Pith. sign in

REVIEW 5 major objections 6 minor 44 references

Novel Physics-Aware Attention-Based Machine Learning Approach for Mutual Coupling Modeling

T0 review · 5 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A physics-aware convolutional LSTM predicts dipole-array mutual impedances from the analytic Green's function, reaching a 7x speedup over full-wave simulation.

desk verdict The paper's physics-aware story rests on a hand-wavy LSTM-to-MoM equivalence that does not hold up; what is left is an empirical surrogate with modest accuracy and no open code. read the letter →

arxiv 2507.09561 v1 pith:I5YZ6UOE submitted 2025-07-13 eess.SP

Can Wang , Wei Liu , Hanzhi Ma , Xiaonan Jiang , Erping Li , Steven Gao This is my paper · ORCID
classification eess.SP
keywords mutualcouplingdipoleantennaarrayportimpedancematrixphysics-awareneuralnetworkconvolutionallongshort-termmemoryattentionmechanismmethodofmomentsGreen'sfunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that a carefully structured neural network can replace full-wave electromagnetic simulation for computing the mutual impedance matrix of dipole antenna arrays, at a fraction of the cost. The proposed PC-LSTM architecture splits the problem: a physics-aware subnetwork learns the Green's function from the analytical expression rather than from labeled data, an attention mechanism fuses the real and imaginary parts of that function, and a convolutional LSTM maps the fused representation to port impedances. Validation on two-element, ten-element, and thirty-element arrays shows impedance predictions that match a commercial antenna toolbox closely and full-wave simulation within engineering tolerance, with up to a 7x speedup. If this holds, array design and beamforming work that currently depends on repeated full-wave solves could be accelerated substantially.

What carries the argument

The load-bearing object is the mapping between the MoM impedance integral and the neural architecture: the Green's function matrix $G(r,r')$ is predicted by a physics-aware neural network whose loss is computed against the analytical expression in Eq. (9), so no labeled training data is required; a self-attention module fuses the real and imaginary Green's function components into $X_{fused}$; a physics-aware convolution kernel with exponential distance decay normalizes to sum to one; and a convolutional LSTM processes the sequence of row vectors and outputs the port impedance matrix via modified nodal analysis. The paper's claimed physical interpretability rests on the identity between the LSTM's expanded output, after linearizing $\tanh$, and the discrete MoM sum $Z_{mn} = \sum w_{mn}G_{mn}$.

What would settle it

Train the proposed PC-LSTM on a fixed array, then extract the learned combination weights $\beta_t$ from the LSTM gates and compare them numerically with the MoM coefficients $w_{mn}$ from Eq. (20) for the same Green's function; if the two sets of numbers do not agree in magnitude and sign pattern, the claimed correspondence between the network and the MoM formula is refuted. A second check is to replace the physics-aware PANN branch with a random or constant Green's function input and measure the resulting impedance error: if accuracy is unchanged, the physics-aware component is not carrying the result.

Watch

Extended reading notes

Core claim

The central claim is that the discrete method-of-moments impedance formula, $Z_{mn} = \sum_{m,n} w_{mn}G_{mn}$, can be reorganized into a neural computation whose layers mirror the physical steps: the Green's function is produced by a physics-aware network constrained by the analytical free-space Green's function, the basis-function integrals are approximated by a physics-aware convolution kernel with distance-decaying weights, and the LSTM's recurrence over rows of the Green's function matrix plays the role of the impedance summation. The paper argues that with a first-order Taylor approximation of the LSTM's tanh nonlinearity, the network output takes the same algebraic form as the MoM sum, so the learned combination weights $\beta_t$ correspond to the physical coefficients $w_{mn}$. On that basis, the authors present numerical results for port impedance matrices and S-parameters of dipole arrays, reporting relative errors near 0.1% against the antenna toolbox and around 9-13% against full-wave simulation, with inference time speedups of 3-3.5x over the toolbox and over 7x over full-wave simulation.

Load-bearing premise

The load-bearing premise is that the LSTM's nonlinear update can be replaced by a first-order linear approximation so that the network's internal weights stand in for the physical method-of-moments coefficients; the paper states this equivalence but does not prove that the approximation holds for the trained network, and if it does not, the physical-interpretability justification collapses, even though the model could still predict impedances accurately as a black-box surrogate.

