Neural Network Approach for Wideband RCS Computation with Wide Incident Angles via Method of Moments

Abstract

In this paper, we present a deep neural network–based approach for computing radar cross section (RCS) over a wide frequency band and a broad range of incident angles. The proposed network, termed WBRCS-Net, is designed to converge to the solution of the method of moments (MoM) formulation by minimizing a mean-squared residual loss without explicitly solving the MoM linear system, thereby avoiding the numerical instabilities commonly encountered in conventional iterative solvers. Moreover, by using only the frequency and incident angle as inputs, WBRCS-Net enables wideband RCS prediction over a broad range of incident angles while substantially simplifying the network architecture. The performance of WBRCS-Net is evaluated on perfectly electrically conducting (PEC) spheres and cubes and is compared with the Maehly approximation based on Chebyshev polynomials, using monostatic RCS over a frequency range of 2–12 GHz and an incident-angle range of 0°∼90°. Experimental results demonstrate that, once trained, WBRCS-Net enables stable wideband RCS computation over a wide range of incident angles with instantaneous inference speed, achieving a minimum mean-squared error (MSE) on the order of ${10}^{-14}$ relative to reference MoM solutions.

1. Introduction

The radar cross section (RCS) is a fundamental metric for radar detection and target identification, and it is a critical design driver for stealth applications. RCS can be computed from the scattered electromagnetic fields obtained by solving electromagnetic scattering problems. Owing to its high accuracy and broad applicability to arbitrarily complex three-dimensional geometries, the method of moments (MoM) is widely used to compute scattering fields [1]. MoM is a numerical technique based on surface discretization, in which continuous electromagnetic integral equations are transformed into a finite-dimensional linear system characterized by an impedance matrix and an excitation vector, from which the surface current vector is obtained. However, achieving higher accuracy requires finer surface discretization, which significantly increases the dimension of the impedance matrix.

Consequently, building and solving large linear systems is a major bottleneck in MoM-based RCS computation [2]. Conventional iterative solvers, such as the Conjugate Gradient (CG) and Biconjugate Gradient (BiCG) methods, have been widely adopted to accelerate the solution process [3,4]. Nevertheless, for high-frequency scenarios and electrically large or geometrically complex targets, the impedance matrix becomes extremely large, leading to high computational complexity and slow or unstable convergence.

The computational burden is further exacerbated in wideband RCS analysis, where RCS must be evaluated over a broad frequency range. In such cases, the impedance matrix must be built and the corresponding linear system solved at each frequency point [2]. To alleviate this cost, various interpolation techniques have been developed to approximate impedance matrices at intermediate frequencies. Quadratic interpolation was introduced in [5], and cubic polynomial interpolation was later shown to improve accuracy [6].

To further reduce the cost associated with repeatedly solving linear systems, several methods have been proposed to approximate the MoM solution vector over frequency. The asymptotic waveform evaluation (AWE) relies on high-order frequency derivatives of the impedance matrix to approximate the MoM solution vector, and the need to compute and store these derivatives can impose significant memory overhead [7]. To address this limitation, Chebyshev polynomial approximation methods have been introduced, which reconstruct the MoM solution vector using samples at Chebyshev nodes [8,9]. Building on this idea, Maehly approximation, a rational-function method based on Chebyshev polynomials, has been proposed to further improve approximation accuracy [10] and has since been widely applied in wideband RCS computation [11,12,13,14].

In addition to frequency dependence, RCS also varies with the incident angle. Consequently, wideband and wide-incident-angle RCS analysis requires repeated MoM solutions over a two-dimensional frequency–angle domain. Each frequency–angle pair corresponds to a distinct MoM linear system, rendering direct computation prohibitively expensive. To reduce this cost, the Maehly approximation has been extended to a two-dimensional frequency–angle formulation for efficient wideband and wide-incident-angle RCS computation [15]. Nevertheless, these interpolation methods necessitate solving large MoM linear systems at every sampled frequency–angle point, leaving the fundamental challenge of large-scale linear system solves unresolved.

Recently, neural networks have been explored for solving large-scale linear systems due to their modeling flexibility and computational efficiency [16,17,18]. In [19], a neural network directly maps discretized geometric grid points to the MoM solution vector by minimizing the residual of the corresponding MoM linear system. While promising, this approach requires a high-capacity network to represent the extremely high-dimensional input–output mapping and does not inherently support interpolation with respect to frequency or incident angle, thereby limiting its applicability to wideband RCS analysis and scenarios involving a broad range of incident angles.

In this paper, we propose WBRCS-Net, an efficient neural-network-based framework for solving large-scale linear systems arising from the method of moments (MoM), specifically tailored for wideband and wide–incident-angle RCS computation. Unlike existing neural-network-based approaches that learn high-dimensional-to-high-dimensional mappings, WBRCS-Net predicts the MoM solution vector conditioned solely on frequency and incident angle by learning a low-dimensional-to-high-dimensional mapping, which substantially simplifies the network architecture. Specifically, the network takes two scalar inputs—the frequency and the incident angle—and outputs the corresponding MoM solution vector. This formulation provides intrinsic interpolation capability across both frequency and incident angle, enabling accurate prediction at intermediate frequency–angle points without additional MoM solves.

