Runze Hu, Vikass Monebhurrun, Ryutaro Himeno, Hideo Yokota, Fumie Costen
©SHUTTERSTOCK.COM/RANJITH RAVINDRAN
Artificial neural networks (ANNs) have appeared as a potential alternative for uncertainty quantification (UQ) in the finite difference time-domain (FDTD) computation. They are applied to build a surrogate model for the computation-intensive FDTD simulation and to bypass the numerous simulations required for UQ. However, when the surrogate model utilizes an ANN, a considerable number of data are generally required for high accuracy, and generating such large quantities of data becomes computationally prohibitive. To address this drawback, a number of adaptations for ANNs are proposed, which additionally improves the accuracy of ANNs in UQ for the FDTD computation while maintaining a low computational cost. The proposed algorithm is tested for application in bioelectromagnetics, and considerable speed up, as well as the improved accuracy of UQ, is observed compared to traditional methods such as the nonintrusive polynomial chaos (NIPC) method.
The FDTD method [1], [2] is a well-proven technique for transient and full-wave numerical simulation of the propagation of electromagnetic waves in inhomogeneous media. It is especially applied in bioelectromagnetics for numerical simulations. However, in FDTD simulations of the human body, input parameters, such as the complex permittivities of human tissues, are determined from measurements of dielectric properties with a typical uncertainty of $\pm{10}{\%}$. The ambiguity of these input parameters results in a degree of uncertainty in the system’s response, and subsequently, the system’s response cannot be precisely determined. Therefore, to increase the reliability and accuracy of FDTD simulations, it is necessary to quantify the impact of uncertainties of input parameters on the system’s responses.
This article addresses the uncertainty analysis of FDTD simulations of the human body, where a digital human phantom (DHP) [3] is utilized as an equivalent human model, and the one-pole Debye model [4] is implemented in the FDTD computation to characterize the complex frequency-dependent behavior of the biological tissues. Each tissue in the DHP is associated with Debye parameters, which are subject to some degree of uncertainty, leading to different frequency responses among the population. This problem is known as forward UQ, or the uncertainty propagation.
Various UQ techniques have been proposed in the last decades. The traditional Monte Carlo method (MCM) [5] proves to be inappropriate for UQ because a considerable number of FDTD simulations are required to reach a reasonable level of convergence. There exist some ideal alternatives to MCM, such as the NIPC expansion method [6], [7] and the stochastic colocation-based method [8]. However, both methods cannot efficiently handle high-dimensional UQ problems due to the curse of dimensionality [9], whereby the number of required simulations substantially grows when the number of random variables increases. Although various techniques, such as the hyperbolic scheme [10] and the analysis of variance decomposition method [11], are applied to reduce the impact of the curse of dimensionality, the computational cost associated with the system simulations remains high.
Many advanced UQ techniques have been proposed over the last decade to alleviate the curse of dimensionality and improve the accuracy of UQ [12], [13], [14]. The ideas behind these techniques include sparse strategies [15], [16], surrogate modeling [17], [18], model order reductions [19], and hierarchical approaches [20], [21]. The surrogate modeling technique, which builds a simpler equivalent model for the original complex system, is an ideal candidate for the UQ of computationally intensive systems. A surrogate model has the potential to accurately predict the system’s outputs, and it allows the bypassing of thousands of otherwise necessary system simulations, thereby significantly improving the efficiency of traditional methods such as MCM and NIPC.
ANNs [22], [23], [24] have become one of the most promising surrogate modeling techniques due to their high flexibility and learning capability. This article studies ANNs in UQ for FDTD computations. An ANN model is trained using data, which include the input samples of Debye parameters and the desired outputs obtained from FDTD simulations. In general, the accuracy of ANNs can be improved by increasing the number of data. However, this approach may lead to computational inefficiency in UQ for computation-intensive systems, such as 3D FDTD simulations. To improve the accuracy of ANNs in UQ for an FDTD computation without increasing the number of system simulations, an activation function of ANNs [25] is proposed.
The proposed activation function is inspired by NIPC expansion and the work in [19]. The quadratic NIPC expansion, i.e., with the order of polynomials restricted to two, was demonstrated to provide sufficient accuracy [26] for the UQ for the FDTD computation. When multiple random variables are considered, it may be useful to treat each random variable as a single uncertain input while the remaining random variables are considered as constants [19]. The proposed activation function is therefore devised using the quadratic NIPC expansion for one random variable. This proposed activation function is also known as the polynomial activation function, which had been utilized on various applications of ANNs. However, it has never been applied to quantify the uncertainty of FDTD results.
The novelty of this article lies in both the application and the method, as follows:
In FDTD calculations, the relationship between electric field E and electric flux density D in the one-pole Debye model is written as [28] ${\boldsymbol{D}} = {\epsilon}_{0}\left[{{\epsilon}_{\infty} + \left({{{\epsilon}_{S}{-}{\epsilon}_{\infty}}\slash{{1} + {\jmath}{\omega}{\tau}}}\right) + \left({{{\sigma}_{S}}\slash{{\jmath}{{\omega}{\epsilon}}_{0}}}\right)}\right]{\boldsymbol{E}}$, where e0 is the permittivity of vacuum, eS is the static permittivity, ${\epsilon}_{\infty}$ is the optical perimittivity, ∼ is the angular frequency, ${\jmath}$ is the imaginary unit satisfying ${\jmath} = \sqrt{{-}{1}},{\sigma}_{S}$ is the static conductivity, and x is the relaxation time. Each tissue in the DHP is associated with four Debye parameters $\left({{\sigma}_{S},{\epsilon}_{\infty},{\epsilon}_{S},{\tau}}\right)$.
