A Deep Neural Network Modeling Methodology for Efficient EMC Assessment of Shielding Enclosures Using MECA-Generated RCS Training Data

We develop a deep neural network (DNN) modeling methodology to predict the radiated emissions of a shielding enclosure in terms of its aperture attributes including aperture shape, size, pitch, and quantity. The target structure is the inside of a 3-D enclosure comprising perfect electric conductor (PEC) boundaries with dimensions of a desktop personal computer (PC) containing thermal dissipation apertures on the surface of its back panel. The DNN model is developed to compute the radar cross section (RCS) as a function of aperture attributes to enable the efficient assessment of the PC's electromagnetic compatibility (EMC). To generate training data for machine learning (ML), we implement the modified equivalent current approximation (MECA) method and validate it against analytical methods and a commercial field-solver. We use MECA to compute RCS data for approximately 55 000 experiments across a wide range of aperture attributes. We examine numerous DNN models across parameters such as number of layers and nodes per layer, activation function, optimization algorithm, loss function, batch size, and epoch, to identify the optimal DNN model based on the following: 1) accuracy, 2) computation time, and 3) memory usage. Results show excellent agreement between MECA and DNN predictions for previously unseen cases.


I. INTRODUCTION
T HE study of EMC has gained significant importance in applied industrial research, driven by the rapid advancement of technology and the increasing proliferation of devices operating across the wide spectrum.Understanding the effects of EM fields on other devices and developing effective strategies to mitigate EM interference (EMI) has become a pressing concern for ensuring EMC across diverse devices and systems [1].
Shielded enclosures are often used as a simple method to mitigate interference between electrical devices and to protect electrical circuits against fault events and even physical damage.These enclosures have various design parameters, such as material properties, thickness, geometry, and structural integrity.However, practical shielded enclosures often include slots and apertures that serve multiple purposes, such as facilitating airflow, providing access to the interior of the enclosure, Manuscript received April 30 2023; revised August 7 2023, accepted September 12 2023.Corresponding author: Ata Zadehgol (email: azadehgol@uidaho.edu).
and allowing for visibility into the enclosure.These apertures enable the ingress and egress of interfering EM waves, which have the potential to disrupt a secondary electronic device by inducing unintended currents and voltages.Therefore, quantifying the EM wave dynamics of apertured enclosures is important to ensuring EMC of the shielded system [2].
The EM behavior of a PEC enclosure with apertured surfaces has been extensively reported in the literature.Analytical methods and formulas, such as those described in [3]- [8] are computationally efficient, however they are often limited to simple structures with specific aperture attributes, i.e., aperture size, shape, pitch, and quantity.On the other hand, computational methods such as finite-difference timedomain [9], finite elements method (FEM), method of moments (MoM) [10], [11], and hybrid methods [12], [13] allow for detailed modeling of EM wave interactions with the apertured enclosure; however, their computational cost may be exorbitant as measured in long compute-times and large memory utilization.Especially for electrically large problems, where the wavelength is much smaller than the structure's size, the computational cost of EMC analysis is often prohibitive.
Classical integral equation (IE) methods, like method of moments (MoM) [14, sec.12.2], determine equivalent currents through an integral equation and enforce continuity of the tangential components of electric and magnetic fields at the boundaries.However, this approach necessitates substantial computational time for inverting the impedance matrix associated with electrically large scatterers.On the other hand, high frequency techniques like physical optics (PO) [14, sec.7.10], can provide faster alternatives to this analysis while maintaining an acceptable level of accuracy.Moreover, the modified equivalent current approximation (MECA) method [15] has been introduced as an extension of PO method to approximate the equivalent current densities for the case of PEC materials as well as lossy materials with complex effective permittivity.The technique employed in this method differs from the one described in the Stationary Phase method [16,Appx. VIII].In this method, the surface is divided into flat triangular facets, characterized by a constant amplitude current distribution and linear phase variation.Consequently, the facets can be relatively large in size, allowing for the analytical solution of the integral.Therefore, the MECA method emerges as a more efficient solution in terms of computational speed and memory usage (RAM), albeit with slightly lower accuracy when compared to classical integral equation (IE) ML has shown great potential for modeling for various complex tasks, including computer vision [17], text recognition, and speech analysis [18], [19].To reduce the computational cost associated with full-wave EM field-solvers, numerous studies such as artificial neural networks (ANNs) [20]- [23], extreme learning machine [24], [25], and Reinforcement learning-based method [26] have explored ML methods to learn the relationship between EM response and structural parameters.However, to the best of our knowledge, none of the previous works utilize an ML-based approach for assessing the EMC performance of an enclosure in terms of its aperture attributes.Fig. 1 illustrates the motivations of this work by comparing the advantages and disadvantages of some existing methods for EMC analysis of shielding enclosures.
