Phy2-ExposNet: A Physics-Informed Neural Network for EMF Exposure Mapping in Complex Urban Environments

Joe Wiart; Shanshan Wang; Shuangning Li; Yarui Zhang

arxiv: 2605.03207 · v1 · submitted 2026-05-04 · 📡 eess.SP

Phy2-ExposNet: A Physics-Informed Neural Network for EMF Exposure Mapping in Complex Urban Environments

Shuangning Li , Yarui Zhang , Shanshan Wang , Joe Wiart This is my paper

Pith reviewed 2026-05-08 17:17 UTC · model grok-4.3

classification 📡 eess.SP

keywords EMF exposure mappingphysics-informed neural networkurban propagationtransformer refinementradio mapelectromagnetic fieldswireless network planning

0 comments

The pith

Phy2-ExposNet maps urban EMF exposure by first enforcing two physical constraints then refining residuals with a transformer, cutting error and model size.

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

The paper introduces Phy2-ExposNet to produce accurate electromagnetic field exposure maps for city environments. It splits the task into an initial stage that estimates fields while obeying two physical constraints and a second stage that uses a transformer to correct the remaining spatial patterns. A sympathetic reader would care because reliable exposure maps support wireless network planning, environmental health checks, and digital twin systems that represent signal behavior at every point. Experiments show the approach reduces relative error by around 15 percent compared with strong baselines while requiring over 80 percent fewer parameters than typical physics-informed networks. Ablation studies indicate that the physics stage is especially important for handling boundaries and shadow zones where pure regression fails.

Core claim

Phy2-ExposNet decouples exposure mapping into a physics-informed estimation stage that respects two physical constraints on field behavior and a transformer-based residual refinement stage that captures long-range interactions and complex propagation patterns, producing lower estimation error with substantially reduced model complexity.

What carries the argument

The two-stage architecture consisting of physics-constrained field estimation followed by transformer residual refinement.

If this is right

Exposure maps become more reliable in boundary and shadow regions where conventional regression struggles.
Model size drops enough to allow deployment on resource-limited devices for real-time radio mapping.
The same decoupling pattern can be applied to other wave-propagation tasks that mix known physics with complex scattering.
Digital-twin systems gain a lighter-weight method for maintaining up-to-date signal coverage layers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach may generalize to outdoor-to-indoor transitions or time-varying scenarios if the physical constraints are extended accordingly.
Parameter efficiency could enable on-device calibration of exposure maps using local sensor data.
If the two constraints prove incomplete for certain frequencies, the residual stage might still compensate but at the cost of losing interpretability.

Load-bearing premise

The two unspecified physical constraints in the first stage correctly describe electromagnetic propagation through complex urban geometry, and the transformer stage adds only real missing effects rather than new distortions.

What would settle it

Running the network on a controlled urban simulation where the output fields violate one of the two physical constraints in a measurable way, or where removing the physics stage yields equal or better accuracy than the full model.

Figures

Figures reproduced from arXiv: 2605.03207 by Joe Wiart, Shanshan Wang, Shuangning Li, Yarui Zhang.

**Figure 1.** Figure 1: Overview of the proposed two stage architecture for EMF exposure estimation. view at source ↗

**Figure 2.** Figure 2: Visualization comparison of different methods on the RadioMapSeer dataset. The predicted values are normalized view at source ↗