We assume that there are ${\cal{K}}$ independent Debye parameters of interest satisfying the normal distribution and ${\cal{M}}$ sets of input samples to an ANN. These ${\cal{K}}$ Debye parameters are considered uncertain inputs, and they are expressed in the vectorial form of ${\xi} = {\left[{{\xi}_{1},{\xi}_{2},\ldots,{\xi}_{K}}\right]}^{T}$. We define ${\xi}^{{(}{m}{)}}$ as the mth sample of ${\xi}$ for ${1}\,{≤}\,{m}\,{≤}\,{\cal{M}}$. Apart from these ${\cal{K}}$ Debye parameters, input parameters in the FDTD simulation, such as the location of the excitation, are treated as constant. As the specific absorption rate (SAR) is usually of interest in bioelectromagnetics applications, the square of the electric field ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$ is considered the output of the FDTD simulation calculated as in ${\left\vert{\boldsymbol{E}}\right\vert}^{2} = {\Sigma}_{\hat{n}}\left({{\left\vert{{E}_{x}^{\hat{n}}{(}{\hat{i}},{\hat{j}},{\hat{k}}{)}}\right\vert}^{2} + {\left\vert{{E}_{y}^{\hat{n}}{(}{\hat{i}},{\hat{j}},{\hat{k}}{)}}\right\vert}^{2} + {\left\vert{{E}_{z}^{\hat{n}}{(}{\hat{i}},{\hat{j}},{\hat{k}}{)}}\right\vert}^{2}}\right)$, where ${(}{\hat{i}},{\hat{j}},{\hat{k}}{)}$ is the point location of the observation and ${\hat{n}}$ is the FDTD time step.
Figure 1 presents an example of how the uncertainty of the Debye parameters of eS and ${\epsilon}_{\infty}$ for muscle affects the FDTD response ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$ in the 3D FDTD simulations in which eS and ${\epsilon}_{\infty}$ are varied from −10 to 10% simultaneously by steps of 2%. The details of the experimental setup for Figure 1 are presented in the “Numerical Experiments Setting for UQ” section. It is observed that the uncertainties of eS and ${\epsilon}_{\infty}$ induce a variation of up to 16% for the FDTD response ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$. Furthermore, it is important to note that the numerical experiments of Figure 1 consider only the uncertainties of Debye parameters from one tissue, i.e., muscle. When multiple uncertain Debye parameters of tissues are considered, the FDTD response cannot be simply determined, and it is therefore crucial to estimate the uncertainty of the FDTD response induced by these uncertain Debye parameters. This article proposes an adaptive ANN to quantify the uncertainty of the FDTD simulation in which a number of adaptations are introduced to the ANN, aiming at improving the performance of the ANN in UQ. The details of the proposed method are presented in the following section.
Figure 1. The impact of uncertain Debye parameters of muscle on the system’s response.
An ANN is a machine learning algorithm utilized to model the underlying relationships between the input parameters and a system’s output. An ANN typically has three types of layers of input, hidden, and output layers. In this article, the neurons in the input layer represent the uncertain Debye parameters, and the neuron in the output layer indicates the output of the ANN, which is a single scalar of the prediction of ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$.
Each neuron in the hidden layer receives its input ${\cal{T}}$ from the previous layer and converts ${\cal{T}}$ to the input of the successive layer by an activation function ${f}{\left({\cal{T}}\right)}$. We form a set of data pairs from the input ${\xi}^{{(}{m}{)}}$ and the desired output ${\left\vert{{E}^{(m)}}\right\vert}^{2}$ and train the ANN to optimize the weights of the ANN, which enables the ANN to make an accurate prediction of ${\left\vert{{E}^{(m)}}\right\vert}^{2}$ for any input ${\xi}^{{(}{m}{)}}$.
This article proposes a polynomial activation function to improve the accuracy of ANNs in predicting the output of the FDTD computation. We define this activation function as \[{f}_{p}{\left({\cal{T}}\right)} = {\cal{T}} + {\cal{T}}^{2}{.} \tag{1} \]
This polynomial activation function is inspired by [19] and the NIPC expansion method. The work in [19] quantifies the uncertainty of the system ${\cal{K}}$ times when a system has ${\cal{K}}$ parameters. The authors in [19] consider only one parameter as an uncertain input at one time, and the remaining ${\cal{K}}{-}{1}$ parameters are treated as constants. Thus, the number of polynomials is calculated based on the case of one parameter instead of ${\cal{K}}$ parameters, thereby alleviating the curse of dimensionality.
The quadratic NIPC expansion method [26] achieves high accuracy in estimating UQ for the FDTD computation. When a system has one random Debye parameter satisfying the normal distribution, ${\psi}_{0}{(}{\xi}_{k}{)} = {1},{\psi}_{1}{(}{\xi}_{k}{)} = {\xi}_{k},{\psi}_{2}{(}{\xi}_{k}{)} = {\xi}_{k}^{2}{-}{1}$ are the quadratic Hermite polynomials. The NIPC expansion is formed by three polynomials, as in ${\Sigma}_{{\alpha} = {0}}^{2}{d}_{\alpha}{\psi}_{\alpha}({\xi}_{k})$, where ${d}_{\alpha}$ is a coefficient of ${\psi}_{\alpha}({\xi}_{k})$. We set ${d}_{\alpha}$ for ${0}\,{≤}\,{\alpha}\,{≤}\,{2}$ to one for the sake of simplicity. Replacing pk with ${\cal{T}}$, we design (1) as in ${f}_{p}{\left({\cal{T}}\right)} = {\Sigma}_{{\alpha} = {0}}^{2}{\psi}_{\alpha}{(}{\cal{T}}{)}$.
An ANN with one hidden layer may underperform on the accuracy of the prediction of the system’s output when it is used to model a complex system. In such circumstances, multiple hidden layers are required. This article defines an ANN with two hidden layers of the first hidden layer with G1 neurons and the second hidden layer with G2 neurons.
Training an ANN with multiple hidden layers mainly involves three stages of forward propagation, backpropagation, and the update of weights. Let ${\xi}_{k}^{{(}{m}{)}}$ be the mth sample of pk used for the FDTD computation, and ${\cal{X}}{(}{\cal{M}}{)} = {\{}{\tilde{\xi}}_{k}^{{(}{m}{)}},{m} = {1}{∼}{\cal{M}},{k} = {1}{∼}{\cal{K}}{\}}$ be a normalized sample set, where ${\tilde{\xi}}_{k}^{{(}{m}{)}}$ is the normalized ${\xi}_{k}^{{(}{m}{)}}$. In the forward propagation stage, we calculate the output of an ANN by \[{\hat{\boldsymbol{E}}} = {f}_{p}{(}{f}_{p}{(}{\cal{X}}{(}{\cal{M}}{)}{\cal{W}}_{1}{)}{\cal{W}}_{2}{)}{\cal{W}}_{3}, \tag{2} \] where the ${\cal{W}}_{1} = {\{}{\cal{W}}_{{1}_{({kg}_{1})}},{k} = {1}{∼}{\cal{K}},{g}_{1} = {1}{∼}{G}_{1}{\}}$ matrix consists of the weights between the input and first hidden layers. ${\cal{W}}_{{1}_{({kg}_{1})}}$ indicates the weight between the kth neuron in the input layer and the g1th neuron in the first hidden layer. ${\cal{W}}_{2} = {\{}{\cal{W}}_{{2}_{({g}_{1}{g}_{2})}},{g}_{1} = {1}{∼}{G}_{1},{g}_{2} = {1}{∼}{G}_{2}{\}}$ refers to the weights between the first and second hidden layers, respectively, and ${\cal{W}}_{3} = {\left[{{\cal{W}}_{{3}_{(1)}},{\cal{W}}_{{3}_{(2)}},\ldots,{\cal{W}}_{{3}_{({G}_{2})}}}\right]}^{T}$ indicates the weights between the second hidden and output layers, respectively. Furthermore, when the input to fp in (1) is a matrix or vector, the operation in fp is elementwise.