In this work, we make two main contributions: (1) We implement the MECA method [15] as an alternative to numerical field-solvers for computing the radiated emissions from apertures of an enclosure.The MECA method utilizes equivalent currents approximation based on reflections at media boundaries and surface discretization (meshing) instead of volume discretization even for 3D problems, while the mesh sizes can be relatively large and the integral can be solved analytically.These features result the significant reduction in number of mesh elements as well as improving computational efficiency, making MECA suitable for simulating a large number of examples with different aperture attributes.Thus, we utilize MECA's speed to simulate approximately 55,000 experiments involving enclosures of varying aperture attributes.The enclosure is assumed to be that of a desktop PC with backside airflow apertures that houses a microprocessor operating at 3.6 GHz.We compute the RCS data of these enclosures to generate training data for deep neural network (DNN) models, and (2), we explore numerous DNN models with different parameters such as number of layers, number of nodes (neurons) per layer, activation function, optimization algorithm (optimizer), loss function, batch size, and epoch.We examine the (a) data accuracy as measured in terms of mean squared error (MSE), (b) computation time in seconds (s), and (c) random access memory (RAM) usage of these models to identify the optimal DNN model for our data.Once the optimal DNN model is found, we use it to predict the RCS for a given set of parameters that describe the aperture attributes, and compare the predicted results against the labeled simulated values.Although the MECA algorithm is computationally efficient relative to comparable full-wave volumetric fieldsolvers, the proposed DNN model outperforms MECA in terms of computational speed for predicting the RCS as a function of aperture attributes.As a result, the proposed model in this work could serve as a valuable tool for guiding the analysis and design optimization of apertured enclosures to achieve optimal EMC performance.
The remainder of this paper is organized as follows.In section II, we provide a brief overview of the MECA formulation, review of DNN modeling methodology and pertinent parameters, followed by proposed algorithm employed in this work.Numerical results are provided in section III, where (i) we present several examples to validate MECA through analytical formulas from the literature as well as the commercial field-solver CST Studio Suite [27], (ii) we provide numerical results for DNN studies, including comparisons of various DNN models against each other, as well as against traditional ML algorithms such as linear regression, decision tree, random forest.In section IV we discuss the DNN results, highlighting the connection between improvements in the DNN model and the EM system, including the backward vs. forward propagation error with respect to variable independence, error improvement from column-wise normalization to row-wise (theta-wise) normalization and relationship to learning the EM system's behavior, and DNN performance variation with respect to the number of nodes per layer, number of layers, activation function, optimizer, loss function, etc., as well as a discussion on alternative varieties of neural networks.Finally, in section V, we conclude with closing remarks as we identify potential topics for further investigation.

A. Review of MECA
Although the MECA formulation is outlined in [15] where we found a number of errors, it is prudent to explain the method in some detail.MECA approximates the modified equivalent electric and magnetic currents at the surface boundaries (SB) of lossy materials or dielectrics under oblique incidence, as follows. where, where ȷ = √ −1, and the material permittivity, permeability, and conductivity are respectively ϵ (F/m), µ (H/m), σ (S/m).
θ inc (rad) is the incidence angle, η (Ω) is impedance of the medium, k (rad/m) is the wave vector, k tw (rad/m) is the transverse component of the transmitted wave vector, n is the unit vector normal to the surface, and E inc TE , E inc TM , and êTE are obtained by TE/TM decomposition of the incident field as E inc = E inc TE êTE + E inc TM êTM (V/m).Note that for PEC material R TE = R TM = −1 and the modified equivalent currents reduce to the well-known physical optics (PO) approximation [14, sec. 7.10].
Referring to Fig. 3 in [15], MECA proceeds to discretize the boundary surfaces into triangular mesh elements.Assuming a constant amplitude and linear phase variation dependent on the direction of propagation of the incident wave (i.e., pi ), the electric current J (or the magnetic current M) at the i th element may be written as J i = J i0 e −ȷk1p i •r ′′ i , where J i0 is obtained from (1) at position r i , and r ′′ i is the vector from the i th element's barycenter (given by the position vector r i ) to the source (given by the position vector r ′ i ).The farzone (Fraunhofer [16, p. 32]) scattered electric field E s k and scattered magnetic field H s k due to the contribution of all the mesh elements of a given geometry may be computed as where and which has a closed form solution provided in [28].Note that the above equations are slightly different than those found in [15], as we have made several corrections.By relying on the assumptions of constant amplitude and linear phase variation in J i , M i , the above method enables modeling larger meshes relative to comparable alternative methods, resulting in reduced computational cost and greater overall computational efficiency.