Editorial extensions

If this is right

  • Mutual impedance matrices for uniformly and non-uniformly spaced linear dipole arrays can be obtained without labeled full-wave training data, because the PANN loss is computed from the analytical Green's function.
  • Once trained, inference replaces an iterative MoM solve, giving a speedup of more than 7x relative to full-wave simulation for the tested configurations.
  • The two-element subarray model can be cascaded to synthesize port impedance matrices of larger arrays (10 and 30 elements) under the spacing constraints $0.5\lambda \ge d_1 \ge 0.1\lambda$ and $d_1 + d_2 \ge 0.6\lambda$.
  • Attention-based fusion of the real and imaginary parts improves convergence and accuracy of complex-valued impedance prediction compared with unweighted training.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The asserted equivalence between LSTM weights and MoM coefficients depends on a linearization of $\tanh$ that a trained network will not satisfy; if the equivalence fails, the 'physics-aware' label mostly acts as a training prior, and the model's accuracy is that of an empirical surrogate rather than a true MoM replacement.
  • The method is demonstrated for half-wavelength dipoles and linear arrays; extending the PANN-for-Green's-function strategy to other element types or planar arrays is plausible, but the cascade synthesis relies on spacing constraints that effectively ignore coupling beyond nearby blocks, which would need explicit validation for dense or electrically large arrays.
  • A direct numerical check would be to extract the learned $\beta_t$ from a trained LSTM and compare them to the $w_{mn}$ computed from Eq. (20); a mismatch would not change the reported accuracy but would show that the physical-interpretability argument is not what drives the predictions.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The manuscript proposes a physics-aware convolutional long short-term memory (PC-LSTM) framework for estimating mutual impedance (port impedance) matrices of dipole antenna arrays. A physics-aware neural network (PANN) is trained to reproduce the analytic free-space Green's function, an adaptive loss balances real and imaginary parts, a self-attention module fuses the two components, and a ConvLSTM predicts impedance matrices for two-element subarrays and, by cascading, for larger linear arrays. The authors assert a mathematical correspondence between the LSTM output and the discrete method-of-moments (MoM) impedance expression via a first-order approximation of the tanh activation (Eqs. (29)-(30)). Validation is reported for two-element arrays against CST and MATLAB Antenna Toolbox, and for 10- and 30-element arrays against Antenna Toolbox, with speedups over CST. The abstract concludes that the method provides accurate impedance extraction with up to 7× speedup and enhanced physical interpretability.

Significance. A fast, reasonably accurate surrogate for mutual impedance in dipole arrays would be practically useful, and the paper contains a substantial experimental corpus: multiple array sizes, uniform/non-uniform spacings, and comparisons with several learning-based methods. The reported speedups are concrete. However, the significance hinges on the physics-aware interpretability claim, which rests on a derivation that is not mathematically justified, and on numerical accuracy against CST, which the authors' own tables show to be around 10%. As it stands, the method is presented as a black-box surrogate with a largely decorative 'physics-aware' component.