The stages of backpropagation and update of weights are based on the gradient descent method. The ANN updates its weights to minimize the loss function of ${\boldsymbol{L}} = {1}\slash{2}{(}{\hat{\boldsymbol{E}}}{-}{\cal{E}}{)}^{2}$, where ${\cal{E}} = {\left[{{\left\vert{{\boldsymbol{E}}^{(1)}}\right\vert}^{2},\ldots,{\left\vert{{\boldsymbol{E}}^{{(}{\cal{M}}{)}}}\right\vert}^{2}}\right]}^{T}$ is a vector consisting of ${\cal{M}}$${\left\vert{\boldsymbol{E}}\right\vert}^{2}$ obtained from the ${\cal{M}}$ FDTD simulations. The procedure for forward propagation, backpropagation, and update of weights is called an ANN iteration. Let ${\cal{W}}_{i}^{({j})}$ for ${i} = {1}{∼}{3}$ be the weights of the ANN at the jth ANN iteration. The ANN updates its weights, as in \begin{align*}{\cal{W}}_{i}^{{(}{j} + {1}{)}} & = {\cal{W}}_{i}^{({j})}{-}{\eta}{f}_{p}{\left({{\cal{T}}_{{i}{-}{1}}}\right)}^{{({j})}^{T}}{\delta}_{i}^{({j})}{\text{ for }}{i} = {2},{3} \\ {\cal{W}}_{1}^{{(}{j} + {1}{)}} & = {\cal{W}}_{1}^{({j})}{-}{\eta}{X}{(}{\cal{M}}{)}^{T}{\delta}_{1}^{({j})}, \tag{3} \end{align*} where ${\cal{T}}_{1} = {\cal{X}}{(}{\cal{M}}{)}{\cal{W}}_{1}$ and ${\cal{T}}_{2} = {f}_{p}\left({{\cal{T}}_{1}}\right){\cal{W}}_{2}$ represent inputs to the first and second hidden layers, respectively. h is the learning rate utilized to tune the update of weights. ${\delta}_{i}^{({j})}{\text{ for }}{i} = {1}{∼}{3}$ are the error signals at the jth ANN iteration used to measure how much L varies with the changes of ${\cal{T}}_{i}$ for ${i} = {1}{∼}{2}$. ${\delta}_{i}^{({j})}{\text{ for }}{i} = {1}{∼}{3}$ are written as \begin{align*}{\delta}_{3} & = \frac{\partial{\boldsymbol{L}}}{\partial{\hat{\boldsymbol{E}}}} = {\hat{\boldsymbol{E}}}{-}{\cal{E}} \\ {\delta}_{i} & = \frac{\partial{f}_{p}\left({{\cal{T}}_{i}}\right)}{\partial{\cal{T}}_{i}}{\odot}\left({{\cal{\delta}}_{{i} + {1}}{\cal{W}}_{{i} + {1}}^{T}}\right),{\text{ for }}{i} = {1},{2} \tag{4} \end{align*} where ${\odot}$ denotes the elementwise multiplication of matrices. The ANN iteration is repeated for a certain number of times, each with updated weights, and is terminated when the accuracy of the estimation of the system’s output via the ANN reaches our expectation. The leave-one-out cross-validation (LOOCV) method [29] is utilized to quantify the accuracy.
For a given data set of ${\cal{M}}$ samples, the LOOCV method splits the data set into training ${\cal{X}}{({\cal{M}}}_{\text{tr}})$, validation ${\cal{X}}{(}{\cal{M}}{-}{1}{-}{\cal{M}}_{\text{tr}}{)}$, and test data, where ${\cal{M}}_{\text{tr}}$ is an integer that satisfies ${0}{<}{\cal{M}}_{\text{tr}}{<}{\cal{M}}{-}{1}$, and the test data contain one sample, that is, ${\tilde{{\xi}}}^{{(}{m}{)}}$. The ANN is trained in the “An ANN With Multiple Hidden Layers” section, using ${\cal{X}}{({\cal{M}}}_{\text{tr}})$ instead of ${\cal{X}}{(}{\cal{M}}{)}$. At each ANN iteration, the ANN updates its weights in (3). These updated weights are utilized to calculate the output of the ANN for ${\cal{X}}{(}{\cal{M}}{-}{1}{-}{\cal{M}}_{\text{tr}}{)}$ at the jth ANN iteration, as in ${\hat{\boldsymbol{E}}}_{v}^{({j})} = {f}_{p}{\left({f}_{p}{\left({\cal{X}}{(}{\cal{M}}{-}{1}{-}{\cal{M}}_{\text{tr}}{)}{\cal{W}}_{1}^{(j)}\right)} {\cdot} {\cal{W}}_{2}^{(j)}\right)} {\cdot} {\cal{W}}_{3}^{({j})}$, where ${\hat{\boldsymbol{E}}}_{v}^{({j})} = {\left[{\hat{\boldsymbol{E}}}_{v}^{{(j)}^{(1)}},\ldots,{{\hat{\boldsymbol{E}}}}_{v}^{(j)}\right]}$ is the output of the ANN of ${{\hat{\boldsymbol{E}}}}_{v}^{(\text{j}{)}^{{(}{m}{)}}}$ using ${\cal{X}}{(}{\cal{M}}{-}{1}{-}{\cal{M}}_{\text{tr}}{)}$ at the jth ANN iteration. Let ${\cal{L}}_{v}^{({j})}$ be the validation error ${\cal{L}}_{v}$, as in \[{\cal{L}}_{v}^{({j})} = \frac{\mathop{\sum}\limits_{{m} = {1}}\limits^{{\cal{M}}{-}{1}{-}{\cal{M}}_{\text{tr}}}{{\left({{{\hat{\boldsymbol{E}}}}_{v}^{(j{)}^{(m)}}{-}{\left\vert{{E}^{(m)}}\right\vert}^{2}}\right)}^{2}}}{{\cal{M}}{-}{1}{-}{\cal{M}}_{\text{tr}}}, \tag{5} \] at the jth ANN iteration.
