency range covers representative aviation-related electromagnetic radiation sources and corresponds to wavelengths that are comparable to the characteristic dimensions of the aircraft. Under such conditions, significant electromagnetic scattering and coupling effects are more likely to occur, making the selected frequency range representative for attitude-dependent electromagnetic response analysis. In order to facilitate the analysis of the electromagnetic response characteristics under different frequency conditions, the frequency band is further divided into the middle frequency band (200 MHz–900 MHz) and the high frequency band (900 MHz–2 GHz). The frequency segmentation method is helpful to analyze the influence of attitude angle changes on the internal electromagnetic response law of the body under different frequency bands, and provides a basis for subsequent model prediction performance and stability analysis.

(2)   At each frequency point, the influence of different plane wave incident directions on the electromagnetic response of the body is further considered. In order to ensure the integrity and physical symmetry of the direction coverage, six orthogonal incident directions (front, rear, left, right, top, and bottom) are selected to provide complete spatial coverage and directional symmetry of the electromagnetic exposure conditions. This configuration captures the major radiation-incidence scenarios that may occur in practical environments while maintaining reasonable computational efficiency.

In the random forest prediction model, the input features and output labels are defined as follows:

Input feature vector: x = [θ, γ, ψ]

Output the label vector: y = [E1, E2]

Here, θ, γ, and ψ represent the pitch angle, roll angle, and yaw angle of the body, respectively, and E1 and E2 represent the coupled electric field strength of the two monitoring points inside the body.

In the data exploration stage, the frequency, plane-wave incident direction, and attitude-angle variables were used together as the input features to construct a unified electromagnetic response prediction model across all operating conditions. However, the experimental results showed that the overall prediction accuracy of the unified model was significantly reduced, particularly in attitude-sensitive regions where small attitude variations led to noticeable changes in the coupled electric field. The primary reason is that the electromagnetic response is governed by the complex nonlinear interactions among frequency, incident direction, and attitude angle. Different combinations of frequency and incident direction exhibit distinct electromagnetic response characteristics, resulting in heterogeneous input-output relationships across operating conditions. Consequently, a single random forest model is unable to accurately learn the response patterns of all scenarios simultaneously, especially the local response characteristics associated with small attitude variations. As a result, the fitting performance in attitude-sensitive regions deteriorates, leading to a reduction in the overall prediction accuracy.

Based on the above analysis, this paper adopts the Scenario-based Modeling strategy: Under the fixed frequency interval (200 MHz–2 GHz) and fixed plane wave incidence directions (forward, back, left, right, up, and down), the random forest prediction models were constructed respectively with attitude angles [θ, γ, ψ] as input features. This method decomposes the original high-dimensional and strongly nonlinear global problem into multiple low-dimensional and locally smooth sub-problems, which significantly improves the prediction accuracy and stability of the model for attitude angle changes. After comprehensively considering 19 frequency bands, 6 orthogonal incidence directions, and 1331 sets of attitude angle combinations, a total of 151,734 sets of effective sample data are obtained. Each set of samples corresponds to a defined frequency point, a defined plane wave incidence direction, and a set of attitude angle combinations, and contains the coupled electric field strength of two monitoring points inside the body. In the specific modeling process, the sample data is further divided according to the electromagnetic environment parameters to construct different scene-level data sets.

Under the combination condition of 19 frequency bands and 6 orthogonal incidence directions set in this paper, a determined frequency interval and a determined plane wave incidence direction are jointly defined as an independent electromagnetic environment scene. Based on this definition, a total of 114 scene-level datasets are constructed in this paper. For each scene-level dataset, a random forest prediction model is trained to learn the mapping relationship between the attitude angle [θ, γ, ψ] and the electric field response of the monitoring point inside the body. By constructing the prediction system, the model can maintain high sensitivity and prediction stability to changes in attitude angle under different electromagnetic environment conditions, which provides reliable support for the subsequent optimization and stability analysis of attitude angle. In each scene-level dataset, the samples are divided into a training set and a test set according to a ratio of 90% to 10% for model training and generalization ability verification.

4.3 Principle of the Random Forest Model

Random Forest is a non-parametric supervised learning algorithm based on the idea of ensemble learning. Its core idea is to build multiple independent decision trees and integrate the prediction results of each tree, so as to improve the generalization ability and stability of the model.In regression problems, random forests achieve high-accuracy predictions mainly through the following mechanisms:

(1)   Bootstrap sampling mechanism: multiple different training subsets are constructed by random sampling with replacement from the original training set, which are used to train multiple decision trees.

(2)   Random feature selection mechanism: when each node splits, the optimal split feature is only selected from the feature subset to reduce the risk of a single feature dominating the model.

(3)   Multi-model ensemble output: Each decision tree makes predictions independently, and the final output is the average of the prediction results of all trees.

The random forest model can effectively depict the complex nonlinear relationship between input variables and output response, and is not sensitive to the characteristic scale and data distribution. It is suitable for the highly nonlinear mapping modeling task between attitude angle and electromagnetic response inside the body. The flow chart is shown in Fig. 5.

Figure 5: Random forest prediction model training flowchart.

To establish a proxy model between the aircraft attitude angles and the internal electromagnetic responses, this study generated a total of 151,734 valid samples from 19 frequency bands, 6 orthogonal incident directions, and 1331 combinations of attitude angles. The dataset was randomly divided into a training set and a test set, with proportions of 90% and 10%, respectively.