major comments (5)
  1. [Section II-E, Eqs. (29)-(30)] The claimed equivalence between the LSTM output and the MoM impedance sum is not established. The step from Eq. (29) to Eq. (30) replaces both tanh nonlinearities with the identity map, appealing to a 'first-order approximation'. This is invalid for a trained LSTM: the cell state c_t and gate outputs are not small, and the gating and tanh nonlinearities are essential to the LSTM's ability to model long-range dependencies. Even after linearization, the term β_t [h_{t-1}, g_t] multiplies a concatenated vector, so β_t must be a matrix, whereas w_mn in Eq. (20) is a scalar. Furthermore, h_{t-1} encodes the entire preceding input sequence, so the sum over t is not a weighted sum of individual Green's function samples G_mn. The identification of β_t with w_mn immediately after Eq. (30) is therefore unsupported. This is load-bearing because the paper's 'physics-aware' and 'physical interpretability' contributions depend on it.
  2. [Section II-E, Eqs. (21)-(23)] Equation (19) is stated for the MoM impedance matrix Z, but the network is trained and evaluated on the port impedance matrix Z_port = (M Z^{-1} M^T)^{-1}. This port reduction requires a matrix inverse and a port selection matrix, so Z_port is not a direct weighted sum of Green's function entries. The derivation in Eqs. (19)-(30) never addresses this distinction. Without an argument connecting Z to Z_port, the claimed correspondence between learned weights and physical coefficients does not apply to the quantity being predicted.
  3. [Section III-B, Table II] The abstract claims 'accurate impedance extraction' and a 'fast alternative to full-wave simulations', but Table II shows that the PC-LSTM prediction for Z11 differs from CST by 13.1% (Case 1) and 9.46% (Case 2). The text attributes this to the difference between MATLAB Antenna Toolbox's ideal-dipole model and CST's more physical model; however, Antenna Toolbox itself differs from CST by only about 0.09% and 0.025% in those rows, so the dominant error is introduced by PC-LSTM. Moreover, the 'Relative Error (%)' row in Table II is ambiguous, since it does not state whether the reference is CST or Antenna Toolbox. As reported, the results do not substantiate the accuracy claim in the abstract.
  4. [Section II-B, Eq. (10)] The PANN is trained to match the analytic Green's function expression in Eq. (9), which is a closed-form formula. Describing this as 'unsupervised' or as embedding a 'physical constraint' is misleading: it is standard supervised regression with an analytic target. The unsupervised claim does not extend to the impedance predictor, which is trained on 100 labeled samples from MATLAB Antenna Toolbox (Section III-B and Table III). The paper should either temper the 'minimal reliance on labeled data' claim or clarify that it applies only to the PANN component.
  5. [Section III-C] For the 10- and 30-element arrays, the predicted impedance matrices are validated only against MATLAB Antenna Toolbox, not against CST or any other full-wave solver. The two-element synthesis rule is justified only by the heuristic observation in Fig. 8 and the spacing constraints in Section III-C. Given that the two-element model already has roughly 10% deviation from CST, a full-wave comparison for at least one large-scale case is needed to support the claim that PC-LSTM is a fast alternative to full-wave simulation for mutual coupling characterization.
minor comments (6)
  1. [Abstract] In the sentence beginning 'Also, an attention mechanism is carefully designed to calibrates...', 'calibrates' should be 'calibrate'.
  2. [Section III-A] 'fatest' should be 'fastest' in the sentence describing inference speed.
  3. [Table III] 'antoencoders' should be 'autoencoders'.
  4. [Section III-A and III-B] Section III-A sets the number of discretization segments to N=16 with a 16×16 PANN output, but Section III-B states that the PANN produces 32×32 Green's function matrices; please clarify the relationship between N and the matrix size.
  5. [Section III-B] The input tuple to PANN is listed as 'd_i, l, r, and f', but d_i is not defined in Section II-B; explain how element spacing enters the Green's function prediction.
  6. [Abstract] The abstract mentions 'five benchmarks' but the paper does not enumerate them; please list the benchmarks explicitly in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the final impedance predictor is supervised against Antenna Toolbox data and checked against CST, while the PANN merely regresses to the closed-form Green's function.

full rationale

The paper's final impedance predictions are obtained by a PC-LSTM trained on port impedance matrices from MATLAB's Antenna Toolbox and then compared against CST, an external full-wave simulator, so the central validation does not reduce to the model's own inputs. The PANN is trained via an MSE loss against the analytical Green's function expression in Eq. (9); this is a regression to a known closed form rather than a circular derivation of a new physical result. The LSTM-to-MoM equivalence in Section II-E is asserted through a first-order Taylor linearization of tanh and a claim that the learned weights beta_t correspond to the MoM coefficients w_mn, but this is an unproven analogy and a correctness risk, not a circular step that makes the prediction equivalent to its inputs by construction. No load-bearing self-citation or author-imported uniqueness theorem is used. Therefore, no circularity is identified.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The method relies on established electromagnetics and ML tools, but introduces several unspecified hyperparameters and a hand-crafted kernel. The most significant burden is the unjustified first-order approximation of tanh in the LSTM-to-MoM derivation, which is an ad-hoc assumption particular to this paper. No new physical entities are introduced.