The training time of WBRCS-Net depends on the size of the MoM system and the number of training points, and may be relatively long. However, once trained, WBRCS-Net computes RCS over a wide range of incident angles with instantaneous inference speed, highlighting a key advantage of the neural network–based approach. This interoperability is a key advantage of the neural network approach, and, to the best of the authors’ knowledge, WBRCS-Net is the first neural network-based RCS computation method that provides such capability.

To evaluate WBRCS-Net, we conduct monostatic RCS experiments on two perfectly electrically conducting (PEC) targets: a sphere and a cube. We first demonstrate the convergence behavior of WBRCS-Net at several representative frequencies. Using these sampled points as training data, we evaluate the prediction performance of WBRCS-Net across wideband frequencies and compare it with the widely used Maehly approximation. The two-dimensional interpolation capability of WBRCS-Net across frequency and incident angle is illustrated by visualizing the resulting 3D RCS surface. We further compare the MoM reference solutions and the WBRCS-Net predictions through cross-sectional slices along fixed-frequency and fixed-angle cuts.

The remainder of this paper is organized as follows. Section 2 reviews the MoM formulation and the Maehly approximation for wideband and wide-angle RCS computation. Section 3 presents the architecture and training strategy of WBRCS-Net and describes the residual-based learning formulation. Section 4 details the experimental settings. Section 5 reports wideband and joint frequency-angle RCS results with comparisons to MoM and Maehly baselines. Finally, Section 6 concludes the paper.

2. Related Works

2.1. Method of Moments for Surface Current

The method of moments (MoM) [1] is a widely used numerical method for computing the induced surface current $\mathbf{J}\left(\mathbf{r}\right)$ on a triangulated surface approximating the target object in electromagnetic scattering problems. The unknown surface current $\mathbf{J}\left(\mathbf{r}\right)$ is represented as a linear combination of given surface basis functions, $\left\{{\mathbf{f}}_q\left(\mathbf{r}\right)\right\}$ (e.g., Rao–Wilton–Glisson (RWG) functions defined on triangular meshes with L edges [20]),

$$\mathbf{J}\left(\mathbf{r}\right)=\sum _{q=1}^L{\mathbf{I}}_q{\mathbf{f}}_q\left(\mathbf{r}\right).$$

(1)

MoM converts the continuous EFIE into a discretized linear system for the surface current coefficient vector ${\mathbf{I}}_q$ (referred to as the MoM solution vector, or simply the solution vector) at the incident-field frequency f and incident angle $\theta$,

$$\mathbf{Z}\left(f\right)\mathbf{I}(f,\theta )=\mathbf{V}(f,\theta ),$$

(2)

where $\mathbf{Z}\left(f\right)\in {\mathbb{C}}^{L\times L}$ is the impedance matrix and $\mathbf{V}(f,\theta )\in {\mathbb{C}}^L$ is the excitation vector given by [21]

$${\mathbf{Z}}_{pq}\left(f\right)=j2\pi f{\mu }_0{\int }_S{\int }_S{\mathbf{f}}_p\left(\mathbf{r}\right)[1-{\displaystyle \frac{1}{k^2}}\nabla ({\nabla }^{\prime }·)]\mathbf{G}(\mathbf{r},{\mathbf{r}}^{\prime })·{\mathbf{f}}_q\left({\mathbf{r}}^{\prime }\right)dSd{S}^{\prime },$$

(3)

$${\mathbf{V}}_p(f,\theta )={\int }_S-{\mathbf{E}}^{\mathrm{inc}}(\mathbf{r},f,\theta )·{\mathbf{f}}_p\left(\mathbf{r}\right)dS,$$

(4)

where $\mathbf{G}(\mathbf{r},{\mathbf{r}}^{\prime })=exp(-jk|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)/\left(4\pi \right|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)$ is the Green’s function, and ${\mathbf{E}}^{\mathrm{inc}}(\mathbf{r},f,\theta )$ is the incident electric field at frequency f and incident angle $\theta$.

Since these computations involve computationally intensive numerical evaluation of the surface integrals in (3) and (4), a finer mesh increases the size of the MoM linear system, thereby significantly increasing the computational cost of building $\mathbf{Z}\left(f\right)$ and solving (2).

2.2. Maehly Approximation

The RCS at a frequency f is computed using the surface current coefficient vector $\mathbf{I}\left(f\right)$ that is obtained from the MoM linear system (2) with respect to f. To compute the RCS over a broad frequency band, the MoM linear system must be built and solved at many frequency points, which is computationally expensive. Maehly approximation [22] is the most popular method to alleviate this burden. The Maehly method accurately interpolates the surface current coefficient vector $\mathbf{I}\left(f\right)$ at a given frequency using a rational function of Chebyshev polynomials [23] obtained from the Chebyshev–Gaussian sampling frequencies [14].