To improve the prediction accuracy and the robustness of the model, a grid search strategy combined with five-fold cross-validation was used to optimize the hyperparameters of the random forest, with the determination coefficient R2 as the evaluation metric. The final model consisted of 1000 decision trees, with a maximum tree depth of 14. The minimum number of samples required for node splitting was set to 4, and the minimum number of samples for each leaf node was set to 2. Additionally, all input features were considered in the splitting selection process. The selected configuration achieved a balance between prediction accuracy, model generalization ability, and computational efficiency. Increasing the number of trees can improve the stability of the prediction and reduce variance, while limiting the depth of the trees helps prevent overfitting. The constraints applied to node splitting and leaf node samples further enhanced the model’s robustness in handling the highly nonlinear relationship between aircraft attitude angles and electromagnetic responses.

4.4 Performance Evaluation

In order to systematically evaluate the prediction accuracy and stability of the constructed random forest prediction model under different electromagnetic environment conditions, this paper comprehensively evaluates the performance of the model from two levels of quantitative error index analysis and visual comparison verification.

4.4.1 Quantitative Performance Index Analysis

In this paper, Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and coefficient of determination (R2) are selected as the performance evaluation indicators of the prediction model. MAE is used to measure the average level of prediction error of the model; RMSE is more sensitive to large prediction errors and can reflect the stability of the model in extreme cases; and R2 is used to describe the fitting ability of the model to the change trend of simulation data.

For the middle frequency (200 MHz–900 MHz) and high frequency (900 MHz–2 GHz) bands, the prediction results of the coupled electric field strength of monitoring points A and B in multiple typical scenarios are statistically analyzed, and the average performance indicators are shown in Table 1.

As can be seen from Table 1, the random forest model shows better prediction performance at different monitoring points under the conditions of medium frequency and high frequency. In general, the R2 values of the model in each scene remain at a high level, indicating that the model can effectively learn the nonlinear mapping relationship between the attitude angle change and the internal electric field response of the body. Further comparison shows that under the same frequency band conditions, the prediction performance of monitoring point A is better than that of monitoring point B, which may be related to the differences in structural positions and electromagnetic coupling mechanisms between the two monitoring points. It is speculated that because the monitoring point A is located in the nose region, the structure continuity is good, and the electromagnetic response shows a smoother change characteristics with the attitude change, while the detection point B is located in the belly region, and due to the discontinuity of the structure and multi-path coupling, its electric field response is more sensitive to the attitude change, which increases the difficulty of model prediction to a certain extent. Nevertheless, the MAE and RMSE of the detection point B remain at a low level under the two types of frequency bands, indicating that the model still has an acceptable prediction accuracy under the sensitive monitoring position. In addition, by comparing the results of intermediate frequency and high frequency, it can be found that the overall prediction error of the model is slightly smaller under the condition of high frequency. This phenomenon indicates that in the high-frequency band, the variation trend of the electromagnetic coupling strength inside the body is smoother due to the change of attitude angle, and the random forest model is easier to capture its local mapping relationship, so as to obtain higher prediction accuracy.

4.4.2 Comparison and Analysis of Predicted Value and Simulation Value

In order to further intuitively verify the fitting ability of the prediction model to the FEKO simulation results, this paper selects representative scenarios with the most balanced prediction performance indicators under the condition of medium frequency (frequency: 300 MHz incidence direction: left) and high frequency (frequency: 1900 MHz incidence direction: right), and draws the scatter plot of the comparison between the predicted value of the random forest model and the FEKO simulation results, as shown in Figs. 6 and 7.

Figure 6: Scatter plots of the predicted and FEKO simulation values in the medium-frequency representative scenario.

Figure 7: Scatter plots of the predicted and FEKO simulation values in the high-frequency representative scenario.

As can be seen from the scatter plot, most of the sample points are distributed around the y = x reference line, indicating good agreement between the model prediction results and the FEKO simulation results in the overall trend. The model does not show obvious systematic overestimation or underestimation under either medium frequency or high frequency conditions, indicating that the constructed prediction model has good generalization ability.

4.4.3 Analysis of Residual Distribution Characteristics

Based on confirming that the overall trend of the predicted value is consistent with the simulation value, the statistical distribution characteristics of the predicted residuals are further analyzed. Figs. 8 and 9 show the histogram distribution of the prediction residuals for representative scenarios in the middle frequency (frequency: 300 MHz, incident direction: left) and high frequency (frequency: 1900 MHz, incident direction: right), respectively.

Figure 8: Histogram of the predicted residual distribution in the medium-frequency representative scenario.

Figure 9: Histogram of the predicted residual distribution in the high-frequency representative scenario.

From the residual distribution diagram, it can be seen that the predicted residuals under the two types of frequency bands are concentrated around zero, and the overall distribution is symmetric, without obvious systematic deviation. This indicates that the prediction results of the random forest model are overall stable under different attitude conditions, and there is no continuous overestimation or underestimation of the coupled electric field strength.

In addition to the prediction accuracy, the computational efficiency of the alternative model was also evaluated. For a single sample of the posture angle, the average computing time of the direct FEKO simulation was approximately 80 s, while the RF model could complete one prediction in just 115 milliseconds, equivalent to an acceleration factor of about 695 times. Although the RF alternative model introduces an offline training stage, the training process only needs to be executed once. During the optimization process, this alternative model can replace thousands of repeated full-wave electromagnetic simulations, thereby significantly reducing the computational burden and meeting the need for a large number of fitness calculations in the iterative optimization process of the AAO-PG algorithm.

In summary, the random forest prediction model based on scene-level data shows high prediction accuracy and good stability in both intermediate frequency and high frequency electromagnetic environments, which can provide a reliable and rapid evaluation method for attitude angle optimization while reducing simulation calculation cost.