free parameters (5)
  • Decay factor alpha in physics-aware convolution kernel
    Controls weight decay with distance in the hand-crafted kernel (Section II-D Step 1); chosen by design, not learned, and no value is reported.
  • Adaptive loss weight parameter alpha
    In Eq. (13), mixes the base weight and the error discrepancy term; no value is provided.
  • Threshold L_bar_l in adaptive loss
    Used in Eq. (13) to decide which component (real or imaginary) receives a larger weight; unspecified.
  • Number of subdivisions N = 16 in Section III-A, 32 in Section III-B
    Determines the size of the Green's function matrix and PANN output; inconsistent across sections, and no sensitivity analysis is given.
  • Architectural hyperparameters
    Number of LSTM layers is stated as four, but hidden sizes, learning rate, batch size, and optimizer are not reported, leaving the model underspecified.
assumptions (6)
  • standard math Free-space scalar Green's function (Eq. 8) is valid for dipole segments
    Standard electromagnetic expression used as the supervision target for PANN (Section II-B).
  • standard math MoM discretization with rooftop basis functions (Eq. 6) is accurate for thin dipoles
    Standard method of moments formulation used to motivate the mapping (Section II-A).
  • domain assumption Mutual impedance is negligible when spacing exceeds 0.6 lambda
    Section III-C states this to justify the two-element cascade synthesis for large arrays, but no numerical evidence is provided for the threshold.
  • domain assumption Large-array synthesis via cascaded two-element units with constraints 0.5 lambda >= d1 >= 0.1 lambda and d1 + d2 >= 0.6 lambda is valid
    Section III-C introduces these constraints to ensure coupling assumptions hold; they restrict the method to a subset of possible array configurations.
  • standard math LSTM can approximate any nonlinear mapping (universal approximation)
    Cites [40] and is used to justify employing LSTM for the impedance mapping (Section II-E).
  • ad hoc to paper First-order Taylor approximation of tanh in Eq. (30) is accurate enough to equate LSTM output to the MoM sum
    This is the key step in the derivation; it is not justified and is generally false for the nonlinear tanh, making the physical mapping claim unsupported.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Novel Physics-Aware Attention-Based Machine Learning Approach for Mutual Coupling Modeling." pith.science (2026). https://pith.science/paper/I5YZ6UOE

@misc{pith2026250709561,
  author       = {Pith},
  title        = {Pith review of: Novel Physics-Aware Attention-Based Machine Learning Approach for Mutual Coupling Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5YZ6UOE}},
  note         = {Machine review of arXiv:2507.09561}
}
read the original abstract

This article presents a physics-aware convolutional long short-term memory (PC-LSTM) network for efficient and accurate extraction of mutual impedance matrices in dipole antenna arrays. By reinterpreting the Green's function through a physics-aware neural network and embedding it into an adaptive loss function, the proposed machine learning-based approach achieves enhanced physical interpretability in mutual coupling modeling. Also, an attention mechanism is carefully designed to calibrate complex-valued features by fusing the real and imaginary parts of the Green's function matrix. These fused representations are then processed by a convolutional long short-term memory network, and the impedance matrix of the linear antenna array can be finally derived. Validation against five benchmarks underscores the efficacy of the proposed approach, demonstrating accurate impedance extraction with up to a 7x speedup compared to CST Microwave Studio, making it a fast alternative to full-wave simulations for mutual coupling characterization.

Figures

Figures reproduced from arXiv: 2507.09561 by the authors.

Figure 1
Figure 1. Illustration of the proposed PC-LSTM architecture. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The correspondence between the MoM formulation and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Framework of the proposed self-attention mechanism. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Comparison of real part (a) and imaginary part (b) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: Loss function of PC-LSTM [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: It can be seen that, within the frequency range of 2–2.8 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: Variations of the self-impedance and mutual impedance [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: Synthesis of a large-scale antenna array based on a two [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: Loss function curves for arrays with 10 elements and [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 11
Figure 11. Figure 11: Impedance matrices for two 10-element arrays with [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]
Figure 12
Figure 12. Figure 12: Impedance matrices for two 30-element arrays with [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 42 canonical work pages

  1. [1]

    Compact and wideband crossed dipole antenna using coupling stub for circular polarization,

    L. Wen, S. Gao, B. Sanz-Izquierdo, C. Wang, W. Hu, X. Ren, and J. Wu, “Compact and wideband crossed dipole antenna using coupling stub for circular polarization,” IEEE Trans. Antennas Propag. , vol. 70, no. 1, pp. 27–34, 2021

    work page 2021

  2. [2]

    R. L. Haupt, Antenna arrays: a computational approach . John Wiley & Sons, 2010

    work page 2010

  3. [3]

    A novel calculation-based parasitic decoupling technique for increasing isolation in multiple-element MIMO antenna arrays,

    M. Li and S. Cheung, “A novel calculation-based parasitic decoupling technique for increasing isolation in multiple-element MIMO antenna arrays,” IEEE Trans. Veh. Technol., vol. 70, no. 1, pp. 446–458, 2020

    work page 2020

  4. [4]