B. Review of DNNs
DNNs are a class of artificial neural networks that are capable of automatically learning patterns embedded in raw data through multiple layers of interconnected nodes (neurons), mimicking the structure and function of the human brain [29].An example of the DNNs structure and interconnections across layers and nodes are depicted in Fig. 2.
Below, we provide a brief overview of the DNN parameters that shape the behavior and performance of our proposed model.We rely on the Keras [30] and TensorFlow [31] libraries for the Python programming language [32] to accomplish the training, validation, and testing of our DNN models.
DNN Model Architecture: An important aspect of a DNN is its architecture which may be characterized by the number of layers, the number of nodes per layer, and the type of activation function.Deeper networks with more layers are generally able to learn more complex features by providing higher degrees-of-freedom, but may also suffer from vanishing or infinite gradients during training [33]- [35].The number of nodes in each layer determines the capacity of the model, where more capacity usually implies potential to learn more complex patterns, although the model may also be more at risk of being over-fitted.The choice of linear or non-linear activation functions, such as sigmoid, tanh, relu (rectified linear unit), exponential, or elu (exponential linear unit) affects the non-linearity of the model; this is important for capturing complex relationships in data.Different activation functions have different properties and may be suitable for different types of data.Proper selection of the activation function is an important aspect of designing an effective DNN architecture.Additionally, there are other non-linear functions, such as softmax, softplus, and softsign; these functions may be better suited for a classification task, whereas here we formulate our problem as a regression task [36].
Pre-processing input data: A DNN often requires preprocessing of its input data for training; this may include data cleaning, data splitting, feature scaling (or normalization), and data augmentation to ensure optimal model performance.The choice of pre-processing techniques and their parameters can significantly impact a DNN model's ability to learn meaningful patterns from the data.For normalization of input data some common types of scaling functions are Min-Max, Standard, and Robust [37].
Essentially, a DNN has great potential to serve as a powerful model for various ML tasks, however its performance is heavily influenced by a multitude of parameters.Properly tuning these parameters is crucial for achieving optimal model performance.
Loss function: The loss function, also known as the cost functions or the objective function, is used in a DNN to measure the error or discrepancy between the predicted outputs and the actual (target) outputs during training.The choice of an appropriate loss function is important in training a DNN as it directly affects the optimization process and the quality of the learned model.There are various types of loss functions available for a DNN and the selection of a particular loss function depends on the specific problem and data, such as whether it's a classification or a regression task, etc.Some popular types of loss functions used in regression tasks are Mean Squared Error (MSE), Mean Absolute Error (MAE), Huber, and LogCosh [37].
Optimization algorithm: Another important aspect of a DNN is the optimization algorithms (or optimizer) used for training, which determines how the model's parameters are updated during the training process.These algorithms are designed to minimize the loss function.
There are various optimization algorithms available for a DNN, ranging from relatively simpler choices like Gradient Descent (GD) and its variations, to more advanced choices like Adam, Adamax, Nadam, and RMSprop.These algorithms differ in how they update the parameters based on the gradients of the loss function, and many of them adaptively adjust the learning rate (which controls the step size of parameter updates) to accelerate convergence and improve performance.Properly selecting and tuning an appropriate optimization algorithm is crucial for achieving efficient and effective training of a DNN, as it can significantly impact the model's convergence speed and accuracy [38].