The aim of using the validation data is to prevent overfitting. ${\cal{L}}_{v}$ starts decreasing during the ANN iteration and later on increases when overfitting occurs. Therefore, we terminate the ANN iteration when ${{\cal{L}}_{v}^{{(}{j} + {1}{)}}}\slash{{\cal{L}}_{v}^{({j})}}$ is larger than a certain value s to prevent potential overfitting in the ANN. This method is also known as early stopping.
Aside from early stopping, other techniques such as dropout [30], [31] can be used to avoid overfitting as well. As this article deals with a two-hidden-layer ANN with a linear activation function, the architecture of our ANN is relatively simple. Solely using the technique of early stopping is capable of detecting overfitting.
The training of the ANN is completed at the termination of the ANN iteration, and we obtain ${\cal{W}}_{i}^{({j})}$ for ${i} = {1}{∼}{3}$ as the trained ANN model. The test data are utilized to quantify the error of the trained ANN model. When ${\tilde{{\xi}}}^{{(}{m}{)}}$ is used as the test data composed of one element, the test error ${\cal{L}}_{\text{ts}}^{{(}{m}{)}}$ is calculated, as in \[{\cal{L}}_{\text{ts}}^{{(}{m}{)}} = \left({{f}_{p}\left({{f}_{p}\left({{\tilde{{\xi}}}^{(m)}{\cal{W}}_{1}^{{(}{j}{)}}}\right){\cal{W}}_{2}^{{(}{j}{)}}}\right){\cal{W}}_{3}^{{(}{j}{)}}}\right.{\left.{{-}\left\vert{{\cal{E}}^{(m)}}\right\vert^{2}}\right)}^{2}{.} \tag{6} \]
We call the process of generation of the trained ANN model and calculation of ${\cal{L}}_{\text{ts}}^{{(}{m}{)}}$ an LOO iteration. As we scan m from one to ${\cal{M}}$, there are ${\cal{M}}$ LOO iterations in total. After the ${\cal{M}}$ LOO iterations, the LOO error ${\cal{L}}_{l}$ is calculated, as in \[{\cal{L}}_{l} = \frac{1}{\cal{M}}\mathop{\sum}\limits_{{m} = {1}}\limits^{\cal{M}}{{\cal{L}}_{\text{ts}}^{(m)}}{.} \tag{7} \]
The “UQ of the FDTD Computation” section introduces the proposed adaptations, including the polynomial activation function in the “Polynomial Activation Function” section and the LOOCV method into the ANN method described in the “An ANN With Multiple Hidden Layers” section, to quantify the uncertainty of the FDTD computation. The procedure of the ANN for UQ for the FDTD computaion is as follows.
We use ${\cal{X}}{(}{\cal{M}}{)}$ to produce the final ANN model, then utilize some of ${\tilde{\cal{X}}}{(}{\cal{N}}{)}$ as inputs to the final ANN model to predict ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$. ${\left\vert{{\boldsymbol{E}}^{(m)}}\right\vert}^{2}$ corresponding to ${\xi}^{{(}{m}{)}}$ is obtained from the FDTD simulation, whereas we do not run the FDTD simulation to acquire ${\left\vert{{\boldsymbol{E}}^{(n)}}\right\vert}^{2}$ corresponding to ${{\xi}}^{({n})}$.
We calculate ${\hat{\boldsymbol{E}}}$ in (2) using the final ANN model of ${\cal{W}}_{i}^{{(}{j}{)}}$ for ${i} = {1}\sim{3}$ obtained in step B) of the “UQ of the FDTD Computation” section, replacing ${\cal{X}}{(}{\cal{M}}{)}$ with ${\tilde{\cal{X}}}{(}{\cal{N}}{)}$. ${\hat{\boldsymbol{E}}}$ consists of ${\cal{N}}$ predictions ${\hat{\boldsymbol{E}}}^{({n})}$ of ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$. These predictions are utilized to estimate the mean $\hat{\mu}{(}{\cal{N}}{)}$ of ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$, as in \[\hat{\mu}{(}{\cal{N}}{)} = \frac{1}{\cal{N}}\mathop{\sum}\limits_{{n} = {1}}\limits^{\cal{N}}{{{\hat{\boldsymbol{E}}}}^{(n)}} \tag{11} \] and the standard deviation ${\hat{\sigma}}{(}{\cal{N}}{)}$ of ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$, as in \[{\hat{\sigma}}{(}{\cal{N}}{)}^{2} = \frac{1}{{\cal{N}}{-}{1}}\mathop{\sum}\limits_{{n} = {1}}\limits^{\cal{N}}{{\left({{{\hat{\boldsymbol{E}}}}^{(n)}{-}\hat{\mu}{(}{\cal{N}}{)}}\right)}^{2}}{.} \tag{12} \]
The MCM and the proposed method are utilized to quantify the uncertainty of the FDTD computation. The simulation scenario is depicted in Figure 2, where the FDTD space is 265 × 490 × 601 voxels each, with a resolution of ${1}{\text{ mm}}^{3}$. The excitation is located 17 mm away from the human body, and the observation location is in the middle of the prostate tissue. Ten layers of the complex frequency shifted-perfect matched layers [33], [34] are used to terminate the 3D FDTD space. The DHP used in this work is provided by Rikagaku Kenkyusho (RIKEN) (Saitama, Japan) [35] under a nondisclosure agreement between RIKEN and the University of Manchester. Its usage was approved by an RIKEN ethics committee.
Figure 2. A numerical simulation setup. (Source: RIKEN, used with permission.)