    Sectorized FMCW MIMO radar by modular design with non-uniform sparse arrays,

    C. A. Alistarh, S. K. Podilchak, P. D. H. Re, T. M. Str ¨ober, Y . Pailhas, C. Mateo-Segura, M. Sellathurai, G. Goussetis, Y . R. Petillot, J. S. Thompson, et al., “Sectorized FMCW MIMO radar by modular design with non-uniform sparse arrays,” IEEE Journal of Microwaves , vol. 2, no. 3, pp. 442–460, 2022

    work page 2022

  5. [5]

    Tdm-mimo automotive radar point-cloud detection based on the 2-d hybrid sparse antenna array,

    J. Ding, Z. Wang, W. Ma, X. Wu, and M. Wang, “Tdm-mimo automotive radar point-cloud detection based on the 2-d hybrid sparse antenna array,” IEEE Trans. Geosci. Remote Sens. , vol. 60, pp. 1–15, 2022

    work page 2022

  6. [6]

    Kedar, Sparse Phased Array Antennas: Theory and Applications

    A. Kedar, Sparse Phased Array Antennas: Theory and Applications . Artech House, 2022

    work page 2022

  7. [7]

    M. G. Amin, Sparse Arrays for Radar, Sonar, and Communications . John Wiley & Sons, 2024

    work page 2024

  8. [8]

    Performance of adaptive array antenna with arbitrary geometry in the presence of mutual coupling,

    Q. Yuan, Q. Chen, and K. Sawaya, “Performance of adaptive array antenna with arbitrary geometry in the presence of mutual coupling,” IEEE Trans. Antennas Propag. , vol. 54, no. 7, pp. 1991–1996, 2006

    work page 1991

Show all 44 references
  1. [9]

    Design of maximally sparse antenna arrays in the presence of mutual coupling,

    C. Bencivenni, M. Ivashina, R. Maaskant, and J. Wettergren, “Design of maximally sparse antenna arrays in the presence of mutual coupling,” IEEE Antennas Wirel. Propag. Lett. , vol. 14, pp. 159–162, 2014

    2014

  2. [10]

    Design of sparse antenna array using physics-aware generative adversarial network,

    C. Wang, Y . Zhang, S. Gao, and W. Liu, “Design of sparse antenna array using physics-aware generative adversarial network,” IEEE Trans. Antennas Propag., pp. 1–1, 2025

    2025

  3. [11]

    Coupling-informed data-driven scheme for joint angle and frequency estimation in uniform linear array with mutual coupling present,

    Y . Zhang, W. Xu, A.-L. Jin, M. Li, P. Ma, L. Jiang, and S. Gao, “Coupling-informed data-driven scheme for joint angle and frequency estimation in uniform linear array with mutual coupling present,” IEEE Trans. Antennas Propag., vol. 72, no. 12, pp. 9117–9128, 2024

    2024

  4. [12]

    Design of decoupling and pattern shaping surface for MIMO antennas using the multiport optimization method,

    M. Li, Y . He, C. Zhou, Y . Zhang, and D. Wu, “Design of decoupling and pattern shaping surface for MIMO antennas using the multiport optimization method,” IEEE Trans. Antennas Propag. , vol. 73, no. 5, pp. 2927–2939, 2025

    2025

  5. [13]

    Interaction suppression technique for high-density antenna arrays for mm-wave 5G MIMO systems,

    M. Alibakhshikenari, B. S. Virdee, A. A. Althuwayb, F. Falcone, and E. Limiti, “Interaction suppression technique for high-density antenna arrays for mm-wave 5G MIMO systems,” in 2021 15th European Conference on Antennas and Propagation (EuCAP) , pp. 1–5, 2021

    2021

  6. [14]

    Generalized odd-even mode theory and mode synthesis antenna design approach,

    W.-J. Lu, “Generalized odd-even mode theory and mode synthesis antenna design approach,” Electromagnetic Science, vol. 2, no. 1, pp. 1– 18, 2024

    2024

  7. [15]

    Multimode resonator technique in antennas: A review,

    L. Zhu and N. Liu, “Multimode resonator technique in antennas: A review,” Electromagnetic Science, vol. 1, no. 1, pp. 1–17, 2023

    2023

  8. [16]