5  Simulation Verification and Discussion

This chapter focuses on the optimal attitude angle decision problem for moving aircraft in complex electromagnetic environments, and conducts a systematic simulation verification and performance evaluation for the proposed AAO-PG algorithm. Through multi-scenario optimization experiments, Monte Carlo stability analysis, effectiveness analysis under different incident field strength conditions, and parameter sensitivity and attitude angle adaptability analysis, the AAO-PG, basic PSO, and traditional PSO-GA algorithms are comprehensively compared. The experimental results verify the performance advantages of the proposed algorithm from multiple dimensions such as universality, stability, effectiveness, and robustness, and further reveal the positive effects of genetic operations and adaptive search mechanisms in enhancing the global optimization ability and optimization stability, providing effective method support for aircraft attitude angle optimization decision-making in complex electromagnetic environments.

5.1 Universality Analysis of the Algorithm in Different Electromagnetic Environments

In practical applications, the electromagnetic environment of moving aircraft has significant uncertainty and diversity, and its frequency characteristics, incident directions, and scene combinations often change with the mission environment and external conditions. Therefore, the attitude angle optimization algorithm not only needs to achieve good results in a single scene, but also should have the ability to play a stable role in different electromagnetic environment conditions, that is, it has good universality. In order to verify the adaptability of the improved attitude angle optimization algorithm in complex electromagnetic environments, this paper compares and analyzes its optimization effect from the perspective of multiple scenes. This paper focuses on the ability of the algorithm to suppress the coupled electric field intensity inside the body under different electromagnetic environments and different monitoring points.

Fig. 10 presents the changes in the coupling electric field intensity at monitoring point A under 114 different electromagnetic environment scenarios, including the initial posture, PSO, PSO-GA, and AAO-PG optimizations. It can be seen that the coupling electric field intensity at monitoring point A fluctuates significantly under different scenarios under the initial posture condition. Some scenarios have relatively high peak values, with the maximum reaching 0.9232 V/m, indicating that the external electromagnetic environment changes have a significant impact on the internal electromagnetic response of the aircraft. After optimizing the posture angle using the PSO algorithm, the coupling electric field intensity at monitoring point A decreased significantly overall, and the curve shifted downward compared to the initial state. According to Table 2, the average E1 value decreased from 0.1365 to 0.0732 V/m, with an average optimization reduction of 46.34%, indicating that the particle swarm optimization can effectively weaken the coupling effect between the aircraft and the external electromagnetic field by adjusting the aircraft’s posture. However, in some scenarios, there are still relatively high coupling electric field intensities, with the maximum reaching 0.4279 V/m, suggesting that the standard PSO is still prone to getting stuck in local optima in complex nonlinear search spaces, thereby limiting its further optimization ability. In contrast, the PSO-GA hybrid algorithm shows better optimization results. Its average E1 value further decreased to 0.0589 V/m, with an average optimization reduction of 56.85%. This result indicates that by introducing selection, crossover, and mutation operations of the genetic algorithm, the population diversity is enhanced, and the algorithm’s ability to escape from local optimal regions is improved, thereby achieving better global search performance. Based on this, the AAO-PG algorithm achieved the best optimization result. After optimization, the average E1 value further decreased to 0.0448 V/m, corresponding to an average optimization reduction of 67.15%, and the maximum coupling electric field intensity decreased to 0.2599 V/m. Compared with PSO-GA, AAO-PG not only achieves a lower average electric field level but also can further suppress high coupling responses in extreme scenarios. This result indicates that after introducing an adaptive search mechanism on the basis of genetic operations, the algorithm can more effectively coordinate the global exploration and local development processes, continuously maintain population activity, and thus obtain higher-quality and more stable posture angle optimization solutions.

Figure 10: Comparison of optimization results of coupling electric field strength at monitoring point A under different electromagnetic environment scenarios.

Fig. 11 shows the variation of the coupling electric field intensity at monitoring point B under different optimization strategies in the same 114 electromagnetic environment scenarios. Similar to monitoring point A, E2 also exhibited significant inter-scenario fluctuations under the initial posture conditions, with its maximum coupling electric field intensity reaching 0.4015 V/m and the average value being 0.1268 V/m, indicating that the external electromagnetic environment changes also have a significant impact on the electromagnetic response at monitoring point B. After optimizing the posture angles using the PSO algorithm, the coupling electric field intensity at monitoring point B decreased overall, with the average value dropping to 0.0868 V/m, corresponding to an average optimization reduction of 31.52%. However, as can be seen from Fig. 11, in some complex scenarios, there are still relatively high coupling electric field intensities, with the maximum value reaching 0.3118 V/m, indicating that the optimization ability of the standard PSO algorithm is still limited by the problem of local optimum in the complex nonlinear search space. In contrast, the PSO-GA hybrid algorithm further improved the optimization effect. After optimization, the average value of E2 decreased to 0.0835 V/m, and the average optimization reduction increased to 34.15%. This result shows that the introduction of the genetic algorithm can enhance population diversity and improve the global search ability, thereby improving the optimization effect of posture angle optimization to a certain extent. The AAO-PG algorithm further achieved the best optimization result based on this. After optimization, the average value of E2 decreased to 0.0782 V/m, with an average optimization reduction of 38.29%, and both the minimum and maximum values were better than those of the PSO and PSO-GA algorithms. Among them, the maximum coupling electric field intensity was further reduced from 0.3118 V/m of the PSO algorithm to 0.2654 V/m, indicating that AAO-PG can not only effectively reduce the overall electromagnetic coupling level but also suppress the high coupling response in some complex scenarios. By combining the results of Table 3 and Fig. 11, it can be seen that all three optimization algorithms can improve the electromagnetic environment at monitoring point B, but AAO-PG performed the best in terms of average value, extreme values, and overall fluctuation, indicating that it has strong adaptability and stable optimization performance in different monitoring positions.

Figure 11: Comparison of optimization results of coupled electric field strength at monitoring point B in different electromagnetic environment scenarios.