A given human tissue is considered influential when the ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$ at the observation location varies significantly due to variations of one or more of its four Debye parameters, i.e., ${\sigma}_{S}$, ${\epsilon}_{\infty}$, ${\epsilon}_{S}$, ${\tau}$. The influential tissues for our scenario follows [36], wherein the influence of each of the four Debye parameters on ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$ was investigated, i.e., a sensitivity analysis was performed. The five influential tissues to be considered are fat, skin, muscle, bone, and prostate. Furthermore, the work in [36] indicates that the influences of ${\sigma}_{S}$ and ${\tau}$ on ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$ are negligible. As previously observed in [37], even though the conductivity variation has a negligible effect on ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$, it has a clear impact on the SAR, which is directly proportional to the product of the conductivity and to ${\left\vert{\cal{E}}\right\vert}^{2}$. Our computation with the one-pole Debye model changes the conductivity by varying ${\epsilon}_{\infty}$. Thus, only two out of the four Debye parameters, i.e., ${\epsilon}_{\infty}$ and ${\epsilon}_{S}{-}{\epsilon}_{\infty}\,{≜}\,\Delta{\epsilon}$, are required for the simulations yielding a total of 10 (2 × 5) uncertain input parameters. The procedures of generating samples for the 10 uncertain input parameters are as follows:
Table 1. The mean values and standard deviations of the 10 debye parameters and their corresponding notations.
The following experiments were derived for UQ:
We run the ANN iteration as described in the “An ANN With Multiple Hidden Layers” section. When ${j}\,{≥}\,{I}$, ${\cal{L}}_{v}^{{(}{j}{)}}$ is calculated in (5), there is no overfitting occurring during the first 200 ANN iterations. Thus, the ANN iteration is terminated when $\left({{\cal{L}}_{v}^{{(}{j} + {1}{)}} / {\cal{L}}_{v}^{{(}{j}{)}}}\right)\geq\varpi$, where I and ${\varpi}$ are set to 200 and 1.01, respectively. The setting of I is application dependent, and it can be influenced by many factors, such as learning rate ${\eta}$ and the size of data set ${\cal{M}}$. A large value of ${\eta}$, such as ${\eta}{>}{0.01}$, makes the ANN learn from the data quickly, resulting in overfitting within a small number of ANN iterations, i.e., ${j}\,{≤}\,{50}$. Furthermore, the overfitting might occur when ${\cal{L}}_{v}$ stops decreasing. Setting ${\varpi}$ to 1.01 allows us to detect the potential overfitting. When ${j} = {325}$, ${\cal{L}}_{v}^{{(}{j} + {1}{)}} / {\cal{L}}_{v}^{{(}{j}{)}}$ becomes greater than 1.01. Therefore, we terminate the ANN iteration at the 325th ANN iteration and calculate ${\cal{L}}_{\text{ts}}^{(1)}$ in (6) using ${\cal{W}}_{i}^{(325)}$ for ${i} = {1}\sim{3}$ as the trained ANN model of the first LOO iteration.
${\cal{L}}_{l}$ is calculated in (7) after 10 LOO iterations. The ANN is then retrained using ${\cal{X}}{(}{10}{)}$ as the training data. We terminate the ANN iteration when ${\cal{L}}_{\text{tr}}^{{(}{j}{)}}\,{≤}\,{\cal{L}}_{l}$ and both (9) and (10) are met, where b is set to 0.01. The setting of b is application dependent. In our scenario, 0.01 of b is a reasonable value, enabling us to detect the stable status of ${\cal{L}}_{\text{tr}}$. Equations (9) and (10) are met when ${j} = {415}$. Thus, we save ${\cal{W}}_{i}^{(415)}$ for ${i} = {1}\sim{3}$ as the final ANN model.
The final ANN model is utilized to make the predictions of ${\left\vert{\cal{E}}\right\vert}^{2}$. We use ${10}^{4}$ sets of input samples in the MCM method. Therefore, we set $\text{N}$ to ${10}^{4}$ for comparison. $\hat{\cal{E}}$ is calculated in (2) using ${\cal{W}}_{i}^{(415)}$ for ${i} = {1}\sim{3}$. $\hat{\mu}{(}{N}{)}$ and $\hat{\sigma}{(}{N}{)}$ are obtained in (11) and (12), respectively.
We call the ANN with proposed adaptations an adaptive ANN. The “Results and Discussions” section presents ${\mu}{(}{\cal{M}}{)}$, ${\sigma}{(}{\cal{M}}{)}$, $\hat{\mu}{(}{N}{)}$, and $\hat{\sigma}{(}{N}{)}$ obtained from the MCM and the adaptive ANN. To evaluate the performance of the adaptive ANN, input samples ${\tilde{\cal{X}}}{(}{N}{)}$, which are utilized to make predictions of ${\left\vert{\boldsymbol{E}}\right\vert}^{2}$ in the adaptive ANN, are the same as those used in the MCM of ${\cal{X}}{(}{\cal{M}}{)}$.
The following results were derived:
Figure 3. The ${\mu}{(}{\cal{M}}{)}$ in (13) from the MCM, and the $\hat{\mu}{(}{N}{)}$ in (11) from the adaptive ANN.
Figure 4. The $\hat{\mu}{(}{10}^{4}{)}$ in (11) from the adaptive ANN for 1,000 experiments, and the ${\mu}{(}{10}^{4}{)}$ in (13) from the MCM.
Figure 5. The ${\sigma}{(}{\cal{M}}{)}$ in (14) from the MCM, and the $\hat{\sigma}{(}{N}{)}$ in (12) from the adaptive ANN.
Figure 6. The $\hat{\sigma}{(}{10}^{4}{)}$ in (12) from the adaptive ANN for 1,000 experiments, and the ${\sigma}{(}{10}^{4}{)}$ in (14) from the MCM.
For ANN-based UQ techniques, the accuracy of UQ depends on the accuracy of the prediction of system’s output via the ANN. The novelty of this article is that we modify the activation function and the termination criteria of the ANN to maximize the accuracy in predicting the system’s output via the ANN while maintaining a low computational cost. The following experiment is conducted to compare the proposed adaptive ANN with the traditional ANN from the viewpoint of the accuracy of the prediction of system’s output. We define the traditional ANN for regression analysis as the architecture that comprises two hidden layers. A traditional ANN utilizes the linear activation function of ${f}_{{l}}{\left({\cal{T}}\right)} = {T}$ and terminates the ANN iteration when the training error becomes stable, as follows:
We repeat the procedures of 2) and 3) 100 times to acquire ${100}\,{\cal{A}}_{p}$ and ${100}\,{\cal{A}}_{\text{ln}}$, each of which uses different ${10}{\xi}$ to build the surrogate model based on the adaptive or traditional ANN. Presented in Figure 7 are the obtained ${100}\,{\cal{A}}_{p}$ and ${100}\,{\cal{A}}_{\text{ln}}$. The mean and standard deviation of ${100}\,{\cal{A}}_{p}$ and ${100}\,{\cal{A}}_{\text{ln}}$ are 89.94% ${\pm}$ 4.18% and 44.21% ${\pm}$ 24.65%, respectively. This indicates that the proposed adaptations have the potential to significantly improve the accuracy of the ANN in predicting the output of the FDTD simulation while maintaining a low computational cost.