    FDTD computation of space/time integrated electromagnetic lagrangian: New insights into design of mutually coupled antennas,

    D. Sarkar and Y . M. M. Antar, “FDTD computation of space/time integrated electromagnetic lagrangian: New insights into design of mutually coupled antennas,” IEEE J. Multiscale Multiphysics Comput. Tech., vol. 7, pp. 16–22, 2022

    2022

  9. [17]

    Generalized-scattering-matrix analysis of a class of finite arrays of coupled antennas by using 3-D FEM and spherical mode expansion,

    J. Rubio, M. Gonzalez, and J. Zapata, “Generalized-scattering-matrix analysis of a class of finite arrays of coupled antennas by using 3-D FEM and spherical mode expansion,” IEEE Trans. Antennas Propag. , vol. 53, no. 3, pp. 1133–1144, 2005

    2005

  10. [18]

    Effect of mutual coupling on the performance of adaptive arrays,

    I. Gupta and A. Ksienski, “Effect of mutual coupling on the performance of adaptive arrays,” IEEE Trans. Antennas Propag. , vol. 31, no. 5, pp. 785–791, 1983

    1983

  11. [19]

    The mutual coupling and diffraction effects on the performance of a CMA adaptive array,

    H. Yuan, K. Hirasawa, and Y . Zhang, “The mutual coupling and diffraction effects on the performance of a CMA adaptive array,” IEEE Trans. Veh. Technol., vol. 47, no. 3, pp. 728–736, 1998

    1998

  12. [20]

    Adaptive mutual coupling compensation based on efficient characterization of coupled antenna arrays,

    R. Li, D. Li, J. Ma, Y . Wu, Z. Gu, L. Zhang, H. Ma, H. Chen, and E.-P. Li, “Adaptive mutual coupling compensation based on efficient characterization of coupled antenna arrays,” IEEE Trans. Electromagn. Compat., 2023

    2023

  13. [21]

    Array antenna pattern modeling methods that include mutual coupling effects,

    D. F. Kelley and W. L. Stutzman, “Array antenna pattern modeling methods that include mutual coupling effects,” IEEE Trans. Antennas Propag., vol. 41, no. 12, pp. 1625–1632, 1993

    1993

  14. [22]

    Universal approximation bounds for superpositions of a sigmoidal function,

    A. R. Barron, “Universal approximation bounds for superpositions of a sigmoidal function,” IEEE Trans. Inf. Theory , vol. 39, no. 3, pp. 930– 945, 1993

    1993

  15. [23]

    Rapid estimation method for coupling degree of airborne antenna based on quantum neural network,

    T. Liu, B. Song, F. Meng, W. Yang, J. Li, J. You, and W. Lu, “Rapid estimation method for coupling degree of airborne antenna based on quantum neural network,” IEEE Antennas Wirel. Propag. Lett. , 2024

    2024

  16. [26]

    Prediction of MRI RF exposure for implantable plate devices using artificial neural network,

    J. Zheng, Q. Lan, X. Zhang, W. Kainz, and J. Chen, “Prediction of MRI RF exposure for implantable plate devices using artificial neural network,” IEEE Trans. Electromagn. Compat. , vol. 62, no. 3, pp. 673– 681, 2019

    2019

  17. [27]

    Deep learning inverse analysis of higher order modes in monocone TEM cell,

    D. Li, Y . Gu, H. Ma, Y . Li, L. Zhang, R. Li, R. Hao, and E.-P. Li, “Deep learning inverse analysis of higher order modes in monocone TEM cell,” IEEE Trans. Microw. Theory Tech., vol. 70, no. 12, pp. 5332–5339, 2022

    2022

  18. [28]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” J. Comput. Phys., vol. 378, pp. 686–707, 2019

    2019

  19. [29]

    CST Studio Suite

    “CST Studio Suite.” https://www.cst.com. Available

  20. [30]

    A new Green’s function formulation for modeling homogeneous objects in layered medium,

    Y . P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propag. , vol. 60, no. 10, pp. 4766–4776, 2012. 11

    2012

  21. [31]

    Jin, Theory and computation of electromagnetic fields

    J.-M. Jin, Theory and computation of electromagnetic fields. John Wiley & Sons, 2015

    2015

  22. [32]

    Electromagnetic scattering by surfaces of arbitrary shape,

    S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. , vol. 30, no. 3, pp. 409–418, 1982