For the given frequency range $[f_a,f_b]$, the Chebyshev–Gauss sampling nodes for $N_f$ sample points are given as

$$f_i={\displaystyle \frac{1}{2}}\left[cos\left({\displaystyle \frac{i-0.5}{N_f}}\pi \right)(f_b-f_a)+(f_a+f_b)\right],i=1,2,\dots ,N_f.$$

(5)

The Maehly method approximates the coefficient vector $\mathbf{I}\left(f\right)$ with a rational function of Chebyshev polynomials

$${\mathbf{I}}_q\left(f_i\right)\approx {\displaystyle \frac{a_q^0{T}_q^0\left(\widetilde{f_i}\right)+a_q^1{T}_q^1\left(\widetilde{f_i}\right)+\cdots +a_q^K{T}_q^K\left(\widetilde{f_i}\right)}{b_q^0{T}_q^0\left(\widetilde{f_i}\right)+b_q^1{T}_q^1\left(\widetilde{f_i}\right)+\cdots +b_q^M{T}_q^M\left(\widetilde{f_i}\right)}},$$

(6)

where ${\tilde{f}}_i$ is the normalized frequency of $f_i$, i.e. ${\tilde{f}}_i=cos\left({\displaystyle \frac{i-0.5}{N_f}}\pi \right)$, $T^i$ is a Chebyshev polynomial of degree i [23], and $\{a^i,b^j\}$ are Maehly coefficients that can be recursively computed [14].

Since the RCS computation depends on both frequency and incident angle, the surface current coefficient vector can be treated as a bivariate function $\mathbf{I}(f,\theta )$. Following the two-dimensional approximation approach in [15], each surface current coefficient is approximated over the frequency and angular domains with a rational function of bivariate Chebyshev polynomials constructed on Chebyshev–Gauss sampling nodes:

$${\mathbf{I}}_q(f_i,{\theta }_j)\approx {\displaystyle \frac{a_q^{0,0}{T}_q^0\left({\tilde{f}}_i\right)T_q^0\left({\tilde{\theta }}_j\right)+\cdots +a_q^{K_f,K_{\theta }}{T}_q^{K_f}\left({\tilde{f}}_i\right)T_q^{K_{\theta }}\left({\tilde{\theta }}_j\right)}{b_q^{0,0}{T}_q^0\left({\tilde{f}}_i\right)T_q^0\left({\tilde{\theta }}_j\right)+\cdots +b_q^{M_f,M_{\theta }}{T}_q^{M_f}\left({\tilde{f}}_i\right)T_q^{M_{\theta }}\left({\tilde{\theta }}_j\right)}},$$

(7)

where ${\tilde{\theta }}_j$ is the normalized incident angle of ${\theta }_j$, i.e., ${\tilde{\theta }}_j=cos\left({\displaystyle \frac{j-0.5}{N_{\theta }}}\pi \right)$, with $N_{\theta }$ denoting the number of sampled incident angles, and the Maehly coefficients $\{a_q^{\ell ,m},b_q^{\ell ,m}\}$ can be recursively computed [15].

3. Methods

3.1. Architecture of WBRCS-Net

Figure 1 illustrates the WBRCS-Net, which solves the MoM linear system using a neural network approach. We design the WBRCS-Net to output a solution vector ${\mathbf{I}}^{Net}(f_i,{\theta }_j)$. Since the solution vector is complex-valued, the number of output nodes is set to twice the dimension of the solution vector. Odd-numbered output nodes represent the real part of the solution vector, while even-numbered output nodes represent the imaginary part. We construct the WBRCS-Net using only two residual blocks and fully connected (FC) layers. Each residual block consists of two FC layers with a skip connection:

$$\mathrm{ResBlock}\left(\mathbf{x}\right)=\mathbf{x}+{\mathbf{W}}_2\varphi ({\mathbf{W}}_1\mathbf{x}+{\mathbf{b}}_1)+{\mathbf{b}}_2,$$

(8)

where $\varphi (·)$ denotes the activation function, $({\mathbf{W}}_1,{\mathbf{b}}_1)$ and $({\mathbf{W}}_2,{\mathbf{b}}_2)$ are the weight matrix and bias vector of the first and second FC layers within the residual block, respectively. Residual blocks are adopted to stabilize training and facilitate convergence when learning a high-dimensional solution vector. It has two input nodes that receive the frequency and incident angle. The number of nodes in the hidden layers is expanded step-wise from 2, $L/4$, L, and $2L$, where L is the number of edges in the triangulated object. Specifically, the layer sequence is consisted of a FC $(2\to L/4)$, a ResBlock $(L/4\to L/4)$, a FC $(L/4\to L)$, a ResBlock $(L\to L)$, and a FC $(L\to 2L)$, where activation function is applied after each non-output FC layer and the final layer is linear. This progressive width expansion increases representation capacity when mapping the low-dimensional input $(f,\theta )$ to the $2L$-dimensional output space. Hence, the proposed WBRCS-Net has total ${\displaystyle \frac{35}{8}}{L}^2+6L$ parameters. We use the smooth and bounded hyperbolic tangent activation function to cope with full-range real values.

Figure 1. Architecture of WBRCS-Net.