Fig. 12 shows the changes in the optimal objective function values corresponding to the three optimization algorithms under different electromagnetic environment scenarios. It can be seen that the objective function curves of the PSO-GA and AAO-PG hybrid algorithms are overall below the basic PSO algorithm, indicating that the introduction of genetic operations can effectively enhance the global optimization ability of the algorithm. From the figure, it can be observed that in most scenarios, both PSO-GA and AAO-PG can obtain lower objective function values than PSO, and in some complex electromagnetic environment scenarios, they show more obvious optimization advantages. Combining Table 4, it can be seen that the average objective function value of the PSO-GA algorithm is reduced from −2.0759 of the basic PSO to −2.6805, with an average optimization reduction of 29.12%, verifying the effectiveness of the traditional PSO-GA hybrid optimization strategy in improving the global search ability. On this basis, AAO-PG further reduces the average objective function value to −3.6880, with an average optimization reduction of 77.66%, achieving the optimal comprehensive optimization result in all test scenarios. From the extreme value distribution, the minimum objective function value of AAO-PG reaches −28.8808, which is significantly better than the −25.7708 and −20.6409 of PSO-GA and PSO, indicating that it can discover more potential excellent solutions in complex electromagnetic environments. Meanwhile, the maximum values of the three algorithms are 1.7083, 1.5410, and 1.4756, respectively, with relatively small differences, indicating that AAO-PG can significantly improve the overall optimization performance without introducing additional instability.

Figure 12: Comparison of optimal values of comprehensive objective function under different electromagnetic environment scenarios.

From the results of Figs. 10–12 and Tables 2–4, it can be seen that all three optimization algorithms can reduce the internal electromagnetic coupling level of the body by adjusting the attitude angle. Among them, PSO-GA shows better optimization performance compared to the basic PSO, verifying the effective role of genetic operations in enhancing the global search ability; AAO-PG achieves the optimal result based on PSO-GA, indicating that the proposed adaptive search mechanism can effectively improve the population diversity and reduce the risk of local optimum. Therefore, AAO-PG exhibits better optimization performance and stronger scene adaptability in different electromagnetic environment scenarios and different monitoring positions, further verifying the effectiveness and superiority of the proposed method.

5.2 Monte Carlo Stability Analysis Experiments

To analyze the stability of the improved posture angle optimization algorithm under fixed electromagnetic environment conditions, this paper selects three typical frequency scenarios of 800, 1000, and 1900 MHz, representing the medium frequency, medium high frequency, and high frequency electromagnetic environments, respectively. Monte Carlo simulation experiments are conducted under the same forward incidence condition. While keeping the electromagnetic environment parameters unchanged, the particle swarm position is randomly initialized, and 50 optimization processes are independently run to evaluate the stability and consistency of the algorithm under different initial conditions. The running results of the basic PSO algorithm and the AAO-PG algorithm in the above three electromagnetic scenarios are compared and analyzed. Figs. 13 and 14 and Table 5, respectively show the stability differences of the two algorithms from the perspectives of result distribution characteristics, dispersion degree, and statistical indicators.

Figure 13: Boxplot comparison of the optimal objective function of different algorithms under Monte Carlo simulation experiments.

Figure 14: Distribution histograms of optimal objective functions of different algorithms under Monte Carlo simulation experiments.

Table 5 further presents the statistical verification results of the Monte Carlo simulation experiments under three typical frequency scenarios, including the mean value of the objective function, the optimal value, the worst value, the standard deviation, the 95% confidence interval, and the significance test results. It can be seen that in the scenarios of 800, 1000, and 1900 MHz, the AAO-PG algorithm has achieved lower average objective function values and smaller standard deviations. Specifically, in the 800 MHz scenario, the standard deviation decreased from 5.28 of the PSO algorithm to 3.94; in the 1000 MHz scenario, it decreased from 0.55 to 0.31; and in the 1900 MHz scenario, it decreased from 0.46 to 0.35. Meanwhile, the average value, optimal value, and worst value of AAO-PG in each scenario are superior to those of the basic PSO algorithm, indicating that it can continuously obtain better and more stable optimization results under different random initialization conditions. From the statistical significance test results, the p-values for the three scenarios are 7.99 ×10−8, 2.62 ×10−17, and 2.76 ×10−17, respectively, all of which are far less than the significance level of 0.05, indicating that the performance difference between AAO-PG and the basic PSO algorithm is statistically significant, rather than caused by random initialization fluctuations. Additionally, the 95% confidence intervals of AAO-PG are overall lower than those of the PSO algorithm, and the interval width is relatively small, further demonstrating that its optimization results have better stability and consistency. The fundamental reason for the large fluctuations in the results of the basic PSO algorithm lies in its limited ability to maintain population diversity. In the problem of attitude angle optimization, the objective function is affected by frequency, incident direction, and attitude parameters, showing obvious nonlinearity and multi-peak characteristics. Therefore, there are often multiple local optimal regions in the search space. The standard PSO mainly relies on the individual optimal position (Pbest) and the global optimal position (Gbest) for search guidance. When the population gradually converges in the later iterations, it is prone to a decrease in population diversity and premature convergence. The search trajectories formed under different random initialization conditions may eventually converge to different local optimal solutions, resulting in large discreteness in the optimization results. In contrast, the AAO-PG algorithm can continuously restore population diversity and enhance the ability to escape from local optima through the periodic introduction of genetic algorithm selection, crossover, and mutation operations. At the same time, the improved penalty mechanism in the evaluation function can effectively suppress the interference of invalid attitude adjustments on the search process, making the search direction clearer. Therefore, even under different random initialization conditions, AAO-PG can stably converge to high-quality solutions, significantly reducing result fluctuations and improving the robustness and stability of the algorithm. Based on the above results, it can be concluded that the proposed AAO-PG algorithm has good stability, consistency, and global search ability in different frequency scenarios, and its optimization performance is significantly less dependent on random initial conditions compared to traditional PSO algorithms.