    1982

  23. [33]

    On the testing of the magnetic field integral equation with RWG basis functions in method of moments,

    J. M. Rius, E. Ubeda, and J. Parr ´on, “On the testing of the magnetic field integral equation with RWG basis functions in method of moments,” IEEE Trans. Antennas Propag. , vol. 49, no. 11, pp. 1550–1553, 2001

    2001

  24. [34]

    Neural networks- method of moments (NN-MoM) for the efficient filling of the coupling matrix,

    E. A. Soliman, M. H. Bakr, and N. K. Nikolova, “Neural networks- method of moments (NN-MoM) for the efficient filling of the coupling matrix,” IEEE Trans. Antennas Propag. , vol. 52, no. 6, pp. 1521–1529, 2004

    2004

  25. [35]

    Uncertainty quantification in PEEC method: A physics-informed neural networks-based polynomial chaos expansion,

    Y . Ping, Y . Zhang, and L. Jiang, “Uncertainty quantification in PEEC method: A physics-informed neural networks-based polynomial chaos expansion,” IEEE Trans. Electromagn. Compat., vol. 66, no. 6, pp. 2095– 2101, 2024

    2024

  26. [36]

    Multi-dimensional multiplexed metasurface for multifunctional near-field modulation by physics-driven intelligent design,

    J. L. Su, Z. X. Cai, Y . Mao, L. Chen, X. Y . Yu, Z. C. Yu, Q. Ma, S. Q. Huang, J. Zhang, J. W. You,et al., “Multi-dimensional multiplexed metasurface for multifunctional near-field modulation by physics-driven intelligent design,” Adv. Sci., p. 2503899, 2025

    2025

  27. [37]

    Unsupervised learning,

    H. B. Barlow, “Unsupervised learning,” Neural Comput., vol. 1, no. 3, pp. 295–311, 1989

    1989

  28. [38]

    On estimating regression,

    E. A. Nadaraya, “On estimating regression,” Theory of Probability & Its Applications, vol. 9, no. 1, pp. 141–142, 1964

    1964

  29. [39]

    Smooth regression analysis,

    G. S. Watson, “Smooth regression analysis,” Sankhy¯a: The Indian Journal of Statistics, Series A , pp. 359–372, 1964

    1964

  30. [40]

    Long short-term memory,

    S. Hochreiter and J. Schmidhuber, “Long short-term memory,” Neural Comput., vol. 9, no. 8, pp. 1735–1780, 1997

    1997

  31. [41]

    The modified nodal approach to network analysis,

    C.-W. Ho, A. Ruehli, and P. Brennan, “The modified nodal approach to network analysis,” IEEE Trans. Circuits Syst. , vol. 22, no. 6, pp. 504– 509, 1975

    1975

  32. [42]

    C. A. Balanis, Antenna theory: analysis and design . John wiley & sons, 2015

    2015

  33. [43]

    A novel mutual coupling ann model for mimo antennas with physical preprocessing,

    Y . Jiang, S. S. Yuan, and E. Wei, “A novel mutual coupling ann model for mimo antennas with physical preprocessing,” IEEE Antennas Wirel. Propag. Lett., 2024

    2024

  34. [44]

    Nonlinear mutual coupling compensation operator design using a novel electromagnetic machine learning paradigm,

    A. M. Alzahed, S. M. Mikki, and Y . M. Antar, “Nonlinear mutual coupling compensation operator design using a novel electromagnetic machine learning paradigm,” IEEE Antennas Wirel. Propag. Lett. , vol. 18, no. 5, pp. 861–865, 2019

    2019

  35. [45]

    Machine-learning-based gen- erative optimization method and its application to an antenna decoupling design,

    H. Huang, X.-S. Yang, and B.-Z. Wang, “Machine-learning-based gen- erative optimization method and its application to an antenna decoupling design,” IEEE Trans. Antennas Propag. , vol. 71, no. 7, pp. 6243–6248, 2023

    2023

  36. [46]

    Fully automated design method based on reinforcement learning and surrogate modeling for antenna array decoupling,

    Z. Wei, Z. Zhou, P. Wang, J. Ren, Y . Yin, G. F. Pedersen, and M. Shen, “Fully automated design method based on reinforcement learning and surrogate modeling for antenna array decoupling,”IEEE Trans. Antennas Propag., vol. 71, no. 1, pp. 660–671, 2022

    2022

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.