3.2. WBRCS-Net for MoM Solutions

In this section, we present a neural-network-based approach for solving Method of Moments (MoM) systems using WBRCS-Net. For a given frequency f and incident angle $\theta$, we construct the impedance matrix $\mathbf{Z}\left(f\right)$ and the excitation vector $\mathbf{V}(f,\theta )$. The solution of the resulting linear system is obtained by minimizing the following residual loss via back-propagation:

$$\mathcal{L}\left({\mathbf{I}}_t^{\mathrm{Net}}\right)={\displaystyle \frac{1}{L}}{∥\mathbf{Z}\left(f\right){\mathbf{I}}_t^{\mathrm{Net}}-\mathbf{V}(f,\theta )∥}^2,$$

(9)

where ${\mathbf{I}}_t^{\mathrm{Net}}\in {\mathbb{C}}^L$ denotes the output of WBRCS-Net at the t-th iteration, and L is the dimension of the MoM linear system.

Since the impedance matrix $\mathbf{Z}\left(f\right)$ and the excitation vector $\mathbf{V}(f,\theta )$ remain fixed throughout all iterations, the network input does not encode any physical parameters and can be set to arbitrary dummy values for this single frequency and single incident angle case. WBRCS-Net learns an iterative update rule that progressively reduces the residual of the MoM linear system. In this sense, WBRCS-Net can be interpreted as a learned iterative solver whose update dynamics are optimized through training.

3.3. Training WBRCS-Net for Wideband Frequency and Wide Range Incident Angle

To extend WBRCS-Net to wideband frequencies and a wide range of incident angles, we construct impedance matrices $\left\{\mathbf{Z}\left(f_i\right)\right\}$ and excitation vectors $\left\{\mathbf{V}(f_i,{\theta }_j)\right\}$ over a set of training points $\{f_1,f_2,\dots ,f_{N_f}\}$ and $\{{\theta }_1,{\theta }_2,\dots ,{\theta }_{N_{\theta }}\}$, where $N_f$ and $N_{\theta }$ denote the numbers of training frequencies and incident angles, respectively.

Accordingly, the residual loss in (9) is extended to all training frequency–angle pairs as

$$\mathcal{L}\left({\mathbf{I}}_t^{\mathrm{Net}}\right)={\displaystyle \frac{1}{N_f{N}_{\theta }L}}\sum _{i=1}^{N_f}\sum _{j=1}^{N_{\theta }}{∥\mathbf{Z}\left(f_i\right){\mathbf{I}}_t^{\mathrm{Net}}(f_i,{\theta }_j)-\mathbf{V}(f_i,{\theta }_j)∥}^2.$$

(10)

With this formulation, the output of WBRCS-Net, ${\mathbf{I}}_t^{\mathrm{Net}}(f_i,{\theta }_j)$, explicitly depends on the frequency and incident angle, enabling the network to learn a continuous mapping from $(f,\theta )$ to the MoM solution vector.

At the first step of back-propagation, the weight vector is iteratively updated using

$${\mathbf{w}}_{t+1}={\mathbf{w}}_t-\mu {\displaystyle \frac{\partial \mathcal{L}}{\partial {\mathbf{w}}_t}}={\mathbf{w}}_t-\mu {\displaystyle \frac{\partial \mathcal{L}}{\partial {\mathbf{I}}_t^{Net}}}{\displaystyle \frac{\partial {\mathbf{I}}_t^{Net}}{\partial {\mathbf{w}}_t}}$$

(11)

for all iteration $t=1,\dots ,N_T$, where $N_T$ is the epoch size. Since

$${\displaystyle \frac{\partial \mathcal{L}}{\partial {\mathbf{I}}_t^{Net}}}={\displaystyle \frac{2}{N_f{N}_{\theta }L}}\sum _{i=1}^{N_f}\sum _{j=1}^{N_{\theta }}\left[\mathbf{Z}{\left(f_i\right)}^H\mathbf{Z}\left(f_i\right){\mathbf{I}}_t^{Net}(f_i,{\theta }_j)-\mathbf{Z}{\left(f_i\right)}^H\mathbf{V}(f_i,{\theta }_j)\right],$$

(12)

when we use conventional auto-differentiation, the matrix computation $\mathbf{Z}{\left(f_i\right)}^H\mathbf{Z}\left(f_i\right)$ and $\mathbf{Z}{\left(f_i\right)}^H\mathbf{V}(f_i,{\theta }_j)$ are repeatedly performed $N_T$ times during training. Therefore, if we pre-compute the matrices

$$\mathbf{G}\left(f_i\right):=\mathbf{Z}{\left(f_i\right)}^H\mathbf{Z}\left(f_i\right)$$

(13)

$$\mathbf{S}(f_i,{\theta }_j):=\mathbf{Z}{\left(f_i\right)}^H\mathbf{V}(f_i,{\theta }_j).$$

(14)

and directly implement first stage update using

$${\displaystyle \frac{\partial \mathcal{L}}{\partial {\mathbf{I}}_t^{Net}}}={\displaystyle \frac{2}{N_f{N}_{\theta }L}}\sum _{i=1}^{N_f}\sum _{j=1}^{N_{\theta }}\left[\mathbf{G}\left(f_i\right){\mathbf{I}}_t^{Net}(f_i,{\theta }_j)-\mathbf{S}(f_i,{\theta }_j)\right],$$

(15)

we can save a matrix–matrix and a matrix–vector multiplications for each iteration $t=1,\dots ,N_T$. Table 1 quantifies the resulting training acceleration by comparing training time with the proposed pre-computed gradient against a standard automatic-differentiation in an example setting introduced in Section 5.2 and Section 5.3. As shown, the proposed method achieves a speed-up factor of $1.15$–$1.35$ across the cases, and the benefit becomes more pronounced as $N_T$ increases.