Based on the results of Figs. 13 and 14 as well as Table 5, it can be concluded that the AAO-PG algorithm demonstrates excellent stability, consistency, and global search capabilities in three typical frequency scenarios. Its optimization results have significantly reduced dependence on random initial conditions and can consistently obtain reliable optimization solutions in multiple independent runs.

In addition to the statistical analysis of the final optimization results, in order to further explore the intrinsic reasons why the AAO-PG algorithm is superior to the basic PSO algorithm, based on the 50 Monte Carlo experiments under the 800 MHz forward incidence scenario, this paper statistically calculates the average optimal objective function value for each iteration step and plots the average convergence curve, as shown in Fig. 15.

Figure 15: Comparison of average convergence behaviors of AAO-PG and PSO.

From Fig. 15, it can be seen that both algorithms can quickly reduce the objective function value in the early stage of optimization, indicating that the particle swarm search mechanism can effectively locate the potential excellent solution area. However, during the iteration process, the two algorithms gradually exhibit significantly different convergence characteristics. The basic PSO algorithm basically enters the convergence platform after the 7th to 8th generation, and its average objective function value stabilizes at approximately −24.5; subsequent iterations can hardly achieve further improvement. In contrast, the AAO-PG algorithm can still continuously achieve performance improvement at the same stage and eventually converge to a target function value of approximately −30.0. This phenomenon indicates that the standard PSO is prone to fall into a local optimal region due to the decline in population diversity in the later iterations. Since the particle update only relies on the individual optimal position (Pbest) and the global optimal position (Gbest), as the population gradually gathers, the search directions of different particles tend to be consistent, thereby weakening the ability to further explore the new solution space. Therefore, although PSO can converge quickly, the quality of the obtained solution is limited by the local optimum. In contrast, AAO-PG can continuously generate new candidate solutions and restore population diversity through periodic introduction of selection, crossover, and mutation operations of the genetic algorithm, thereby enhancing the ability to escape from the local optimal region. At the same time, the improved penalty mechanism in the evaluation function can effectively suppress the interference of invalid posture adjustments on the search process, making the search direction clearer. Therefore, after PSO has stopped improving, AAO-PG can continue to explore the better solution area and ultimately obtain a lower target function value. The above results further reveal the intrinsic mechanism by which AAO-PG is superior to the basic PSO algorithm, that is, by maintaining population diversity and enhancing global search ability, it achieves higher-quality optimization results.

5.3 Effectiveness Analysis of the Algorithm under Different Incident Field Strength Conditions

In the above experiment, in order to ensure the comparability of electromagnetic response results under different attitude conditions and reduce the computational burden in the data acquisition stage of large-scale electromagnetic simulation, the external plane wave incident field strength is uniformly set as 1 V/m as a standardized reference condition in this paper. However, in the actual complex electromagnetic environment, the electromagnetic radiation intensity of the aircraft may vary in a large range, and the field intensity in the typical civil and military electromagnetic environment can reach tens to hundreds of V/m. Therefore, it is difficult to fully reflect the applicability of the algorithm in the real environment by only verifying the algorithm under a single field strength condition. In order to further verify the effectiveness and robustness of the proposed attitude angle optimization algorithm under different electromagnetic intensity conditions, the incident field strength is extended to the range of 1–200 V/m, and the performance of the algorithm is compared and analyzed. The following data are obtained under the condition of high frequency (frequency: 1900 MHz, incident direction: right).

As can be seen from Fig. 16, as the incident field strength increases from 1 to 200 V/m, the coupled electric field strength of monitoring point A and monitoring point B shows an obvious upward trend as a whole. This indicates that when other conditions remain unchanged, there is a relatively stable proportional relationship between the internal coupling electric field strength and the external incident field strength, which is consistent with the basic physical law of electromagnetic coupling. At the same time, it can be observed that under all incident field strength conditions, the coupled electric field strength at the monitoring point inside the body can be effectively reduced by attitude angle optimization. Among them, the improved AAO-PG algorithm can obtain lower coupled electric field strength under most field strength conditions, which indicates that the particle swarm optimization algorithm with genetic operation has a stronger global search ability in a complex nonlinear search space, so as to find a better combination of attitude angles to reduce the electromagnetic coupling effect inside the body.

Figure 16: Variation of coupled electric field intensity between monitoring point A and monitoring point B under different incident field strength.

In order to further quantitatively analyze the electromagnetic suppression effects of different algorithms under various incident field strengths, Tables 6 and 7 respectively show the comparison results of the reduction amplitude of the coupled electric field at monitoring point A and monitoring point B. As can be seen from Table 6, under most conditions of incident field strength, the electric field reduction of the AAO-PG algorithm at monitoring point A is better than that of the basic PSO algorithm as a whole, especially under medium and high field strength conditions (such as 20 V/m and above). Table 7 shows that at monitoring point B, the AAO-PG algorithm similarly achieves higher electric field reduction under most field strength conditions, but its suppression effect is slightly lower than that of the basic PSO algorithm under individual field strength conditions. This phenomenon needs to be comprehensively analyzed in combination with the results of attitude angle variation in Fig. 17. It can be seen from Fig. 17 that under most field strength conditions, the attitude angle variation corresponding to the AAO-PG algorithm is smaller than that of the basic PSO algorithm as a whole. This indicates that the improved algorithm does not simply pursue the minimum value of the coupled electric field intensity when searching for the optimal solution, but considers the comprehensive tradeoff between the electric field suppression effect and the attitude maneuver cost. Since the objective function constructed in this paper includes both the coupled electric field reduction benefit and the attitude angle adjustment cost term, in some scenarios, the AAO-PG algorithm may select the attitude solution with a slightly higher electric field but a smaller attitude adjustment, in order to obtain better comprehensive optimization results. In other words, the algorithm pays more attention to the balance between the electromagnetic suppression effect and the flight maneuver cost in the optimization process, rather than only pursuing the local minimum electric field value.