Figure 7. One hundred ${\cal{A}}_{p}$ from the adaptive ANN, and 100 ${A}_{\ln}$ from the traditional ANN.
Furthermore, we compare the planned adaptive ANN with the state-of-the-art ANN-based UQ method proposed in [42]. The work in [42] estimates the uncertainty of the SAR calculation through two NNs, where the first NN is designed based on the autoencoder NN [43] for dimensionality reduction, and the second NN is a traditional ANN for UQ. We conduct the experiment to build surrogate models for the FDTD simulation using the proposed ANN and [42]. Figure 8 shows the prediction accuracy of the surrogate model against the number of training data for these two methods, where the number of training data is varied from 10 to 1,000, as in ${\cal{M}}_{\text{tr}} = {\left\{10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 500, {1000}\right\}}$. For each value of ${\cal{M}}_{\text{tr}}$, we construct a surrogate model using the proposed ANN and [42], respectively, and evaluate the prediction accuracy of the surrogate model on the test data, namely, ${\cal{X}}_{ts}{(}{100}{)}$. From Figure 8, it is observed that our proposed ANN outperforms [42], especially when the quantity of the training data is relatively small. It is important to note that generating one set of training data requires one system run. For computation-intensive systems, it is impractical to utilize a large amount of training data for UQ. Accordingly, the proposed ANN demonstrates a high potential in efficiently handling the UQ problem for computation-intensive systems.
Figure 8. Comparisons between the proposed ANN and [42], with the number of training data ${\cal{M}}_{\text{tr}} = {\left\{10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 500, {1000}\right\}}$.
The proposed adaptive ANN is utilized in UQ for the FDTD computation. We compare the proposed method with other state-of-the-art or classical UQ methods, which include the works in [42], [44], and [19]; the NIPC expansion method and the traditional ANN, from the viewpoints of accuracy in UQ estimations and computational efficiency. In the NIPC expansion method, we utilize the regression method to estimate the coefficients of polynomials, where the highest order of polynomials is set to two. The work published in [44] builds a surrogate model using the sparse grid interpolation. We implement [44] using the sparse grid toolbox from MATLAB [45], where the relative tolerance is set to 0.2 in our scenario. ${\mu}$ and ${\sigma}$ obtained from these UQ techniques are presented in Table 2, where the accuracy of the ${\mu}$ and ${\sigma}$ estimation is calculated by ${1}{-}{((}{u}{\mu}{-}{\mu}{(}{10}^{4}{)}{u}{)} / {\mu}{(}{10}^{4}{))}$ and ${1}{-}{((}{u}{\sigma}{-}{\sigma}{(}{10}^{4}{)}{u}{)} / {\sigma}{(}{10}^{4}{)),}$ respectively.
Table 2. Comparisons of the proposed method with the uq methods of the nipc method, a traditional ann, and [42], [44], and [19].
The adaptive ANN outperforms other UQ techniques in terms of accuracy and computational efficiency, which are measured by the number of FDTD simulations required. In our scenario, generating one set of input samples requires one run of the FDTD simulation with 5,000 FDTD iterations, which takes approximately 1.4 h to complete. Our in-house FDTD code was implemented based on openMP and was executed on an Intel Xeon computer at 2.4 GHz with 128 GB of memory operating on a Red Hat Enterprise Linux 7.3 system, where the number of threads was set to eight, and the memory usage of one FDTD simulation was 9 GB.
The computational cost of a UQ technique is regarded as negligible compared with the one taken for an FDTD simulation. The training process of an ANN with 10 sets of input samples takes fewer than 10 s to complete. Therefore, when we evaluate the efficiency of UQ techniques, only the number of FDTD simulations required is taken into account. In Table 2, the NIPC expansion method demands 66 FDTD simulations and achieves an accuracy of 78.11% for the ${\sigma}$ estimation. Comparatively, our proposed method requires 10 FDTD simulations and achieves an accuracy of 98.83% for the ${\sigma}$ estimation, which is roughly 6.6 times faster and 20% more accurate than the NIPC expansion method.
This article proposed a number of adaptations for an ANN to improve the accuracy of an ANN in UQ for the FDTD computation while maintaining low computational resources. We offered a versatile activation function to enable the ANN to effectively learn from limited information. A series of termination criteria of the ANN were proposed to maximize the accuracy of UQ. The main contribution of this article lies in the significant improvement of the accuracy and the stability of UQ compared with the existing UQ techniques as alternatives to the MCM.
This work involved human subjects or animals in its research. Approval of all ethical and experimental procedures and protocols was granted by RIKEN. This work was completed, in part, with the HOKUSAI-GreatWave Computer System at RIKEN: http://i.riken.jp/download/sites/2/HOKUSAI_system_overview_en.pdf. Additional research data supporting this publication are available at http://dx.doi.org/ repository at 10.17632/vnvmd9vysy.2.
Runze Hu (hrzlpk2015@gmail.com) is with the University of Manchester, Manchester, U.K. He is currently a postdoctoral fellow with Tsinghua University, Beijing, 100084, China. His research interests include visual quality assessment, uncertainty quantification, and high-performance computing.
Vikass Monebhurrun (vikass.monebhurrun@centralesupelec.fr) is with CentraleSupélec, Gif-sur-Yvette, 91192, France. He currently serves as chair of the IEEE Antennas and Propagation Standards Committee and associate editor of IEEE Transactions on Antennas and Propagation and IEEE Antennas and Propagation Magazine. He is a Senior Member of IEEE.
Ryutaro Himeno (r.himeno.tx@juntendo.ac.jp) is currently a specially appointed professor with Juntendo University, Tokyo, 113-8421, Japan. His research interests include high-performance computing, computer simulation, and bioengineering.
Hideo Yokota (hyokota@riken.jp) is an image processing research team leader in the Center for Advanced Photonics RIKEN, Wako¯, Saitama, Japan. He is currently a specially appointed professor in the Faculty of Medicine, University of the Ryukyus, Okinawa, 903-0213, Japan.