Table 1. Training-time comparison between the proposed pre-computed gradient and a standard automatic-differentiation. The speed-up factor is computed as $T_{\mathrm{auto}}/T_{\mathrm{pre}}$.

Target	${\mathit{N}}_{\mathit{T}}$	${\mathit{T}}_{\mathbf{pre}}$	${\mathit{T}}_{\mathbf{auto}}$	Speed-Up Factor
Sphere ($r=3$ cm, $L=1641$)	120,000	334.1	413.1	1.24
	200,000	579.8	752.9	1.30
Sphere ($r=4$ cm, $L=1884$)	120,000	352.9	439.7	1.25
	200,000	609.1	820.3	1.35
Cube ($a=3$ cm, $L=810$)	120,000	286.2	329.1	1.15
	200,000	488.1	566.1	1.16
Cube ($a=6$ cm, $L=1788$)	120,000	349.3	431.0	1.23
	200,000	599.9	795.3	1.33

The training time of WBRCS-Net depends on the number of edges (L) and the number of training points, and is on the order of minutes in our experiments. In contrast, the prediction (inference) runtime for an arbitrary test point, specified by frequency and incident angle, is only ${10}^{-5}$ seconds to compute the solution vector, highlighting a key advantage of the neural network–based approach.

3.4. RCS Computation

Using the trained WBRCS-Net, the surface currents are obtained from (1). Then, the RCS is computed using the obtained surface current. The formula is as follows [21]:

$$\begin{array}{cc}\hfill \sigma \left(f\right)& ={\displaystyle \frac{\pi f^2}{c_0^2}}{\left|{\int }_S[\widehat{\mathit{r}}\times (\mathit{J}\left({\mathit{r}}^{\mathbf{\prime }}\right)\times \widehat{\mathit{r}})]e^{-jk\widehat{\mathit{r}}·{\mathit{r}}^{\mathbf{\prime }}}d{S}^{\prime }\right|}^2,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & \approx {\displaystyle \frac{\pi f^2}{c_0^2}}{\left|\sum _{i=1}^N{A}_i[\widehat{\mathit{r}}\times (\mathit{J}\left({\mathit{r}}_{\mathit{i}}\right)\times \widehat{\mathit{r}})]e^{-jk\widehat{\mathit{r}}·{\mathit{r}}_{\mathit{i}}}\right|}^2,\hfill \end{array}$$

(17)

where $\widehat{\mathit{r}}$ is the unit observation vector that is opposite to the propagation direction of the incident wave in monostatic RCS. $A_i$ and ${\mathit{r}}_i$ denote the area and the center point of the ith triangulated mesh, respectively, and N is the total number of triangulated meshes.

4. Experiment Settings

4.1. MoM Configuration

In order to train WBRCS-Net, impedance matrices $\mathbf{Z}\left(f_i\right)$ and excitation vectors $\mathbf{V}(f_i,{\theta }_j)$ were built at the training sample frequencies using the python-based BEM++ package (v0.4.2) [24]. The surface currents were expanded with Rao–Wilton–Glisson (RWG) basis functions [20] on triangular surface meshes for perfectly electrically conducting (PEC) objects. The incident wave is a plane wave polarized along the x-axis, and its propagation direction is defined by $\widehat{\mathbf{k}}=[sin\theta ,0,cos\theta ]$. In the frequency-only setting, the incident angle is set to zero, i.e., $\theta =0$ (propagation along the +z-axis), while in the joint frequency–angle setting, $\theta$ ranges from 0° to 90°. The RCS is evaluated in a monostatic configuration, where the observation direction is opposite to the propagation direction of the incident wave.

We evaluated two PEC objects: a sphere and a cube. For the sphere, two radii were considered (3 cm and 4 cm), and for the cube, two side lengths were considered (3 cm and 6 cm), resulting in four experimental cases in total. The corresponding surface meshes are illustrated in Figure 2. The number of edges for the four cases are 1641 (sphere, 3 cm), 1884 (sphere, 4 cm), 810 (cube, 3 cm), and 1788 (cube, 6 cm). All cases are evaluated over the frequency range from 2 GHz to 12 GHz.

Figure 2. Triangulated PEC meshes used in the experiments.