Figure 17: Comparison of attitude angle variation under different incident field strength.

From the objective function change results in Fig. 18, it can be found that the optimal objective function value obtained by the AAO-PG algorithm is always lower than that of the basic PSO algorithm in the whole range of incident field strength. Because the objective function reflects the electric field reduction effect and the attitude maneuver cost simultaneously, the lower objective function value indicates that the AAO-PG algorithm is better than the basic PSO algorithm in the overall optimization performance. At the same time, it can be seen from the results under different incident field strengths that the improved algorithm can stably obtain better solutions in the range of typical civil and military electromagnetic environments, which verifies the effectiveness and engineering applicability of the proposed attitude angle optimization method in complex electromagnetic environments.

Figure 18: Comparison of optimal objective function values under different incident field strengths.

To further analyze the change in optimization gain of the improved algorithm compared with the basic algorithm, Table 8 shows the average optimization decrease of the AAO-PG algorithm compared with the PSO algorithm on the objective function (Best_f) under different incident field strength conditions. It can be observed from Table 8 that as the incident field strength continues to increase, the optimization advantage of the improved algorithm over the basic algorithm shows a gradually decreasing trend; that is, it has a significant advantage under low field strength conditions (such as 1 V/m), and this advantage gradually weakens under high field strength conditions. This phenomenon may be closely related to the fitness evaluation function constructed in this paper. Under the standard condition of 1 V/m, the weight relationship between the electromagnetic suppression benefit and the attitude maneuver cost in the objective function can better reflect the optimization objective, so it has strong applicability. However, with the increase of the intensity of the incident field, the intensity of the coupled electric field inside the body increases significantly. Although the weight coefficient in the evaluation function remains unchanged, the absolute value of the electric field term continues to increase, which leads to the gradually dominant influence of the electromagnetic coupling term in the objective function. This change weakens the constraint effect of the attitude maneuver cost term in the optimization process, and makes the difference between different algorithms in the comprehensive objective function be partially compressed, which shows that the optimization drop of the AAO-PG algorithm is gradually reduced compared with the basic PSO algorithm. From a deeper physical perspective, the reduction in optimization gain is not solely caused by the changing contribution of different terms in the objective function. As the incident field strength increases, the electromagnetic response of the aircraft becomes increasingly dominated by the external excitation itself. Under such conditions, the achievable suppression margin through attitude-angle adjustment becomes inherently limited. Consequently, both PSO and AAO-PG tend to converge toward similar feasible attitude configurations, reducing the relative performance gap between the two algorithms. Despite the reduced relative improvement, the proposed method still maintains stable optimization performance and consistently achieves better solutions than the benchmark algorithms.

5.4 Parameter Sensitivity and Attitude Angle Adaptability Analysis

To further evaluate the parameter sensitivity of the proposed AAO-PG algorithm and the applicability of the attitude angles in different attitude search spaces, this section conducts an analysis from two aspects: parameter sensitivity and attitude adaptability. Among them, the parameter sensitivity analysis is used to study the impact of changes in key parameters on the optimization results, while the attitude adaptability analysis is used to verify the optimization performance and applicability of the algorithm in different attitude angle constraint ranges.

Among the parameters involved in the objective function, the degree of influence on the optimization results varies significantly. To enhance the specificity of the analysis, this paper only conducts sensitivity analysis on the attitude adjustment weight coefficient λs. This parameter directly determines the trade-off relationship between the electromagnetic suppression effect and the cost of attitude adjustment, and thus has a direct impact on the final optimization result. In contrast, the smoothing parameter α is mainly used to avoid numerical instability when the attitude changes approach zero, and it is not triggered in all the optimal solutions obtained in this paper; while the enhancement penalty coefficient β and the zero improvement penalty coefficient λ mainly act on infeasible or low-quality solutions that occur during the search process, and their influence is more reflected in the search path and convergence process rather than the final optimal solution. Therefore, λs is selected as a representative parameter to evaluate the sensitivity of the proposed AAO-PG algorithm to changes in key parameters and its robustness.

To assess the influence of the key parameter λs on the optimization results, a sensitivity analysis was conducted under the condition of keeping other parameters unchanged. Different values of λs were selected for the analysis. According to the parameter settings in the previous text, λs = 0.5 was adopted by default in this study. Considering that λs is mainly used to balance the relationship between the electromagnetic suppression effect and the cost of attitude adjustment, λs∈ [0.3, 0.7] was chosen for the test. All experiments were carried out under typical mid-frequency scenarios (with a frequency of 800 MHz and an incident direction of directly above), and the optimization results are shown in Table 9 and Fig. 19.

Figure 19: Sensitivity analysis of λs on optimization performance.

From Table 9 and Fig. 19, it can be observed that as λs increases, the value of the objective function shows a gradually rising trend. This is because λs is used to measure the weight of the cost of attitude adjustment in the objective function. A larger λs will further strengthen the constraint on the cost of attitude maneuver, making the optimization process more inclined to maintain the current flight attitude, thereby reducing the pursuit of the electromagnetic suppression effect. On the contrary, when λs takes a smaller value, the algorithm pays more attention to the weakening of the coupling electric field, thus obtaining a better objective function value. It is worth noting that although there are certain differences in the objective function values when λs varies within the range of 0.3 to 0.7, the change process is smooth and has good monotonicity, without any performance sudden change or significant degradation. Moreover, the optimal attitude parameters obtained under different λs conditions are generally consistent, with only minor fluctuations, indicating that the changes in key parameters do not alter the overall trend of the optimization result. Combined with the aforementioned Monte Carlo stability analysis results, it can be concluded that the proposed AAO-PG algorithm has good robustness to the change of λs, and can maintain stable and reliable optimization performance within a reasonable parameter range.