**Figure 3.** Figure 3: Comparison of ablation models on two test samples. The predicted values are normalized to the range view at source ↗

read the original abstract

Accurate electromagnetic field (EMF) exposure mapping is critical for wireless network planning, environmental monitoring, and the deployment of next generation communication systems. The mapping results can be converted into the form of a radio map, a key technology in digital twin communication systems, used to describe the wireless signal propagation characteristics at every location in a specific area. Existing deep learning approaches treat propagation estimation as a pure regression problem and do not enforce physical consistency in the predicted fields. In this paper, we propose Phy2-ExposNet, a novel neural network framework that decouples exposure mapping into a physics-informed estimation stage and a transformer-based residual refinement stage. It first estimates the fields under two physical constraints and then refines the resulting exposure map by capturing long range interactions and complex spatial propagation patterns. Experiments demonstrate that the proposed method achieves lower estimation error while significantly reducing model complexity compared to existing approaches. It achieves around 15% relative error reduction over strong baselines, while using over 80% fewer parameters than conventional physics-informed models. Ablation results further reveal that the physics-informed design is crucial for capturing complex propagation effects, particularly in boundary and shadow regions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The two-stage design (physics-informed estimation then transformer residual) is a reasonable practical step for urban EMF radio maps, but the unspecified constraints and thin experimental details leave the 15% gain and parameter savings hard to trust.

read the letter

The paper splits EMF mapping into a first stage that applies two physical constraints and a second stage that uses a transformer to fix residuals and capture long-range urban effects. That split is the actual novelty here, and it targets a real pain point in digital-twin tools where pure data-driven models ignore propagation rules and full physics solvers are too slow or complex for city-scale work. The reported outcome—roughly 15% lower error than strong baselines while cutting parameters by more than 80%—would be useful for network planners if it survives scrutiny, and the ablation note on boundary and shadow regions shows they at least checked where physics matters most. The architecture itself is straightforward and the motivation is clear. The soft spot is that the two physical constraints are never written down. The abstract and available text only say they are enforced; there are no loss terms, no link to the wave equation or ray-tracing priors, and no boundary conditions. Without those, it is impossible to tell whether the constraints actually encode urban EMF physics or simply act as mild regularizers that the transformer later corrects. The experiments also give no dataset description, no baseline implementation details, and no error bars, so the performance numbers stay difficult to reproduce or compare. This is aimed at engineers who build radio maps for 5G planning and environmental monitoring. A reader who needs a working method to plug into simulation pipelines could extract value once the constraints and data are made explicit. It deserves peer review because the idea is grounded enough and the empirical direction is honest, even though the current version needs the missing formulations filled in before the claims can be taken at face value.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes Phy2-ExposNet, a physics-informed neural network for mapping electromagnetic field (EMF) exposure in complex urban environments. The framework decouples the mapping into a physics-informed estimation stage that applies two physical constraints and a subsequent transformer-based residual refinement stage to capture long-range interactions and complex propagation patterns. Experiments are reported to show approximately 15% relative error reduction over strong baselines and over 80% reduction in model parameters compared to conventional physics-informed models, with ablations indicating the importance of the physics-informed component particularly in boundary and shadow regions.

Significance. Should the physical constraints prove to be correctly derived from EMF propagation physics and the empirical results hold under scrutiny, this approach could offer a more efficient alternative for generating radio maps in digital twin systems for wireless networks. The emphasis on reducing model complexity while improving accuracy addresses practical deployment challenges in urban EMF monitoring and 5G/6G planning. The decoupling of physics enforcement and residual learning is a promising architectural choice that merits further investigation if details are provided.

major comments (3)

The two physical constraints used in the estimation stage are never explicitly formulated, with no loss functions, boundary conditions, or derivations provided (e.g., no reference to the wave equation, impedance boundaries, or ray-tracing priors). This is a load-bearing issue because the central claims of 15% error reduction and 80% parameter savings rest on these constraints correctly enforcing urban EMF physics rather than serving as generic regularizers; without them, it is impossible to assess whether the transformer stage compensates for biases introduced by weak or incorrect constraints.
No descriptions of the datasets (e.g., simulation parameters, urban geometries, measurement setups), baseline implementations, error bars, or statistical significance tests are provided for the reported performance numbers. This leaves the strongest empirical claim unverifiable from the manuscript text.
The ablation results claim that the physics-informed design is crucial for capturing complex effects, but without explicit constraint formulations, it is unclear what specific components are being ablated and how their removal affects physical consistency.