Fumie Costen (fumie.costen@manchester.ac.uk) is with the University of Manchester, Manchester, M13 9PL, U.K. She received two research awards in noninvasive microwave imaging and metacomputing and filed four patents. She has also received teaching excellence awards. She is a Senior Member of IEEE.
[1] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA, USA: Artech House, 2005.
[2] K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302–307, May 1966, doi: 10.1109/TAP.1966.1138693.
[3] M. Abalenkovs, F. Costen, J.-P. Bérenger, R. Himeno, H. Yokota, and M. Fujii, “Huygens subgridding for 3-D frequency-dependent finite-difference time-domain method,” IEEE Trans. Antennas Propag., vol. 60, no. 9, pp. 4336–4344, Sep. 2012, doi: 10.1109/TAP.2012.2207039.
[4] K. Tekbas, F. Costen, J.-P. Bérenger, R. Himeno, and H. Yokota, “Subcell modeling of frequency-dependent thin layers in the FDTD method,” IEEE Trans. Antennas Propag., vol. 65, no. 1, pp. 278–286, Jan. 2017, doi: 10.1109/TAP.2016.2628712.
[5] A. Arduino, M. Chiampi, F. Pennecchi, L. Zilberti, and O. Bottauscio, “Monte Carlo method for uncertainty propagation in magnetic resonance-based electric properties tomography,” IEEE Trans. Magn., vol. 53, no. 11, pp. 1–4, Nov. 2017, doi: 10.1109/TMAG.2017.2713984.
[6] D. Xiu and G. E. Karniadakis, “Modeling uncertainty in flow simulations via generalized polynomial chaos,” J. Comput. Phys., vol. 187, no. 1, pp. 137–167, 2003, doi: 10.1016/S0021-9991(03)00092-5.
[7] A. C. Austin and C. D. Sarris, “Efficient analysis of geometrical uncertainty in the FDTD method using polynomial chaos with application to microwave circuits,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 12, pp. 4293–4301, Dec. 2013, doi: 10.1109/TMTT.2013.2281777.
[8] D. Xiu and J. S. Hesthaven, “High-order collocation methods for differential equations with random inputs,” SIAM J. Sci. Comput., vol. 27, no. 3, pp. 1118–1139, 2005, doi: 10.1137/040615201.
[9] I. Jeong, B.-G. Gu, J. Kim, K. Nam, and Y. Kim, “Inductance estimation of electrically excited synchronous motor via polynomial approximations by least square method,” IEEE Trans. Ind. Appl., vol. 51, no. 2, pp. 1526–1537, Mar.-Apr. 2015, doi: 10.1109/TIA.2014.2339634.
[10] G. Blatman and B. Sudret, “Adaptive sparse polynomial chaos expansion based on least angle regression,” J. Comput. Phys., vol. 230, no. 6, pp. 2345–2367, 2011, doi: 10.1016/j.jcp.2010.12.021.
[11] I. M. Sobol, “Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates,” Math. Comput. Simul., vol. 55, nos. 1–3, pp. 271–280, 2001, doi: 10.1016/S0378-4754(00)00270-6.
[12] P. Rocca, N. Anselmi, A. Benoni, and A. Massa, “Probabilistic interval analysis for the analytic prediction of the pattern tolerance distribution in linear phased arrays with random excitation errors,” IEEE Trans. Antennas Propag., vol. 68, no. 12, pp. 7866–7878, Dec. 2020, doi: 10.1109/TAP.2020.2998924.
[13] L. Tenuti, N. Anselmi, P. Rocca, M. Salucci, and A. Massa, “Minkowski sum method for planar arrays sensitivity analysis with uncertain-but-bounded excitation tolerances,” IEEE Trans. Antennas Propag., vol. 65, no. 1, pp. 167–177, Jan. 2016, doi: 10.1109/TAP.2016.2627548.
[14] L. Poli, P. Rocca, N. Anselmi, and A. Massa, “Dealing with uncertainties on phase weighting of linear antenna arrays by means of interval-based tolerance analysis,” IEEE Trans. Antennas Propag., vol. 63, no. 7, pp. 3229–3234, Jul. 2015, doi: 10.1109/TAP.2015.2421952.
[15] M. Ahadi and S. Roy, “Sparse linear regression (SPLINER) approach for efficient multidimensional uncertainty quantification of high-speed circuits,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 35, no. 10, pp. 1640–1652, Oct. 2016, doi: 10.1109/TCAD.2016.2527711.
[16] X. Yang, M. Choi, G. Lin, and G. E. Karniadakis, “Adaptive ANOVA decomposition of stochastic incompressible and compressible flows,” J. Comput. Phys., vol. 231, no. 4, pp. 1587–1614, 2012, doi: 10.1016/j.jcp.2011.10.028.
[17] C. Cui and Z. Zhang, “Stochastic collocation with non-Gaussian correlated process variations: Theory, algorithms, and applications,” IEEE Trans. Compon., Packag., Manuf. Technol. A) (1994–1998), vol. 9, no. 7, pp. 1362–1375, Jul. 2019, doi: 10.1109/TCPMT.2018.2889266.
[18] R. Hu, V. Monebhurrun, R. Himeno, H. Yokota, and F. Costen, “A statistical parsimony method for uncertainty quantification of FDTD computation based on the PCA and ridge regression,” IEEE Trans. Antennas Propag., vol. 67, no. 7, pp. 4726–4737, Jul. 2019, doi: 10.1109/TAP.2019.2911645.
[19] X. Cheng and V. Monebhurrun, “Application of different methods to quantify uncertainty in specific absorption rate calculation using a CAD-based mobile phone model,” IEEE Trans. Electromagn. Compat., vol. 59, no. 1, pp. 14–23, Feb. 2017, doi: 10.1109/TEMC.2016.2605127.
[20] A. Litvinenko, A. C. Yucel, H. Bagci, J. Oppelstrup, E. Michielssen, and R. Tempone, “Computation of electromagnetic fields scattered from objects with uncertain shapes using multilevel monte Carlo method,” IEEE J. Multiscale Multiphys. Comput. Tech., vol. 4, pp. 37–50, Feb. 2019, doi: 10.1109/JMMCT.2019.2897490.