C. Proposed Algorithm
We have provided a comprehensive and intricately detailed algorithm that navigates through a series of well-defined steps to achieve a superior-performing DNN model.This model is particularly fine-tuned to facilitate the aperture design of PEC enclosures, with a focused emphasis on optimizing EMC performance.Algorithm 1 encapsulates the steps utilized to determine the most optimal DNN model, which will be applied in section III for EMC assessment of the apertures of a PEC enclosure with dimensions of a regular desktop PC.

A. MECA validation
In this section, we provide numerical examples to show the validity of the MECA through comparisons against analytical formula and CST simulations, all at a frequency of 3.6 GHz.Fig. 3(a) shows the initial structure comprised of two square PEC plates with length 2λ, positioned in parallel to the x-y plane at z = ±λ and represented with triangular meshes for processing by MECA; note, wavelength is λ (m).Figures 4(a) and 4(b) compare the RCS at scattered angles ϕ s = π/2 and variable θ s between MECA and the analytical formula for uniform plane wave illumination at incidence angles θ inc = 30 • and θ inc = 90 • , respectively, while ϕ inc = −π/2.The incidence and scattered angles are shown in Fig. 3(d).For validation, we also derive the analytical RCS by combining the scattering equations for a single PEC plate (σ single ) from equations (11.44), (11.38a), and (11.38b) in [14], with the twoelement array factor (AF) for infinitesimal dipoles in [16, sec.6.2], as follows where the phase constant β = 2π/λ (rad/m), and d (m) is the distance between plates.The total RCS is obtained as σ = σ single × |AF|  tube open-ended in z-direction.This configuration may be considered as two sets of parallel plates.For validation, we also derive the analytical RCS formula using a similar approach described above while applying appropriate coordinate transformations; thus the total RCS is given by where a (m) and b (m) are the plate dimensions, and Figures 4(c) and 4(d) compare the RCS from MECA simulations and the analytical formula at ϕ s = π/2, for uniform plane wave illumination at ϕ inc = −π/2 and incidence angles θ inc = 30 • and θ inc = 90 • , respectively.
As evident in Figures 4(a) through 4(d), RCS from MECA is in excellent agreement with analytical formula for the parallel plates and the cubic tube.
Next, we explore a more complex structure of an aperture within a finite PEC plate.We simulate a square plate with a length of 2λ and an aperture size of λ × 0.4λ, as depicted in Fig. 3(c).The plate is illuminated with a uniform plane wave at ϕ inc = −π/2 at two incident angles θ inc .Given the complexity of deriving an analytical formula for this finite structure, we rely on CST simulations to validate the MECA results.Figures 4(e) and 4(f) compare the RCS of MECA and the CST simulations at ϕ s = π/2 for incidence angles θ inc = 30 • and θ inc = 90 • , respectively.
Figures 4(e) and 4(f) show good agreement between MECA and CST simulations, particularly in the regions close to the maximum RCS.The discrepancy in other regions may be attributed to the fact that the CST simulation takes into account the plate's thickness and edge reflections, whereas MECA neglects these factors to achieve faster computation.We can improve the MECA results by incorporating plate thickness and edge reflections, however that would require a higher resolution mesh, increased number of elements, requiring larger RAM and increased computational cost.
The final structure is a PEC cube closed on all six sides with length 5λ, centered at the origin, illuminated with a uniform plane wave at ϕ inc = −π/2 and θ inc = 30.Fig. 5 shows close agreement in RCS results at the ϕ s = π/2 plane between MECA and CST simulation, with negligible discrepancies due to the edge reflection considerations in CST simulations.
Considering the above validation data, we use MECA to simulate approximately 55,000 PC enclosures with dimensions 1.5λ×4λ×3λ operating at a processor frequency of 3.6 GHz, resulting λ = 8.33 cm, and vary aperture attributes, including shape, size, pitch values (i.e., the distance between aperture centers), and quantity at each direction of aperture plane.We collect results of these experiments into a dataset for training of DNN models which will be discussed in the proceeding