minor comments (2)

The performance claims in the abstract ('around 15% relative error reduction', 'over 80% fewer parameters') should reference specific tables or figures for precision.
Ensure all acronyms (e.g., EMF, Phy2-ExposNet) are defined on first use in the main text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The comments highlight important areas where the manuscript can be strengthened for clarity and verifiability. We address each major comment below and commit to substantial revisions that directly respond to the concerns raised.

read point-by-point responses

Referee: The two physical constraints used in the estimation stage are never explicitly formulated, with no loss functions, boundary conditions, or derivations provided (e.g., no reference to the wave equation, impedance boundaries, or ray-tracing priors). This is a load-bearing issue because the central claims of 15% error reduction and 80% parameter savings rest on these constraints correctly enforcing urban EMF physics rather than serving as generic regularizers; without them, it is impossible to assess whether the transformer stage compensates for biases introduced by weak or incorrect constraints.

Authors: We agree that the explicit formulation of the two physical constraints requires additional detail to allow proper evaluation. In the revised manuscript we will insert a new subsection (likely in Section III-B) that derives the constraints from the underlying EMF propagation physics, provides the exact loss-function expressions, states the boundary conditions employed, and includes references to the wave equation and urban ray-tracing priors. This addition will make clear that the constraints are not generic regularizers but are specifically tailored to urban EMF exposure mapping. revision: yes
Referee: No descriptions of the datasets (e.g., simulation parameters, urban geometries, measurement setups), baseline implementations, error bars, or statistical significance tests are provided for the reported performance numbers. This leaves the strongest empirical claim unverifiable from the manuscript text.

Authors: We acknowledge the need for greater experimental transparency. The revised version will expand the Experiments section with: (i) complete simulation parameters and urban geometry descriptions for all datasets, (ii) implementation details and hyper-parameter settings for each baseline, (iii) error bars computed as standard deviations over repeated runs, and (iv) results of statistical significance tests (paired t-tests) comparing Phy2-ExposNet against the baselines. These additions will render the reported 15 % error reduction and 80 % parameter savings fully verifiable. revision: yes
Referee: The ablation results claim that the physics-informed design is crucial for capturing complex effects, but without explicit constraint formulations, it is unclear what specific components are being ablated and how their removal affects physical consistency.

Authors: We will revise the ablation study to cross-reference the newly added constraint formulations. Each ablation variant will be described in terms of which specific loss terms or boundary conditions are removed, and we will report quantitative measures of physical consistency (e.g., residual violations of the wave-equation constraint and field continuity in shadow regions) alongside the usual error metrics. This will clarify the contribution of the physics-informed stage to the observed performance gains. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical architecture evaluated on external benchmarks

full rationale

The paper presents Phy2-ExposNet as an architectural proposal that decouples EMF mapping into a physics-informed estimation stage (using two physical constraints) followed by transformer residual refinement. Claims of ~15% error reduction and >80% parameter savings are supported solely by experimental comparisons to baselines and ablation studies, with no derivations, equations, or self-citations that reduce the outputs to fitted inputs by construction. No load-bearing steps invoke self-referential definitions, fitted parameters renamed as predictions, or uniqueness theorems from overlapping prior work. The method is self-contained against external data via direct empirical testing, satisfying the criteria for a non-circular finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the physics constraints are mentioned but not formulated.

pith-pipeline@v0.9.0 · 5512 in / 1090 out tokens · 44896 ms · 2026-05-08T17:17:49.570402+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

S. Faye, R. Camino, G. Rziga, P. A. Sarvari, N. Al-Naffakh, J. C. Estrada-Jim ´enez, E. Pardo, and D. Khadraoui. A survey on EMF-aware mobile net- work planning.IEEE Access, 11:84568–84597, 2023

work page 2023
[2]

Guidelines for limiting exposure to electro- magnetic fields (100 kHz to 300 GHz).Health Phys., 118(5):483–524, 2020