4.2. Data and Training Setting

For the sphere and cube, we consider several different numbers of training frequency samples. We set the number of training epochs separately for each training frequency-sample setting, increasing the number of epochs for larger training frequency sets. The training frequencies are uniformly sampled over the given frequency band. For the Maehly approximation, the Chebyshev–Gauss nodes described in Section 2.2 are used for the sampling frequencies. All frequencies are converted to GHz and used as inputs to the neural network. All incident angles are normalized from the range of 0° to 90° to the range of 0 to 1 and used as inputs to the neural network. Training is performed in a full-batch setting, where all training samples are processed simultaneously in each optimization step using the Adam optimizer with a learning rate of 0.001. All experiments are carried out on an NVIDIA H100 GPU.

5. Experiment Results

5.1. Demonstration of WBRCS-Net for MoM

We conducted experiments to demonstrate the feasibility of the proposed WBRCS-Net for MoM. The experiment was conducted on a unit sphere, triangulated with $L=714$, at a frequency of 100 MHz with the incident angle fixed at $\theta =0°$. During training, the network input was set to a constant dummy value of 1 for the frequency and 0 for the incident angle.

Figure 3 presents the training loss over epochs, demonstrating stable and consistent convergence. Table 2 provides a quantitative comparison against the Mie-theory reference [25] by summarizing the differences in the real part of the surface-current vector (min/max and mean/std). The statistics confirm that WBRCS-Net attains accuracy comparable to MoM. Figure 4a,b visualizes the real component of the surface currents obtained from the MoM solution and the WBRCS-Net, respectively. The surface currents predicted by WBRCS-Net closely match the MoM surface currents. The corresponding RCS values computed from the MoM and WBRCS-Net surface currents are 4.15204 and 4.150207, respectively, indicating near-identical RCS results. After 5000 iterations, WBRCS-Net achieves a residual error of approximately ${10}^{-13}$ with a runtime of 5.35 s on an NVIDIA H100 GPU.

Figure 3. Training loss over epochs.

Table 2. Statistics of the differences between the real component of surface-current vectors and the Mie-theory reference.

Metric	WBRCS-Net-Mie		MoM-Mie
	(min, max)	(mean, std)	(min, max)	(mean, std)
$dx$	($-5.22\times {10}^{-3}$, $5.89\times {10}^{-3}$)	($1.38\times {10}^{-4}$, $1.39\times {10}^{-3}$)	($-5.22\times {10}^{-3}$, $5.78\times {10}^{-3}$)	($1.32\times {10}^{-4}$, $1.38\times {10}^{-3}$)
$dy$	($-7.31\times {10}^{-3}$, $6.81\times {10}^{-3}$)	($-1.37\times {10}^{-5}$, $2.51\times {10}^{-3}$)	($-7.31\times {10}^{-3}$, $6.83\times {10}^{-3}$)	($-1.40\times {10}^{-5}$, $2.50\times {10}^{-3}$)
$dz$	($-6.59\times {10}^{-3}$, $5.34\times {10}^{-3}$)	($1.84\times {10}^{-5}$, $2.09\times {10}^{-3}$)	($-6.54\times {10}^{-3}$, $5.33\times {10}^{-3}$)	($1.79\times {10}^{-5}$, $2.08\times {10}^{-3}$)
$\parallel d\parallel =\parallel J-J_{\mathrm{Mie}}\parallel$	($3.88\times {10}^{-4}$, $7.72\times {10}^{-3}$)	($3.26\times {10}^{-3}$, $1.39\times {10}^{-3}$)	($4.11\times {10}^{-4}$, $7.73\times {10}^{-3}$)	($3.26\times {10}^{-3}$, $1.39\times {10}^{-3}$)

Figure 4. Surface currents on the unit sphere at 100MHz: (a) MoM. (b) WBRCS-Net.

5.2. Wideband RCS Computation on the Sphere

In Figure 5, we compare the wideband RCS obtained using the Maehly approximation and WBRCS-Net under different numbers of sampled frequencies. Figure 5a,b presents the results for a sphere with a radius of 3 cm ($L=1641$) when the number of sampled frequencies is $N_f=18$ and $N_f=30$, respectively. The black, blue, and red curves correspond to the RCS computed by MoM, Maehly, and WBRCS-Net, respectively. For WBRCS-Net, the training runtime is 334.1 s for $N_f=18$ (120,000 epochs) and 579.8 s for $N_f=30$ (200,000 epochs), while the inference time is only ${10}^{-5}$ s per test point. For the 3 cm sphere, both methods exhibit good interpolation performance.

Figure 5. Monostatic RCS on the sphere (r = 3 cm) with $N_f$ sampled points: (a) $N_f=18$. (b) $N_f=30$.

Figure 6 presents an experiment designed to evaluate interpolation performance under a more rapidly changing RCS response. We increase the sphere radius to 4 cm ($L=1884$), which yields an RCS curve with more rapid changes. We consider two training frequency-sample settings, $N_f=18$ and $N_f=30$; for WBRCS-Net, the training runtime is 352.9 s for $N_f=18$ (120,000 epochs) and 609.1 s for $N_f=30$ (200,000 epochs), while the inference time is only ${10}^{-5}$ s per test point. When the RCS varies more rapidly, the Maehly approximation exhibits unsatisfactory interpolation performance even as the number of sampled frequencies increases from $N_f=18$ to $N_f=30$. In contrast, WBRCS-Net maintains reliable interpolation performance even with $N_f=18$, and with $N_f=30$, the computed wideband RCS closely matches the MoM reference. WBRCS-Net produces a smooth and well-behaved wideband response without abrupt spikes, indicating improved robustness to rapidly varying RCS behavior compared with the Maehly approximation.