B. Results of DNN Models
In this section, we elucidate the process of selecting the optimal architecture and parameters for the DNN models, by presenting results obtained from multiple models with various architectures and parameter configurations.
We begin by exploring different activation functions for the output layer in simple DNN models without hidden layers, and report the validation errors measured as MSE in Table II.It is notable that the validation error is a metric used to assess the performance of the model on a separate validation dataset during the training process, which helps to estimate how well the trained model generalizes to new unseen data.The validation error, which is often represented using a loss function, quantifies the difference between the model's predicted outputs and the true labels in the validation dataset.For example, the validation error in terms of MSE for multi-label regression modeling, which is our case, can be shown as follows [37].
where, N is the total number of samples in the validation dataset, M is the total number of target labels (output dimensions), y ij is the true target value for the i th sample and j th target label, and ŷij is the predicted target value by the DNN model for the same sample and same target label.Considering that the MSE represents the average squared error, and our training data is scaled (normalized) between 0 and 1, we should be aware that even a slight validation error could significantly affect the prediction results.For instance, 2.6% error of relu function in Table II, means that the average error (mean error) is √ 2.6 × 10 −2 = 0.161, which is about 16% of normalized data.Therefore, when we transform the scaled predicted data to non-scaled predicted data, the final predicted pattern of RCS can be significantly affected, even by this small validation error.
Based on the data in Table II, we select the sigmoid activation function for the output layer due to its superior performance.Next, we sequentially add the hidden layers, one by one.After adding the first hidden layer and fine-tuning its parameters (i.e., activation function and number of nodes (N )), the best performance is achieved with relu activation and N 1 = 1000.Results are further improved by adding and tuning a second hidden layer with relu activation and N 2 = 1000.This procedure is repeated multiple times, and the best model architecture is determined to have five hidden layers with N 1 = 1000, N 2 = 1000, N 3 = 2000, N 4 = 2000, N 5 = 2000, and relu and sigmoid activation functions for hidden and output layers, respectively.
Note that further adding nodes and layers degrades the model's performance in this case, which is likely due to overfitting.So far, we have identified the optimal architecture for the DNN model, resulting in significant improvement in performance with MSE = 2.1 × 10 −4 , which is approximately ten times better performing compared to the best performing simple DNN model reported in Table II.
During determination of the model's architecture, we conduct additional experiments to assess the impact of normalization.The input training data for the model consists of various features with different scales.For instance, the aperture length in the x-and z-directions (w and t) are within the range of [ λ 20 , λ 4 ], the pitch (p) values, representing the distance between aperture centers, are in the range of [ λ 10 , λ], the number of apertures in the x-and z-direction (N x and N z ) are constrained respectively within the ranges of [1,15] and [1,30], the RCS values for each θ s are bounded within a minimum and maximum value in dB, and to ensure consistency in data types the aperture shapes are categorized using shape values ranging in 0, 1, 2, 3 corresponding respectively to rectangle, circle, triangle, and diamond shapes.
For training the DNN models with such input data, normalizing the input data to a consistent range, typically between [0, 1] or [−1, 1] helps to prevent certain features from artificially dominating others during the learning process, and ensures that the model's weights and biases are updated more evenly while improving the convergence rate and stability of the training process.Normalization in DNN is typically done column-wise, also known as feature-wise normalization.Each column or feature of the input data is independently scaled to a consistent range based on its own minimum and maximum values.Therefore, for normalizing the independent features such as w, t, p, N x , N z , and shape value to a range of [0, 1], we use a Min-Max Scaler in a column-wise manner.However, since the RCS values for each column (i.e., each θ) may not be independent of other columns, we have the choice of either column-wise or row-wise normalization for these values.
During the tuning of the DNN model architecture, we experiment with both normalization options on a 2-layered DNN model with relu activation function and N 1 = N 2 = 1000.The data shows an improvement in validation error from 1.2×10 −3 to 4×10 −4 by applying row-wise normalization to RCS values, instead of column-wise normalization.This finding leads us to select row-wise normalization for RCS values and columnwise normalization for other independent features during the To identify the optimal loss function for each type of optimizer in our problem, we train the DNN models with all the 160 parameter combinations and compare the validation errors and the training time for different loss functions for each optimizer, as the epoch number increases.These comparisons are shown in Fig. 6.
Further investigation of Fig. 6(a) to 6(e), indicates that different optimizers have varying performance with different loss functions.The data reveal that the Adam optimizer achieves better performance when paired with LogCosh loss function.On the other hand, the Adamax optimizer yields better results when combined with the MSE loss function.Furthermore, the Nadam optimizer exhibits improved performance with the Huber loss function.Finally, the SGD and RMSprop optimizers demonstrate superior performance when utilized in combination with the MAE loss function.Additionally, Fig. 7 presents a comparison of the validation errors and the training time among different optimizers, along with their respective best-performing loss functions.
Table IV shows the validation error, required time, and memory usage of classical ML algorithms such as linear regression, decision tree, and random forest, versus the DNN models with the obtained optimal architectures (comprised of five hidden layers) and different optimizers with their respective best-performing loss functions, at one epoch.

IV. DISCUSSION
We have constructed and compared various DNN models by applying algorithm 1 to predict the RCS of a regular desktop PC containing different aperture configuration.Based on the results in Fig. 7 and Table IV, it is evident that the combination of Nadam optimizer with Huber loss function results in the lowest validation error for our dataset, albeit at the cost of longer training time.There is a trade-off between   computational cost and model performance, therefore selection of a DNN model with the optimal parameter combination depends on the specific dataset and objectives.Based on the desired accuracy in our problem (validation error ≤ 10 −5 ) and the DNN models' performance, we may choose either the Nadam optimizer with the Huber loss function or the Adam optimizer with the LogCosh loss function, and 40 epochs.Our final DNN model parameters are as follows: • Number of hidden layers: 5 • Activation: relu for hidden layers, sigmoid for output The obtained DNN model's architecture and hyperparameters can be explained based on the physics of the problem described in this work.Since the nature of EM fields are continuous, which leads to a continuous RCS pattern, the RCS prediction problem is considered as a regression task rather than a classification problem.Given that we are seeking to correlate RCS value to multiple features, including w, t, p, N x , N z , and shape value, it is apparent that our DNN model will require more than a single layer to effectively learn these complex features and provide the necessary degrees of freedom.A DNN model with five hidden layers provided enough degree of freedom for our problem.Due to the intricate and complex nature of the relationship between aperture configurations and RCS patterns, it is necessary to employ a DNN model with a substantial number of nodes per layer.To establish correlations between the inputs and outputs effectively, we found that uti-  [39], is very effective for datasets with sparse gradients [40].In our specific problem, the variations of RCS values at each scattering angle are heterogeneous, with minimal variations observed at nulls.Consequently, the use of Nadam optimizer is more suitable for effectively handling these sparse gradients and facilitating faster convergence of the model.Moreover, the Huber loss function is well-suited for datasets that contain mix of outliers and normally distributed data points.It strikes a balance between robustness and smoothness, providing an alternative to loss functions like MSE or MAE.
In the context of our problem, where outliers are present in the RCS pattern, particularly as nulls, the Huber loss function effectively reduces the impact of these outliers, leading to a more stable and robust training process [37,Ch. 10].
To find the mentioned best-performing model, we trained 160 DNN models which took approximately 18 hours on 96 cores across two Xeon 6248R CPUs each running at 3.0 GHz [41].Training each DNN model utilizes approximately 14.43 MB of RAM.Using the best-performing DNN model with minimum validation error of 3.1 × 10 −5 , we predict the RCS values of about 10,000 previously unseen test data composed of random aperture configurations, and the average prediction time for each experiment is about 2.6 ms which is very fast compared to any other numerical algorithms.Table V compares the computational costs of proposed DNN model vs. MECA method and CST simulation in terms of calculation time and required memory usage (RAM) for RCS calculation of example presented in Fig. 8.According to Table V, our optimum DNN model outperforms the other methods in terms of prediction time and memory usage, while it retains accuracy with negligible average error of 3.1 × 10 −5 .Fig. 8 shows the normalized predicted RCS pattern versus ground truth (real) RCS pattern of a sample experiment, with four diamond apertures with attributes w = 0.125λ, t = 0.175λ, p = 0.4λ, N x = 2, and N z = 2.As anticipated from the validation error, they are in excellent agreement.

A. Nexus between the DNN and the EM System
Here, we discuss several points of interest in which the electromagnetic system and the resulting dataset impacts our choices in design-optimization of the DNN model.
Column-wise vs. Row-wise Normalization: we are faced with a choice in normalizing the RCS data in either a columnwise or a row-wise manner.Column-wise normalization is more popular in published DNN methodologies, however our experiments reveal that row-wise normalization of RCS data yields better results.The rationale for this lies in the fact that column-wise normalization assumes the features are independent from each other.In contrast, row-wise normalization assumes the existence of inter-feature connections, thus applying row-wise normalization to independent features may create additional spurious neural connections and ultimately results in inferior DNN performance.
Considering EM wave dynamics, we note that RCS values at different θ are strongly correlated.Column-wise normalization adversely interferes with this inherent correlation and leads to degraded DNN performance.In our case, there is dependency between RCS values across θ s , thus choosing row-wise normalization for RCS values and column-wise normalization for other independent features (i.e., aperture attributes) leads to improved DNN performance.
Forward vs. backward propagation: Our DNN formulation uses a forward propagation mode (FPM), that is the DNN model's inputs are the attributes of the apertured enclosure, and its output is the RCS(θ).Our DNN model is optimized for the FPM; however, a backward propagation mode (BPM) may also be of interest, where the RCS(θ) is provided as input and the aperture attributes are the output of the DNN model.It is worth noting that the FPM-optimized DNN model may not be automatically optimized for operation in the BPM without retraining (i.e., backward training) and tuning to find the optimal DNN model.In fact, comparing the validation errors of FPMoptimized DNN has a val.error = 3.1 × 10 −5 whereas the same DNN operated in the BPM has a val.error = 3.9×10 −2 .Additionally, we find that the FPM is relatively easier to train than the BPM.This is likely due to two main factors: (a) the inputs of FPM (i.e., aperture attributes) are independent while the outputs (i.e., RCS values) are strongly correlated, whereas the inputs of the BPF (i.e., RCS values) are strongly correlated while the outputs (i.e., aperture attributes) are uncorrelated.Therefore it is reasonable that the ML algorithm should find it more challenging to predict the outputs of BPM, and (b) each experiment in FPM has only 6 input parameters, whereas the BPM has 181 input parameters for RCS constructed at onedegree increments in the range [0, 180].

B. Alternative Varieties of Neural Networks
In general, three commonly used and popular types of neural networks are Deep Neural Network (DNNs), Recurrent Neural Networks (RNNs), and Convolutional Neural Network (CNN).Each of these networks is specifically designed with unique characteristics to serve different purposes.DNNs are the simplest type among them, consisting of an input layer, one or more hidden layers, and an output layer.Information flows in a unidirectional manner, from the input layer to the output layer.DNNs are well-suited for handling structured data, performing complex nonlinear transformations, where each input is processed independently without considering temporal relationships.On the other hand, CNNs are specifically designed to handle grid-like data, such as images.They employ convolutional layers to extract features and pooling layers to reduce spatial dimensions.Typically, multiple convolution and pooling layers are applied alternately, with each convolution layer followed by a pooling layer and vice versa.CNNs excel in computer vision tasks like image classification, object detection, and image segmentation, and they are able to automatically learn hierarchical representations of visual features.In contrast to traditional neural networks with independent layers, RNNs possess the ability to leverage sequential information.This makes them highly suitable for processing data that has a sequence, such as natural language processing, speech recognition, and time series analysis.RNNs are equipped with recurrent connections that allow them to capture information from previous steps in the sequence.Additionally, their capability to retain internal memory proves invaluable for tasks involving context and temporal dependencies.In summary, DNNs are versatile networks for structured data processing, RNNs excel in sequential data processing with temporal dependencies, and CNNs are specialized for grid-like data processing in computer vision tasks.It is worth mentioning that CNNs and RNNs have more intricate architectures compared to DNNs, which typically leads to an increased requirement for longer training times [42]- [44].
As our MECA-generated dataset is structured without any temporal dependency and lacks grid-like data, we initially employed DNN modeling, which served as a fundamental neural network architecture and yielded satisfactory outcomes.Nevertheless, considering the interdependence of inputs during backward propagation, we may require exploring the other neural networks such as RNNs to achieve the desired results in this case, which is considered as potential future research for the authors.

V. CONCLUSION
We developed an optimal DNN model with relevant parameters by following key steps.(a) We implemented the modified equivalent current approximation (MECA) method [15]  It is important to highlight that this study goes beyond mere hyperparameter optimization of a DNN model.Instead, it involves the design of multiple DNN models with different architectures, fine-tuning each model to determine their optimal parameters, and ultimately identifying the best-performing model among the optimized models.While optimization algorithms are typically employed to tune the hyperparameters of a predefined DNN network, this work starts with undefined model, undertakes the creation of several DNN models, fine-tunes each model to determine the optimal parameters, compares the various optimized models, and selects the most effective one.Furthermore, it is worth noting that this work introduces a novel concept of employing DNN models for EMC analysis of shielding enclosures through RCS prediction via training with a MECA-generated dataset, enabling rapid simulation of a large number of examples.
The optimal DNN model outperforms any other traditional numerical method in computational efficiency, especially for electrically large problems.The proposed DNN modeling methodology provides a powerful and efficient alternative tool for analysis and design-optimization of EMC performance in apertured enclosures.Some potential topics for future research include: (a) exploring the optimal DNN model for aperture attributes involving different aperture shapes with non-uniform sizes, (b) exploring DNN model parameters to find the optimal DNN for backward propagation mode, and (c) applying the proposed methodology to the problem of susceptibility of electronics inside shielding enclosures being affected by sources outside the enclosure.

2 Fig. 2 :
Fig. 2: Example of the DNN structure used in our work.

Step 11 :
Fig.3(b)  shows the next structure which is a square PEC tube open-ended in z-direction.This configuration may be considered as two sets of parallel plates.For validation, we also derive the analytical RCS formula using a similar approach described above while applying appropriate coordinate transformations; thus the total RCS is given by

Fig. 3 :
Fig. 3: The structures being illuminated by a uniform plane wave are: (a) two parallel plates, (b) cubic tube, (c) an aperture inside a finite plate, where all plates are assumed PEC.(d) illustration of incidence and scattering angles.

Fig. 7 :
Fig. 7: Comparison of optimizers with optimal loss functions in terms of: (a) Validation error, (b) Required training time.
as an alternative to numerical field solvers for computing the radiated emissions from apertured enclosures.(b) We used MECA to simulate approximately 55,000 experiments with varying aperture attributes to study the EMC performance of airflow apertures in desktop PCs operating at a processor frequency of 3.6 GHz.(c) We used the MECA-generated RCS data to train, validate, and test numerous DNN models across number of layers, number of nodes (neurons) per layer, activation function, optimization algorithm, loss function, batch size, and epoch.(d) We identified the optimal DNN model based on accuracy (MSE), compute time, and RAM usage.(e) We demonstrated the DNN model's performance in computing RCS data for several examples.

TABLE I :
Information of generated dataset.

TABLE II :
Validation errors of output layer's activation function in simple DNN models.

TABLE III :
DNN model error vs. batch size variation.After establishing the optimal DNN model architecture, we conduct further experiments to investigate the impact of batch size on model performance.Table III presents the validation error and the training time for the aforementioned optimal DNN model with five hidden layers, when trained with different batch sizes while keeping the optimizer and loss function unchanged at one epoch.Table III demonstrates that a batch size of 50 is a reasonable choice, considering the training time and accuracy of the model.So far, we have determined the appropriate normalization type, DNN model architecture, and batch size.The next step involves selecting the optimizer, loss function, and the epoch number.We pre-select five optimizers, namely Adam, Adamax, Nadam, SGD, and RMSprop, along with four types of loss functions, including Huber, LogCosh, MSE, and MAE.Additionally, we consider eight different epoch numbers {1, 2, 4, 10, 20, 30, 40, 50}.

TABLE IV :
Classical vs. DNN models: various optimizers and optimal loss functions.

TABLE V :
Computational costs for RCS prediction methods.=N 2 =1000 and N 3 =N 4 =N 5 =2000 provided sufficient capacity for the DNN model.Although it is possible to increase the number of nodes per layer beyond these values, such increments do not yield significant improvements in accuracy.Instead, they substantially increase the required training time without proportional gains in performance.Due to the substantial number of nodes per layer, is beneficial to not activate all the neurons (nodes) at the same time for computational efficiency.The main advantage of using the relu function over other activation functions is that it does not activate all the neurons at the same time.This makes relu as the best choice for the activation function of hidden layers, allowing the model to converge very quickly.Nadam, a combination of Adam with Nesterov accelerated gradient (NAG)