[21] Z. Zhang, T.-W. Weng, and L. Daniel, “Big-data tensor recovery for high-dimensional uncertainty quantification of process variations,” IEEE Trans. Compon., Packag., Manuf. Technol. A) (1994–1998), vol. 7, no. 5, pp. 687–697, May 2017, doi: 10.1109/TCPMT.2016.2628703.
[22] F. Rosenblatt, “The perceptron: A probabilistic model for information storage and organization in the brain,” Psychol. Rev., vol. 65, no. 6, pp. 386–408, 1958, doi: 10.1037/h0042519.
[23] R. J. Schalkoff, Artificial Neural Networks, vol. 1. New York, NY, USA: McGraw-Hill, 1997.
[24] Y. Kim, S. Keely, J. Ghosh, and H. Ling, “Application of artificial neural networks to broadband antenna design based on a parametric frequency model,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 669–674, Mar. 2007, doi: 10.1109/TAP.2007.891564.
[25] E. Soria-Olivas, J. D. Martn-Guerrero, G. Camps-Valls, A. J. Serrano-Lopez, J. Calpe-Maravilla, and L. Gomez-Chova, “A low-complexity fuzzy activation function for artificial neural networks,” IEEE Trans. Neural Netw.) (1990–2011), vol. 14, no. 6, pp. 1576–1579, Nov. 2003, doi: 10.1109/TNN.2003.820444.
[26] R. Hu, V. Monebhurrun, R. Himeno, H. Yokota, and F. Costen, “An adaptive least angle regression method for uncertainty quantification in FDTD computation,” IEEE Trans. Antennas Propag., vol. 66, no. 12, pp. 2131–2134, Dec. 2018, doi: 10.1109/TAP.2018.2872161.
[27] G. N. Karystinos and D. A. Pados, “On overfitting, generalization, and randomly expanded training sets,” IEEE Trans. Neural Netw.) (1990–2011), vol. 11, no. 5, pp. 1050–1057, Sep. 2000, doi: 10.1109/72.870038.
[28] K. Tekbas, F. Costen, J.-P. Bérenger, R. Himeno, and H. Yokota, “Subcell modeling of frequency-dependent thin layers in the FDTD method,” IEEE Trans. Antennas Propag., vol. 65, no. 1, pp. 278–286, Jan. 2016, doi: 10.1109/TAP.2016.2628712.
[29] A. M. Molinaro, R. Simon, and R. M. Pfeiffer, “Prediction error estimation: A comparison of resampling methods,” Bioinformatics, vol. 21, no. 15, pp. 3301–3307, 2005, doi: 10.1093/bioinformatics/bti499.
[30] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: A simple way to prevent neural networks from overfitting,” J. Mach. Learn. Res., vol. 15, no. 1, pp. 1929–1958, 2014.
[31] R. Hu, V. Monebhurrun, R. Himeno, H. Yokota, and F. Costen, “A general framework for building surrogate models for uncertainty quantification in computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 99, p. 1, Sep. 2021, doi: 10.1109/TAP.2021.3111333.
[32] R. Hu, Y. Liu, Z. Wang, and X. Li, “Blind quality assessment of night-time image,” Displays, vol. 69, p. 102,045, Sep. 2021, doi: 10.1016/j.displa.2021.102045.
[33] J.-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 114, no. 2, pp. 185–200, 1994, doi: 10.1006/jcph.1994.1159.
[34] S. D. Gedney, G. Liu, J. A. Roden, and A. Zhu, “Perfectly matched layer media with CFS for an unconditionally stable ADI-FDTD method,” IEEE Trans. Antennas Propag., vol. 49, no. 11, pp. 1554–1559, Nov. 2001, doi: 10.1109/8.964091.
[35] “Media parameters for the Debye relaxation model,” RIKEN, Tokyo, Japan. Accessed: Sep. 3, 2018. [Online] . Available: http://cfd-duo.riken.jp/cbms-mp/
[36] R. Hu, “Uncertainty quantification for FD-FDTD computations for the human body,” Ph.D. dissertation, Univ. Manchester, Manchester, U.K., 2020.
[37] V. Monebhurrun, C. Dale, J.-C. Bolomey, and J. Wiart, “A numerical approach for the determination of the tissue equivalent liquid used during SAR assessments,” IEEE Trans. Magn., vol. 38, no. 2, pp. 745–748, Mar. 2002, doi: 10.1109/20.996193.
[38] A. M. Abduljabbar, M. E. Yavuz, F. Costen, R. Himeno, and H. Yokota, “Continuous wavelet transform-based frequency dispersion compensation method for electromagnetic time-reversal imaging,” IEEE Trans. Antennas Propag., vol. 65, no. 3, pp. 1321–1329, Mar. 2017, doi: 10.1109/TAP.2016.2647594.
[39] K. Binder, D. Heermann, L. Roelofs, A. J. Mallinckrodt, and S. McKay, “Monte Carlo simulation in statistical physics,” J. Comput. Phys., vol. 7, no. 2, pp. 156–157, 1993, doi: 10.1063/1.4823159.
[40] P. Schratz, J. Muenchow, E. Iturritxa, J. Richter, and A. Brenning, “Hyperparameter tuning and performance assessment of statistical and machine-learning algorithms using spatial data,” Ecological Model., vol. 406, pp. 109–120, Jun. 2019, doi: 10.1016/j.ecolmodel.2019.06.002.
[41] I. Guyon, “A scaling law for the validation-set training-set size ratio,” AT&T Bell Laboratories, Berkeley, CA, USA, 1997. [Online] . Available: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.33.1337&rep=rep1&type=pdf
[42] X. Cheng, C. Henry, F. P. Andriulli, C. Person, and J. Wiart, “A surrogate model based on artificial neural network for RF radiation modelling with high-dimensional data,” Int. J. Environ. Res. Public Health, vol. 17, no. 7, p. 2586, 2020, doi: 10.3390/ijerph17072586.
[43] G. E. Hinton and R. R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” Science, vol. 313, no. 5786, pp. 504–507, 2006, doi: 10.1126/science.1127647.
[44] J. S. Ochoa and A. C. Cangellaris, “Random-space dimensionality reduction for expedient yield estimation of passive microwave structures,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 12, pp. 4313–4321, Dec. 2013, doi: 10.1109/TMTT.2013.2286968.
[45] Alexander, “Sparse grid interpolation,” MathWorks, https://www.mathworks.com/matlabcentral/fileexchange/54289-sparse-grid-interpolation (accessed Sep. 3, 2019).
Digital Object Identifier 10.1109/MAP.2022.3143428