Figure 6. Monostatic RCS on the sphere (r = 4 cm) with $N_f$ sampled points: (a) $N_f=18$. (b) $N_f=30$.

To provide a quantitative comparison, Figure 7 reports the mean squared error (MSE) with respect to the RCS, directly computed by MoM, evaluated at test frequencies spaced every 100 MHz. Figure 7a summarizes the results for the 3cm sphere by sweeping the number of training frequencies from 9 to 30. In this less rapidly changing case, the MSE gap between the Maehly approximation and WBRCS-Net remains relatively small, indicating comparable interpolation performance. However, as shown in Figure 7b, for the more rapidly changing 4cm sphere, WBRCS-Net achieves lower MSE than the Maehly approximation for almost all sampling configurations. Overall, these results suggest that WBRCS-Net provides more accurate and reliable wideband interpolation when the RCS exhibits more rapid changes with frequency.

Figure 7. MSE for each number of samples: (a) sphere(r = 3 cm). (b) sphere(r = 4 cm).

5.3. Wideband RCS Computation on the Cube

In Figure 8, we compare the wideband RCS of the cube with an edge length of 3 cm ($L=810$) computed using the Maehly approximation and WBRCS-Net. For WBRCS-Net, the training runtime is 286.2 s for $N_f=18$ (120,000 epochs) and 488.1 seconds for $N_f=30$ (200,000 epochs), while the inference time is only ${10}^{-5}$ s per test point. The RCS curve is relatively smooth across the frequency band, showing a single broad change followed by a gradual increase toward higher frequencies. When $N_f=18$, the Maehly approximation shows a small spike around 12 GHz. When $N_f$ is increased to 30, both Maehly and WBRCS-Net yield accurate and stable wideband interpolation, closely following the MoM reference.

Figure 8. Monostatic RCS on the cube (a = 3 cm) with $N_f$ sampled points: (a) $N_f=18$. (b) $N_f=30$.

In Figure 9, we evaluate a more challenging case by increasing the cube edge length to 6 cm ($L=1788$). Compared with the 3 cm cube, the wideband RCS curve exhibits more frequent changes across the frequency band. For WBRCS-Net, the training runtime is 349.3 s for $N_f=18$ (120,000 epochs) and 599.9 s for $N_f=30$ (200,000 epochs), while the inference time is only ${10}^{-5}$ s per test point. In Figure 9a with $N_f=18$, the Maehly approximation is consistent with the MoM reference over most of the band. Interestingly, when the number of training frequencies is increased to $N_f=30$ in Figure 9b, the Maehly result exhibits a slightly local mismatch around 12 GHz. In contrast, WBRCS-Net remains stable and closely aligned with the MoM reference for both $N_f=18$ and $N_f=30$, demonstrating accurate and robust wideband interpolation even for an RCS curve with more frequent variations.

Figure 9. Monostatic RCS on the cube (a = 6 cm) with $N_f$ sampled points: (a) $N_f=18$. (b) $N_f=30$.

To provide a quantitative comparison, Figure 10 reports the mean squared error (MSE) relative to the MoM reference, evaluated at test frequency every 100 MHz. Figure 10a summarizes the results for the 3 cm cube by varying the number of training frequencies from 9 to 30. The MSE difference between the Maehly approximation and WBRCS-Net remains small across this range, indicating comparable interpolation fidelity; in some cases, the Maehly approximation even slightly outperforms WBRCS-Net. Figure 10b shows that for the 6 cm cube, whose RCS response varies more rapidly with frequency, WBRCS-Net exhibits comparable MSE at lower sampling counts, while delivering lower MSE as the number of sampled frequencies increases. Overall, these results indicate that WBRCS-Net provides more accurate and reliable wideband interpolation when the RCS curve exhibits more frequent variations across frequency.

Figure 10. MSE for each number of samples: (a) cube (a = 3 cm). (b) cube (a = 6 cm).

Maehly relies on Chebyshev polynomial interpolation, which is computationally inexpensive in the offline stage, but because it uses a fixed polynomial basis, its accuracy depends on how smoothly the target function varies; when the RCS changes rapidly, a higher polynomial degree is typically required and the interpolation error can increase [26,27]. By contrast, WBRCS-Net incurs a higher offline cost due to network training, yet its nonlinear, high-capacity representation can model more complex responses and thus better capture rapidly varying RCS patterns, leading to more reliable wideband interpolation in our experiments.