ICNIRP. Guidelines for limiting exposure to electro- magnetic fields (100 kHz to 300 GHz).Health Phys., 118(5):483–524, 2020

work page 2020
[3]

Chan- nel knowledge maps for 6g wireless networks: Con- struction, applications, and future challenges.arXiv preprint arXiv:2505.24151, 2025

Xingchen Liu, Shu Sun, Meixia Tao, et al. Chan- nel knowledge maps for 6g wireless networks: Con- struction, applications, and future challenges.arXiv preprint arXiv:2505.24151, 2025

work page arXiv 2025
[4]

Overview of the first pathloss radio map prediction challenge.IEEE Open Journal of Signal Processing, 2024

Cagkan Yapar et al. Overview of the first pathloss radio map prediction challenge.IEEE Open Journal of Signal Processing, 2024

work page 2024
[5]

Electromagnetic field expo- sure mapping: A taxonomy and survey.Computers, 2025

Yiannis Kiouvrekis et al. Electromagnetic field expo- sure mapping: A taxonomy and survey.Computers, 2025

work page 2025
[6]

Efficient design of electromagnetic field exposure maps with multi-method evolutionary ensembles.Environmen- tal Research, 278:121636, 2025

Jorge Guill ´en-Pina, Jorge P ´erez-Aracil, Ricardo Chocano-del Cerro, Roc ´ıo S´anchez-Montero, Pablo- Luis L´opez-Esp´ı, and Sancho Salcedo-Sanz. Efficient design of electromagnetic field exposure maps with multi-method evolutionary ensembles.Environmen- tal Research, 278:121636, 2025

work page 2025
[7]

Expos- net: A deep learning framework for emf exposure pre- diction in complex urban environments.arXiv preprint arXiv:2503.02966, 2025

Yarui Zhang, Shanshan Wang, and Joe Wiart. Expos- net: A deep learning framework for emf exposure pre- diction in complex urban environments.arXiv preprint arXiv:2503.02966, 2025

work page arXiv 2025
[8]

A recent survey on radio map estimation methods for wireless networks.Electronics, 14(8):1564, 2025

Bin Feng, Meng Zheng, Wei Liang, and Lei Zhang. A recent survey on radio map estimation methods for wireless networks.Electronics, 14(8):1564, 2025

work page 2025
[9]

Design of an integrated platform for mapping residential exposure to rf-emf sources.In- ternational Journal of Environmental Research and Public Health, 17(18):6775, 2020

Claire Regrain, Roger Lobjois, Abdelhamid Hadjem, and Joe Wiart. Design of an integrated platform for mapping residential exposure to rf-emf sources.In- ternational Journal of Environmental Research and Public Health, 17(18):6775, 2020

work page 2020
[10]

Z. Yi, Z. Liu, R. Huang, J. Wang, W. Xie, and S. Liu. Feature extraction in reference signal received PREPRINT 12 power prediction based on convolution neural net- works.IEEE Commun. Lett., 25(6):1751–1755, June 2021

work page 2021
[11]

Lee and A

J.-H. Lee and A. F. Molisch. A scalable and general- izable pathloss map prediction.IEEE Trans. Wireless Commun., 2024

work page 2024
[12]

Saeizadeh, M

A. Saeizadeh, M. Tehrani-Moayyed, D. Villa, J. G. Beattie, P. Johari, S. Basagni, and T. Melodia. AIRMap: AI-generated radio maps for wireless digital twins.arXiv preprint, 2025

work page 2025
[13]

Overview of the first pathloss radio map prediction challenge

C ¸ a˘gkan Yapar, Kehai Qiu, Tao Cheng, et al. Overview of the first pathloss radio map prediction challenge. IEEE Open Journal of Signal Processing, 5:588–605, 2024

work page 2024
[14]

Levie, C

R. Levie, C. Yapar, G. Kutyniok, and G. Caire. