Measurement-Based Neural Network Technique for Modeling the Low-Frequency Electric Field Radiated Behavior of Satellite Units

Abstract

In this work, a machine learning technique employing a measurement-driven artificial neural network (ANN) architecture is proposed as a solution to the precise determination of the position and the moment of equivalent electric dipoles for unit characterization. These dipoles are used to match the generated electric field from various sources inside the spacecraft, during space exploration missions. Various methodologies for unit characterization have been proposed in the literature, the most common being the heuristic approaches, least squares variants, method of auxiliary sources, etc. Contrary to the previous time-consuming post-process methodologies, the proposed electric dipole neural network (EDMnet) can offer a real-time characterization of the measured unit (Device Under Test) after a proper training stage, especially as a fast pre-compliance method. The network uses the electric field vector, measured at 14 discrete locations, as input and reports the position and moment of the electric dipole that best matches the measured fields. In this work, various ANN architectures are tested and compared in order to select the optimal EDMnet parameters for accurate source identification. It is shown that the size of the artificial training data affects the performance of the network. The proposed EDMnet can provide accuracy in mm-scale, with respect to dipole positioning, greater than 99% in dipole moment prediction.

1. Introduction

Space missions have always been at the forefront of scientific and technological advancement. These missions, whether they involve satellite deployments, deep space explorations, or future manned missions to the Moon and Mars, require extremely high levels of precision and reliability. One of the critical factors ensuring the success of these missions is maintaining electromagnetic cleanliness in instrument locations for accurate and noise-free data acquisition. Electric cleanliness refers to the state of minimizing or eliminating unwanted electric fields and currents within a spacecraft or satellite. These unwanted electric phenomena can arise from various sources, including internal electric components, the spacecraft’s interaction with the space environment, or even other onboard systems. Maintaining electric cleanliness is crucial because electromagnetic interference (EMI) can disrupt the operation of sensitive instruments and systems, leading to malfunctions or failures that could jeopardize the entire mission [1]. Moreover, the extremely low-frequency (quasi-static) electric field emitted by planetary and various objects, is measured by specialized instrumentation that presents innate sensitivity to electric fields. In order to provide accurate field measurements and capture the electric signature of these celestial bodies, it is necessary to operate in zero-field conditions. Thus, it is required to measure, model, and minimize the EMI generated from the equipment inside the spacecraft or satellite to the sensor’s position at these frequencies [2,3]. Many space programs, such as Bepi Colombo, Juice, and Solar Orbiter, have studied and developed methods to adhere to the strict electric cleanliness requirements. In these methods, the electric field measurements are carried out either at the system level, providing the electric signature of the entire spacecraft, or at the unit level, with each spacecraft unit measured and modeled separately [4,5,6,7].

When the measurements are carried out at the unit level, the quasi-static electric near-field signatures from all individual units are utilized to evaluate the total electric field. Field probes are placed in various angles/orientations near the unit under test in order to capture its spatial electric signature in the frequency range of interest [2].

A common method to characterize the unit from the collected data is multiple dipole modeling (MDM). MDM relies on the fact that the measured signature can be reproduced from a set of virtual dipoles assigned to the measured unit and by solving in this way an inverse electromagnetic problem. In the low frequencies, usually, one dipole is sufficient to accurately express the unit behavior. The parameters that are required to describe the dipole source, for the electric field in this case, are the electric moment vectors and the electric dipole position vectors inside the unit spatial boundaries [6].

The MDM method can be implemented by either deterministic or heuristic techniques, which predict the electric signature with adequate accuracy [3,8]. These techniques, for simple dipole sources, have a satisfactory convergence speed; however, they are post-process methodologies due to the time that the algorithm needs to explore the solution space and converge to the best candidate solution. As the number of sources comprising the unit increases, so does the number of unknowns, and thus the solution speed gradually decreases [6,9].

In recent years, the advent of advanced computational techniques, particularly neural networks, has opened new avenues for addressing the challenges of electric cleanliness in space missions [10,11,12]. Neural networks, a subset of artificial intelligence (AI) and machine learning (ML), are computational models inspired by the human brain’s structure and function. They excel in pattern recognition, learning from data, and making predictions, making them highly suitable for complex, data-intensive tasks.

Neural networks have been successfully applied in various domains, such as image and speech recognition, natural language processing, and autonomous driving. Their ability to model complex relationships and learn from large datasets makes them a powerful tool for addressing the intricacies of electric cleanliness in space missions. By leveraging neural networks, researchers and engineers can develop more efficient and effective methods for predicting, monitoring, and mitigating electric interference in real-time scenarios.

In this work, neural networks (Electric Dipole Model Networks—$EDMnet$) were used to approximate the inverse function of the MDM electric field problem, as they constitute an efficient method of solving complex and non-linear equations. The problems of (i) the electromagnetic characterization of actual space units and (ii) the development of accurate models able to virtually represent these actual units with equivalent dipole sources, are addressed herein. This is very important at the early steps of a space mission design, in order for the engineers to optimally locate the equipment on the spacecraft hull, ensuring minimum electromagnetic interference issues. In this way, (i) a pre-compliance process was taken for each unit under test and (ii) dedicated measurements (14 measurements for different angles and unit orientations) were performed according to the referred herein process. Using this measurement dataset the solution of the inverse electromagnetic problem yields the best dipole that accurately represents the unit’s electromagnetic behavior; i.e., matches the measurement dataset. As stated, this solution is obtained either heuristically or statistically. Both these methods are post-process, due to their time-consuming nature. Additionally, this work presents a machine learning methodology (ANN), as depicted in Figure 1, to find the optimum inverse function for the specific measurement setup based on simulated data. Thus, when the ANN is optimally trained for the specific measurement setup, the inverse calculation (for the actual measurement dataset) is a matter of nanoseconds, instead of a couple of minutes needed for the heuristic approach.

Neural networks can learn to approximate functions of arbitrary complexity based on training data, i.e., the inputs and outputs of the function. The neural network training was performed by analyzing artificially generated datasets from the electric field equation in vector form at various positions, as discussed in the next section. The inputs of the proposed $EDMnet$ were the electric field vector values, for the specific measurement setup, and the outputs were the parameters of the dipole sources (electric dipole position, and electric moment vectors). During training, the weights of the network are evaluated iteratively in order to minimize a predefined loss function that compares the target (artificial dipole) and predicted output values, i.e., predicted dipole parameters. In contrast to heuristic approaches, in which the exploration of a solution space is time-consuming as the algorithm attempts iteratively to identify the set of dipole parameters that best fit the measurement dataset, a pre-trained neural network offers a direct, instant, real-time evaluation of the solution (straightforward calculation of the dipole parameters) during the measurement campaign, enabling system level predictions regardless of the system architecture complexity. The core idea of this work is that for a custom measurement setup, such as the one presented herein and studied in the authors’ previous work [2], an ANN (i.e., the inverse dipole field equation for the specific measurement setup) pre-trained with simulated data, can instantly and accurately predict the radiating source based on field measurements. Moreover, in a continuous measurement data stream the ANN architecture is capable of providing a direct stream of results. It should be noted that for different measurement setups and boundary conditions (i.e., measurement points and variables domain) other ANNs should be re-trained entirely, starting with the generation of new simulated data, since those would be totally different problems. In the following sections, the measurement methodology is briefly presented along with the ANN architecture and results comparison.

2. Materials and Methods

2.1. Mathematical Formulation

Electric Field Representation

In the quasi-static regime, the frequencies under examination take values in the sub-MHz range, which is the main frequency area of interest for planetary science space missions. In this way, close to the spacecraft area, the near-field approximation on dipole field equations is valid. The electric field produced by an electric dipole oscillating at a single frequency, located at (x, y, z), is expressed in cartesian representation (electric field components) at the measurement point (x_m, y_m, z_m), with $\mathrm{r}$ as the magnitude of the distance vector between the measurement point and dipole location, as in [2]:

$${\mathrm{E}}_{\mathrm{x}}={\displaystyle \frac{1}{4\mathsf{\pi }{\mathsf{\epsilon }}_{\mathsf{o}}}}\left[{\displaystyle \frac{3\left({\mathrm{x}}_{\mathrm{m}}-\mathrm{x}\right)·\mathrm{C}}{{\mathrm{r}}^5}}-{\displaystyle \frac{{\mathrm{p}}_{\mathrm{x}}}{{\mathrm{r}}^3}}\right]$$

(1)

$${\mathrm{E}}_{\mathrm{y}}={\displaystyle \frac{1}{4\mathsf{\pi }{\mathsf{\epsilon }}_{\mathsf{o}}}}\left[{\displaystyle \frac{3\left({\mathrm{y}}_{\mathrm{m}}-\mathrm{x}\right)·\mathrm{C}}{{\mathrm{r}}^5}}-{\displaystyle \frac{{\mathrm{p}}_{\mathrm{y}}}{{\mathrm{r}}^3}}\right]$$

(2)

$${\mathrm{E}}_{\mathrm{z}}={\displaystyle \frac{1}{4\mathsf{\pi }{\mathsf{\epsilon }}_{\mathsf{o}}}}\left[{\displaystyle \frac{3\left({\mathrm{z}}_{\mathrm{m}}-\mathrm{x}\right)·\mathrm{C}}{{\mathrm{r}}^5}}-{\displaystyle \frac{{\mathrm{p}}_{\mathrm{z}}}{{\mathrm{r}}^3}}\right]$$

(3)

where

$$\mathrm{C}={\mathrm{p}}_{\mathrm{x}}·\left({\mathrm{x}}_{\mathrm{m}}-\mathrm{x}\right)+{\mathrm{p}}_{\mathrm{y}}·\left({\mathrm{y}}_{\mathrm{m}}-\mathrm{y}\right)+{\mathrm{p}}_{\mathrm{z}}·\left({\mathrm{z}}_{\mathrm{m}}-\mathrm{z}\right)$$

(4)

and then the total electric field magnitude for every measurement point is [2]:

$$\left|{\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}}\right|=\sqrt{{{\mathrm{E}}_{\mathrm{x}}}^2+{\mathrm{E}}_{\mathrm{y}}^2+{{\mathrm{E}}_{\mathrm{z}}}^2}$$

(5)

Equations (1)–(3) are used in data generation procedure, as described in the next section, in order to calculate the electric field components E_X, E_y, and E_z ($\mathrm{E}\mathrm{D}\mathrm{M}\mathrm{n}\mathrm{e}\mathrm{t}$ inputs) in the 14 measurement points.

2.2. Methodology

2.2.1. Measurement Setup

The proposed $EDMnet$ is targeted to provide rapid unit characterization for units measured according to [13] i.e., for pre-compliance purposes. In this measurement, setup the electric field emissions of the unit were captured in 14 distinct measurements. A measurement dataset is capable, according to previous studies of the authors, of providing the necessary information for the algorithm to solve accurately the inverse problem. Measurement procedure is briefly presented in the next paragraph and the graphical representation is depicted in Figure 2.

In [13], 6 of the 14 measurements were carried out by rotating the unit under test with an angular step of 60° and clockwise. Then the unit was rotated 90° around the x-axis and the next 8 measurements were performed starting with an angular offset of 30° and rotating the unit clockwise with a step of 45°. The fixed electric field probe position was ($x_m,y_m,z_m$) = (1, 0, 0). Figure 2 depicts this measurement topology with all the parameters and derails as previously studied in [13].

In order to train the proposed $\mathrm{E}\mathrm{D}\mathrm{M}\mathrm{n}\mathrm{e}\mathrm{t}$, the artificial dataset was populated with randomly generated dipoles located in a predefined volume (unit), and the corresponding field component values were calculated according to the measurement procedure discussed herein.

2.2.2. Artificial Data Generation

Specifically, the unit under test is considered to have a cuboid geometrical shape, with dimensions $L\times W\times H$ ($0.5 \mathrm{m}\times 0.5 \mathrm{m}\times 0.5 \mathrm{m}$). So in order to generate random dipoles, the six necessary parameter values were drawn via a uniform distribution bounded in the intervals $[-L/2,L/2]$, $[-W/2,W/2]$, and $[0,H]$ for the x, y, and z dipole positions and in the interval $\left[{-p}_{max},p_{max}\right]$ with $p_{max}=5·e^{-14}C·m$ for the electric moment [6]. With this in hand, employing Equations (1)–(3), the corresponding $3\times 14$ electric field values ($\mathrm{E}\mathrm{D}\mathrm{M}\mathrm{n}\mathrm{e}\mathrm{t}$ inputs) were calculated for each dipole. These field values, along with the 3 electric dipole position values and the 3 electric dipole moment values (6 $\mathrm{E}\mathrm{D}\mathrm{M}\mathrm{n}\mathrm{e}\mathrm{t}$ output values), assembled a 48-element vector as a sample of the dataset.

3. Simulation Results and Discussions

3.1. Artificial Neural Network Setup

Different datasets with 10,000, 20,000, 40,000, and 100,000 samples were initially created in order to train the artificial neural networks. 90% of each dataset was used as training data, and the remaining 10% was used as test data to evaluate the performance of the neural network. Moreover, the training data were further divided into 90% training data, 5% test data, and 5% validation data. Initially, a set of 10,000 data samples was selected to evaluate the $\mathrm{E}\mathrm{D}\mathrm{M}\mathrm{n}\mathrm{e}\mathrm{t}$ architecture. According to the previous analysis, the $EDMnet$ was configured with 42 neurons at the input layer and 6 neurons at the output layer. With the intention of determining the most appropriate ANN for this problem, a large number of simulations were performed, taking into account various parameters, i.e., the number of hidden layers, the number of neurons per layer, the type of training algorithm, the type of training transfer function at each layer, and various learning rates. Simulations were performed in a MATLAB environment using the NN toolbox.

The optimal number of hidden layers was found to be four, with 580, 280, 180, and 80 neurons at each layer, respectively, and the most appropriate training algorithm was Resilient Backpropagation (trainrp). After testing all available transfer functions, the best results were obtained by applying the logistic sigmoid transfer function (logsig), which is expressed by f(x) = 1/(1 + e^−x), to the first and fourth hidden layers. The normalized radial basis transfer function (radbasn) was selected for the second and third hidden layers. This is equivalent to radbasn given by f(x) = e^−x/2, except that output vectors were normalized by dividing by the sum of the pre-normalized values [14]. Moreover, the linear transfer function was applied to the output layer. Also, simulations were performed for two learning rates (0.01, 0.001) and the best results were given by the learning rate $LR=0.01$. In Figure 3 the neural network architecture is presented [6].

The performance during the training procedure for all neural network candidates was evaluated via the MSE (Mean Squared Error). At this point, it should be mentioned that the present problem had neural networks with multi-element outputs with different ranges. In order to avoid prioritizing the relative accuracy of the outputs with the larger range, the parameter ‘normalization’ of the MSE was set to ‘percent’. By this adjustment, errors were normalized between −1 and 1 [15].

A brief description of the methodology followed in this work is presented through the flow chart in Figure 4. In the left block, enclosed in the dashed lines, the steps followed from the measurement setup to the training data creation are depicted. Thus, initially, the 14 measurement points were defined and the equations of the electric field were formulated. Then, the size of the EUT (equipment under test) was determined, and based on this, the limits of the position of the electric dipole. At the same time, the appropriate boundaries of the electric moments p_x, p_y, and p_z were selected. Hence, through the aforementioned steps, the artificial data were generated in a MATLAB R2022b environment. These data, as shown in the right block of the flow chart, were fed into neural networks, where the electric field values were the inputs and the six dipole parameters (x, y, z, p_x, p_y, p_z) were the target outputs. Afterward, the necessary adjustments to the neural network were carried out, such as the selection of the appropriate training algorithm, the activation function, and the performance function. The next step was the definition of the number of hidden layers as well as the number of neurons. Then the trainings were performed. It should be mentioned that for each specific number of hidden layers, several neuron number tests were performed in order to find the optimal number of neurons in each hidden layer. When the training and test error goal was achieved, the training was stopped and the neural network solution and results were stored; otherwise, a different number of hidden layers and neurons were tested.

3.2. Neural Network Performance Evaluation

The performance evaluation of the neural networks was carried out by using statistical measures such as Mean Absolute Error ($\mathrm{M}\mathrm{A}\mathrm{E}$) for the electric dipole position and for the corresponding electric moments along with the Relative Error ($\mathrm{R}\mathrm{E}$) for each vector of the electric field ($\mathrm{E}x,\mathrm{E}y,\mathrm{E}z$) at the 14 measurement points across all of the training and testing data samples for all scenarios, as follows:

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{\mathrm{N}}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\left|{\mathrm{s}}_{\mathrm{n}}-{\mathrm{g}}_{\mathrm{n}}\right|$$

(6)

$$\mathrm{R}\mathrm{E}={\displaystyle \frac{\sqrt{{\sum }_{\mathrm{n}=1}^{\mathrm{N}}{\left({\mathrm{E}}_{\mathrm{n},\mathrm{p}\mathrm{r}}-{\mathrm{E}}_{\mathrm{n},\mathrm{a}\mathrm{c}\mathrm{t}}\right)}^2}}{\sqrt{\sum _{\mathrm{n}=1}^{\mathrm{N}}{\mathrm{E}}_{\mathrm{n},\mathrm{a}\mathrm{c}\mathrm{t}}^2}}}$$

(7)

Regarding $MAE$ loss function, $n=1,2,\dots ,N$ stands for the number of the test or training data samples, $s_n$ is the EDMnet’s predicted values of the electric dipole parameters, and $g_n$ is the corresponding artificially generated ones. Moreover, as a second metric for better understanding the effect of the deviations of the dipole parameter values on the loss function ($MAE$), the $RE$ metric for the field values was also calculated for each measurement point. Thus, for $k=1,2,\dots ,14$ and for each sample, the electric field vector $E_{n,pr}$ was calculated from the EDMnet’s predicted values ($x,y,z,p_x,p_y,p_z)$ at the 14 measurement points and $E_{n,act}$ the corresponding electric field vector from the artificially generated datasets.

3.3. Verification Results

Table 1. Test Mean Absolute Error for the different datasets-learning rate $LR=0.01$.

Data Sample Size	Test MAEx / MAEy / MAEz (m)	Test MAEpx / MAEpy / MAEpz $(\times {10}^{-16}\mathbf{C}·\mathbf{m}$)
10,000	0.0024 / 0.0023 / 0.0026	2.6177 / 2.9975 / 2.8126
20,000	0.0016 / 0.0017 / 0.0017	2.2935 / 2.2030 / 2.2338
40,000	0.0016 / 0.0017 / 0.0016	2.0314 / 2.1723 / 2.0643
100,000	0.0016 / 0.0016 / 0.0016	2.2639 / 2.6554 / 2.2791

below tabulates the best simulation results obtained with the network settings mentioned in the previous section, for all different datasets (10,000, 20,000, 40,000, and 100,000 data samples). Specifically, the loss function (MAE) results for the $x,y,z$ electric dipole position and the $p_x,p_y,p_z$ electric moments for corresponding test datasets (10,000, 20,000, 40,000, and 100,000 data samples) and the learning rate $LR=0.01$ is presented.

Table 1 indicates that the best overall results were obtained when the 40,000 samples dataset was applied. As mentioned in Section 2.2.2, the electric dipole position variables x, y, and z were bounded in the intervals [−0.25, 0.25] m, [−0.25, 0.25] m, and [0, 0.5] m, respectively, and thus the reached Mean Absolute Errors of 0.0016 m, 0.0017 m, and 0.0016 m presented in Table 1 for the 40,000 data samples case constitute acceptable and satisfactory position errors. Correspondingly, in the case of the electric moments p_x, p_y, and p_z, which are all bounded in the interval $\left[-5·e^{-14}, 5·e^{-14}\right]\mathrm{C}·\mathrm{m}$, the achieved Mean Absolute Errors are 2.0314$·{10}^{-16}\mathrm{C}·\mathrm{m}$, 2.1723$·{10}^{-16}\mathrm{C}·\mathrm{m}$ and 2.0643$·{10}^{-16}\mathrm{C}·\mathrm{m}$ for each moment parameter, which are also satisfactory training outcomes.

Table 2. Test results for various hidden layers and neurons in the case of 40,000 data samples.

[L1 L2 L3 L4 L5]	Test MAEx / MAEy / MAEz (m)	Test MAEpx / MAEpy / MAEpz $(\times {10}^{-16}\mathbf{C}·\mathbf{m}$)
[550 250 50]	0.0027 / 0.0030 / 0.0029	3.4642 / 3.5243 / 3.6122
[350 150 100 50]	0.0025 / 0.0025 / 0.0026	3.3937 / 3.5066 / 3.5379
[580 280 180 80]	0.0016 / 0.0017 / 0.0016	2.0314 / 2.1723 / 2.0643
[470 370 270 170 70]	0.0018 / 0.0019 / 0.0019	2.3513 / 2.4934 / 2.3510
[350 250 150 100 50]	0.0018 / 0.0019 / 0.0018	2.5087 / 2.5930 / 2.4311

presents the best results achieved in the case of 40,000 data samples for various numbers of hidden layers and neurons at each layer. As can be observed, the MAE criterion for all the variables was reduced when four and five hidden layers were applied to the neural network structure, and the optimal solution was given for four hidden layers. This way, the minimization of MAE yielded the optimal configuration.

Table 3. Results of the learning rates LR = 0.01, LR = 0.001 for the optimal neural network structure (580 280 180 80) and for the 40,000 dataset.

Learning Rate	Test MAEx / MAEy / MAEz (m)	Test MAEpx / MAEpy / MAEpz $(\times {10}^{-16}\mathbf{C}·\mathbf{m}$)
0.01	0.0016 / 0.0017 / 0.0016	2.0314 / 2.1723 / 2.0643
0.001	0.0019 / 0.0019 / 0.0018	2.3739 / 2.4802 / 2.3651

presents the results obtained by applying during the training procedure the learning rates $LR=0.01$, $LR=0.001$ in the optimal neural network architecture of 4 layers and 580, 280, 280, and 80 neurons at each layer, respectively, for the 40,000 data case. The learning rate $LR=0.01$ contributed to obtaining better overall results.

The test relative error of the electric field vector at the 14 measurement points and for the case of 40,000 data (4000 test data) and learning rate $LR=0.01$ is depicted in Figure 5. From this Figure, it is readily apparent that the electric field vector can be accurately reproduced from the EDMnet’s predicted values ($x,y,z,p_x,p_y,p_z)$, as the relative error between the generated and predicted values at the 13 from the 14 measurement points is less than 2%.

Figure 6 illustrates the deviation between the test generated and calculated (output of the neural network) values of the position and electric moment, of the test samples (10%) for the case of 40,000 data samples and the learning rate $LR=0.01$. Figure 7 presents the corresponding differences in the electric moment values. It should be noted that in the majority of the test samples, the difference between the predicted and the initially generated samples’ position is smaller than 5 mm, while more than 40% of the test samples had a maximum difference in position of 2 mm per axis. Moreover, the difference in the electric moments of the generated dipoles and the predicted ones was also smaller than $5\times {10}^{-16}\mathrm{C}·\mathrm{m}$, while 40% of the test sample had a difference of less than $2\times {10}^{-16}\mathrm{C}·\mathrm{m}$, rendering a notable accuracy.

Moreover, a complexity calculation was performed indicatively for (i) the DE algorithm of [2], and (ii) the proposed ANN, according to [16]. For a fully connected feedforward network (also known as a multi-layer perceptron), the time complexity of the inference process is O(N), where N is the total number of weights in the network. This is because each weight participates in exactly one multiplication and one addition operation during the forward propagation. This way, assuming that all layers have the maximum number of 580 neurons, the worst-case complexity of our network is O(L × n × m) = 97,440, where L is the number of layers, n is the number of neurons, and m is the input dimension.

On the other hand, for the DE, the time complexity is O(D × NP × GMAX)= 2,688,000, where D is the input dimension (3 × 14 = 42), NP is the population number (80), and GMAX is the number of iterations (800). Based on these worst-case calculations, the proposed ANN achieves more than 25 times faster source prediction than the DE algorithm.

Table 4. Comparison results between the 10 first values of the generated and calculated values of x, y, and z electric dipole position of the test sample (of 40,000 data samples).

x Dipole Position Generated (m)	x Dipole Position Calculated (m)	y Dipole Position Generated (m)	y Dipole Position Calculated (m)	z Dipole Position Generated (m)	z Dipole Position Calculated (m)
0.0470	0.0484	−0.2232	−0.2202	0.3311	0.3324
−0.0327	−0.0328	0.0078	0.0086	0.3529	0.3521
0.1072	0.1060	−0.1533	−0.1527	0.2686	0.2687
0.0756	0.0755	−0.0037	−0.0033	0.3827	0.3834
−0.2035	−0.2034	−0.0147	−0.0172	0.2883	0.2925
0.1973	0.1999	−0.0176	−0.0153	0.1350	0.1319
0.0501	0.0512	0.2456	0.2536	0.4867	0.4886
0.1801	0.1800	−0.1872	0.1871	0.2836	0.2857
−0.0687	−0.0676	−0.2210	−0.2178	0.2271	0.2263
−0.0458	−0.0464	0.1549	0.1542	0.3903	0.3931

and

Table 5. Comparison results between the 10 first values of the generated and calculated values of p_x, p_y, and p_z electric dipole moment of the test sample (of 40,000 data samples).

px Electric Dipole Moment Generated $(\mathbf{C}·\mathbf{m}$)	px Electric Dipole Moment Calculated $(\mathbf{C}·\mathbf{m}$)	py Electric Dipole Moment Generated $(\mathbf{C}·\mathbf{m}$)	py ELECTRIC Dipole Moment Calculated $(\mathbf{C}·\mathbf{m}$)	pz Electric Dipole Moment Generated $(\mathbf{C}·\mathbf{m}$)	pz Electric Dipole Moment Calculated $(\mathbf{C}·\mathbf{m}$)
3.4288 × 10−14	3.4236 × 10−14	−8.3500 × 10−15	−8.4496 × 10−15	3.7943 × 10−14	3.8256 × 10−14
3.3400 × 10−15	3.2049 × 10−15	3.1184 × 10−14	3.1345 × 10−14	−3.5865 × 10−14	−3.5803 × 10−14
−1.9000 × 10−15	−1.6746 × 10−15	4.4120 × 10−15	4.4138 × 10−15	2.3214 × 10−15	2.3311 × 10−14
3.8461 × 10−14	3.8419 × 10−14	−1.5352 × 10−14	−1.5187 × 10−14	−2.2280 × 10−14	−2.2396 × 10−14
1.9769 × 10−14	1.9707 × 10−14	−4.7800 × 10−15	−4.8572 × 10−15	4.4728 × 10−14	4.4841 × 10−14
4.7139 × 10−14	4.7292 × 10−14	−3.7521 × 10−14	−3.7648 × 10−14	−4.6177 × 10−14	−4.6269 × 10−14
2.5831 × 10−14	2.6678 × 10−14	3.9439 × 10−14	3.8626 × 10−14	−3.8829 × 10−14	−4.1089 × 10−14
−8.7290 × 10−15	−8.5717 × 10−15	4.3612 × 10−14	4.3438 × 10−14	1.8381 × 10−14	−1.8322 × 10−14
2.2363 × 10−14	2.2490 × 10−14	4.4925 × 10−14	4.4639 × 10−14	−3.0269 × 10−14	−3.0059 × 10−14
−9.2940 × 10−15	−9.2646 × 10−15	−2.0527 × 10−14	−2.0534 × 10−14	−2.3873 × 10−14	−2.3781 × 10−14

present the generated and the calculated (by the neural network) 10 first values of x, y, and z electric dipole position values of the test sample for the case of 40,000 data samples and for the optimal neural network structure with four layers and 580, 280, 180, and 80 neurons at each layer. Τhe aforementioned errors in Table 1 can be easily observed in the presented examples.

Next, Figure 8 depicts the neural network performance (Mean Square Error) for the training, test, and validation data during the training. As can be observed, training was completed in 76,102 epochs and the three curves (training, validation, and test) converge almost in the same value of the Mean Square Error.

4. Conclusions and Future Works

In this work, authors deal with the problem of electromagnetic cleanliness, targeted but not limited to space missions from the perspective of a fast pre-compliance test measurement setup with the aid of AI and ML algorithms. For this technique, various architectures were tested in order to find the best configuration of the neural network. The results of the training and testing procedure of this neural network for various datasets were presented with the mandate to solve the inverse electromagnetic problem and thus allocate an equivalent dipole source to a device under test subjected to a predefined measurement process. This new perspective, in the state of the art of this domain, relies on the fast response of a pre-trained NN, in comparison to the post-process methodology that the heuristic or deterministic algorithms offered until now. This methodology offers fast and accurate unit characterization, during the unit’s measurement campaign, as the first step to system electromagnetic cleanliness.