To verify the applicability of the proposed AAO-PG algorithm in different posture search spaces, this paper selects three typical scenarios (with frequencies of 800, 1000, and 1900 MHz, and all incident directions being positive leftward) and expands the search range of posture angles from ±10∘ to ±20∘. The optimization results are compared, and the results are shown in Table 10.

As can be seen from Table 10, in the three test scenarios, after the search range is expanded from ±10∘ to ±20∘, the optimal posture parameters and the target function values remain consistent. This indicates that under the current electromagnetic environment conditions, the optimal solutions are all within the ±10∘ range. Further expanding the search space does not bring additional optimization benefits. From the perspective of the optimization mechanism, the influence of the change in posture angles on the electromagnetic coupling effect mainly concentrates within a limited posture adjustment range. When the search range is expanded, although the algorithm gains a larger search space, no better solution appears in the new area, so the final optimization result remains unchanged. This result shows that the ±10∘ posture constraint range selected in this paper can cover the main effective solution area, and it also takes into account the optimization efficiency and engineering feasibility while satisfying typical flight maneuver constraints.

In addition, the experimental results show that the AAO-PG algorithm can still converge to the same optimal solution stably after the search space is expanded, indicating that the proposed method has good posture adaptability and search stability. Its optimization performance will not be significantly affected by changes in the search space within a reasonable range.

6  Conclusion

Aiming at the problem that aircraft is vulnerable to electromagnetic coupling interference in complex electromagnetic environments and that traditional protection methods are difficult to adapt dynamically, this paper proposes an improved attitude angle optimization (AAO-PG) method based on particle swarm optimization and genetic algorithm (PSO) under the framework of digital twin modeling, and constructs a scene-level electromagnetic coupling interference data set based on Feko simulation platform. The random forest algorithm is used to establish a fast prediction model between the attitude angle and the internal electric field strength of the body, and the efficient solution of the attitude angle optimization decision is realized.

Firstly, based on FEKO full-wave simulation, an electromagnetic response dataset covering 19 frequency bands, 6 incident directions, and 1331 attitude combinations is constructed, totaling 151,734 groups of sample data, which are further divided into 114 scene-level sub-datasets. The dataset has good coverage and representation in the dimensions of frequency, spatial direction, and attitude angle, which can fully describe the influence of attitude angle change on the electromagnetic coupling effect inside the body, and provide a reliable data basis for subsequent machine learning modeling and optimization analysis. Secondly, the random forest prediction model based on scene-level data shows good prediction performance under medium and high frequency conditions, where the coefficient of determination R2 is higher than 0.872 (up to 0.957), and the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) are kept at a low level. At the same time, the predicted value and the simulation value have good consistency in the overall trend, and the residual distribution is concentrated without obvious systematic deviation, which indicates that the model can accurately describe the nonlinear mapping relationship between the attitude angle and the coupled electric field strength. In the optimization process, the prediction model can effectively replace the high-cost electromagnetic simulation and realize the rapid evaluation of the objective function, so as to significantly improve the computational efficiency of the algorithm and meet the requirements of engineering applications. Finally, through multi-scene simulation experiments, Monte Carlo stability analysis, and comparison experiments under different incident field strength conditions, the universality, stability, and effectiveness of the proposed algorithm in complex electromagnetic environments are verified. Under multi-scenario conditions, the average electric field optimization reduction of the AAO-PG algorithm at monitoring points A and B reaches 67.15% and 38.29%, respectively, which is about 20.81% and 6.77% higher than that of the basic PSO algorithm. In terms of the comprehensive objective function index, the average optimization reduction reaches 77.66%, which shows a significant global optimization ability. In the Monte Carlo experiments, the mean value of the objective function of the improved algorithm is lower, and the standard deviation is reduced from 5.28 to 3.94, which shows that it has better robustness and stability to random initial conditions. In addition, the proposed algorithm can stably obtain better solutions under different incident field strengths of 1–200 V/m, which verifies its applicability in different electromagnetic intensity environments.

Although the proposed AAO-PG algorithm outperforms the benchmark PSO algorithm and the traditional PSO-GA hybrid algorithm, it is still necessary to acknowledge that it has several limitations. Firstly, as the incident field strength increases, the performance improvement of AAO-PG relative to the benchmark algorithm decreases, but it can still obtain stable optimization results. This phenomenon is mainly attributed to the fact that in high field strength conditions, the dominant role of the electromagnetic coupling term in the objective function increases, thereby reducing the relative contribution of the adaptive optimization mechanism. Secondly, this study uses a simplified helicopter model with a PEC surface. Although this model retains the main geometric scattering characteristics required for attitude angle optimization analysis, it fails to fully reflect the structural details, material properties, antennas, and onboard subsystems of a real aircraft platform. Therefore, the quantitative optimization results may differ from those obtained with the actual aircraft configuration. Thirdly, all the data used in this study were generated through electromagnetic simulation. Although the data based on simulation provide a controllable environment for the development and evaluation of the algorithm, they do not consider experimental measurements and a broader electromagnetic scenario. Therefore, the verification of the generalization ability of the proposed framework under real operating conditions still needs to be further carried out. Future work will focus on expanding this method to make it applicable to more realistic aircraft geometries, incorporating complex material properties and onboard subsystems, and verifying the effectiveness of the framework through experimental electromagnetic measurements.

Acknowledgement: None.

Funding Statement: This work was supported by the National Key Laboratory on Electromagnetic Environment Effects Foundation under Grant 6142205240201.