5.4. Comparison of Offline and Online Runtime

We report the offline and online runtime in Table 3. Here, the offline time includes one-time costs required to enable each approach (e.g., building Z and V at sampling points, estimating Maehly coefficients, or training WBRCS-Net), whereas the online time corresponds to the runtime required to generate wideband RCS results for a given task after the offline stage is completed. All experiments in this runtime comparison were conducted on an NVIDIA H100 GPU. In this experiment, both Maehly and WBRCS-Net used 18 training frequency points and 101 test frequency points, respectively. For direct MoM, the online cost is dominated by repeatedly building the MoM system and solving the linear system at all test frequencies. In contrast, both Maehly and WBRCS-Net substantially reduce the online runtime once their offline stage is completed. However, WBRCS-Net incurs a higher offline cost than Maehly due to network training.

Table 3. Offline/online runtime for wideband RCS computation.

Case	Method	Offline			Online
		Building $(\mathit{Z},\mathit{V})$	Solving	Training	Building $(\mathit{Z},\mathit{V})$	Solving	Inference
Sphere (3 cm)	Direct MoM	-	-	-	569.64	10.5545	-
	Maehly	101.52	1.881	0.1595	-	-	0.06
	WBRCS-Net	101.52	-	334.1	-	-	0.001
Sphere (4 cm)	Direct MoM	-	-	-	631.25	15.2409	-
	Maehly	112.5	2.7162	0.1779	-	-	0.079
	WBRCS-Net	112.5	-	352.9	-	-	0.001
Cube (3 cm)	Direct MoM	-	-	-	397.94	2.4745	-
	Maehly	70.92	0.441	0.0816	-	-	0.028
	WBRCS-Net	70.92	-	286.2	-	-	0.001
Cube (6 cm)	Direct MoM	-	-	-	590.85	13.9481	-
	Maehly	105.3	2.4858	0.1671	-	-	0.064
	WBRCS-Net	105.3	-	349.3	-	-	0.001

5.5. Wideband RCS Computation with Incident Angle on the Cube

Following the experimental setup in [15] for joint interpolation over frequency and incident angle, we consider a PEC cube with an edge length of 0.5 cm, triangulated with $L=1464$, over a frequency band of 10–50 GHz and an incident angle range of $\theta \in [0°, 90°]$. We construct a training set using $N_f=21$ frequencies and $N_{\theta }=21$ incident angles, yielding a total of 441 training points. WBRCS-Net is trained for 500,000 epochs, with a total training runtime of approximately 52 min, while the inference runtime is ${10}^{-5}$ s per test point. For evaluation, we test on a dense grid with a 100 MHz frequency spacing and a $1°$ angular spacing, resulting in $401\times 91$ test points. Figure 11 illustrates the predicted frequency–angle RCS as a 3D surface (in dB), providing an overall view of the joint variation with respect to frequency and incident angle. Table 4 reports quantitative metrics and distribution statistics over the entire frequency–angle grid, supporting the validity of the WBRCS-Net predictions against the MoM reference.

Figure 11. 3D Monostatic RCS on the cube (a = 0.5 cm) predicted by WBRCS-Net.

Table 4. Statistical comparison of 3D monostatic RCS (cube, $a=0.5$ cm) between MoM and WBRCS-Net.

Metric	|MoM-WBRCS-Net|
Min	$2.1071\times {10}^{-12}$
Max	$1.9068\times {10}^{-6}$
Mean	$1.8272\times {10}^{-7}$
Std	$2.2783\times {10}^{-7}$
MSE	$8.5297\times {10}^{-14}$
RMSE	$2.9205\times {10}^{-7}$

To validate the surface prediction against the MoM reference, we take representative one-dimensional cuts along each axis. Specifically, Figure 12a plots the RCS versus frequency at a fixed incident angle of $\theta =45°$, and Figure 12b plots the RCS versus incident angle at a fixed frequency of $f=30$ GHz. In both cuts, WBRCS-Net remains consistent with the MoM response, indicating accurate and stable interpolation in both frequency and angle. These results demonstrate that WBRCS-Net enables accurate and stable joint interpolation over frequency and incident angle.

Figure 12. Monostatic RCS on the cube (a = 0.5 cm): (a) Incident angle, $\theta =45°$. (b) Incident frequency, f = 30 GHz.

6. Conclusions

In this work, we propose WBRCS-Net, an efficient neural network for solving large-scale linear systems arising in MoM-based RCS computation. By minimizing the residual error as a loss function, WBRCS-Net converges to the accurate solution of the underlying linear system. Moreover, by taking the frequency and incident angle as inputs, WBRCS-Net enables RCS interpolation at arbitrary frequency–angle points, thereby eliminating the need for repeated impedance matrix construction and corresponding linear solves. The effectiveness of WBRCS-Net is demonstrated on PEC spheres and cubes across a wide range of frequencies and incident angles, highlighting both its accuracy and interpolation capability. Although validated on canonical PEC targets, the proposed residual-minimization framework is not tied to a specific geometry (including more complex, electrically large, or non-convex targets) and can be applied to other MoM-based scattering problems by training with the corresponding MoM system. Moreover, the interoperability of the proposed network allows it to be applied to other parameter-dependent scattering problems. For example, it can be extended to time-varying scenarios such as in [28,29], where time is treated as an input parameter. Exploring such extensions constitutes an important direction for future research.