Author Contributions: Conceptualization, Min Huang; methodology, Min Huang; software, Min Huang; validation, Min Huang; formal analysis, Min Huang; investigation, Cuichen Zhou; resources, Cuichen Zhou; data curation, Cuichen Zhou; writing—original draft preparation, Cuichen Zhou ; writing—review and editing, Cuichen Zhou; visualization, Yuming Wang; supervision, Yuming Wang; project administration, Yuming Wang; funding acquisition, Yuming Wang. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: Due to collaboration reasons, this article cannot directly provide publicly available data.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflict of interest.

References

1. Volkova T, Ou J. General issues of ensuring electromagnetic compatibility of aircraft on-board equipment. E3S Web Conf. 2024;515:02004. doi:10.1051/e3sconf/202451502004. [Google Scholar] [CrossRef]

2. Zhu M, Cazzella L, Linsalata F, Magarini M, Matteucci M, Spagnolini U. Toward real-time digital twins of EM environments: computational benchmark for ray launching software. IEEE Open J Commun Soc. 2024;5:6291–302. [Google Scholar]

3. Du M. Simulation study on electromagnetic coupling effects of aircraft in high-intensity radiated field environment [Ph.D. thesis]. Xi’an, China: Xi’an Shiyou University; 2022. (In Chinese). [Google Scholar]

4. Hu G, Liu Q. Characteristics of HEMP radiation field on typical transport aircraft with frequency sweep method. Appl Sci. 2021;11(19):9284. doi:10.3390/app11199284. [Google Scholar] [CrossRef]

5. Zhou J, Yu D, Chai M, Bai Y, Du J, Li T, et al. Analysis of electromagnetic effects and coupling characteristics of UAV system-level cables. High Power Laser Part Beams. 2025;37(2):023001. (In Chinese). [Google Scholar]

6. Liang T, Xie YZ. Determining incidence and polarization of electromagnetic field for maximal/minimal coupling to transmission line system. IEEE Microw Wirel Compon Lett. 2020;30(11):1021–4. doi:10.1109/lmwc.2020.3021695. [Google Scholar] [CrossRef]

7. Zhang Z, Zhou Y, Zhang Y. Strong electromagnetic interference and protection in UAVs. Electronics. 2024;13(2):393. doi:10.3390/electronics13020393. [Google Scholar] [CrossRef]

8. Mordachev V, Sinkevich E, Yatskevich Y. Simulation of nonlinear interference in aircraft systems operating in complex electromagnetic environment created by land-based and air-based wireless systems. In: Proceedings of the International Symposium on Electromagnetic Compatibility (EMC Europe); 2017 Sep 4–8; Angers, France. p. 1–6. [Google Scholar]

9. Weng R. Research on efficient electromagnetic computation methods for complex electrically large targets [Ph.D. thesis]. Nanjing, China: Southeast University; 2023. (In Chinese). [Google Scholar]

10. Bhosale VA. The digital twin in aviation: a review of core technologies, lifecycle applications, and future challenges. TechRxiv. 2025. doi:10.36227/techrxiv.176524820.07513276/v1. [Google Scholar] [CrossRef]

11. Mohebbi A, Achiche S, Baron L. Integrated and concurrent detailed design of a mechatronic quadrotor system using a fuzzy-based particle swarm optimization. Eng Appl Artif Intell. 2019;82:192–206. doi:10.1016/j.engappai.2019.03.025. [Google Scholar] [CrossRef]

12. Zhao TT. Nonlinear model predictive control based on improved particle swarm optimization [Ph.D. thesis]. Fuxin, China: Liaoning Technical University; 2015. (In Chinese). [Google Scholar]

13. Shami TM, El-Saleh AA, Alswaitti M, Al-Tashi Q, Summakieh MA, Mirjalili S. Particle swarm optimization: a comprehensive survey. IEEE Access. 2022;10:10031–61. [Google Scholar]

14. Gad AG. Particle swarm optimization algorithm and its applications: a systematic review. Arch Comput Methods Eng. 2022;29(5):2531–61. [Google Scholar]

15. Zhou HD. Research on multi-model joint simulation method for airborne phased array antenna [Ph.D. thesis]. Chengdu, China: University of Electronic Science and Technology of China; 2024. (In Chinese). [Google Scholar]

16. Shi L, Gong J, Zhai C. Application of a hybrid PSO-GA optimization algorithm in determining pyrolysis kinetics of biomass. Fuel. 2022;323:124344. doi:10.2139/ssrn.3998957. [Google Scholar] [CrossRef]

17. Gao X, Zhang Y, Wang B. The optimal strategies of maneuver decision in air combat of UCAV based on the improved TD3 algorithm. Drones. 2024;8(9):501. doi:10.3390/drones8090501. [Google Scholar] [CrossRef]

18. Ren W, Zou B, Zhu X. Research on UAV path planning algorithm under electromagnetic interference environment. In: Proceedings of the International Conference on Computer Application and Information Security (ICCAIS 2024); 2024 Dec 20–22; Wuhan, China. p. 388–93. [Google Scholar]

19. Wang Z, Han Z, Tayyaba S. Adaptive control for uncrewed aerial vehicles based on communication information optimization in complex environments. PeerJ Comput Sci. 2024;10(3–4):e1920. doi:10.7717/peerj-cs.1920. [Google Scholar] [PubMed] [CrossRef]

20. Breiman L. Random forests. Mach Learn. 2001;45(1):5–32. doi:10.1023/a:1010933404324. [Google Scholar] [CrossRef]

21. Jiang XT, Zhang HH, Guo KT, Zheng Y, Liang JM. Mesh-based deep learning surrogate model for rapid wideband 3D electromagnetic scattering computation. IEEE Trans Antennas Propag. 2026;74(5):4546–61. doi:10.1109/tap.2026.3661600. [Google Scholar] [CrossRef]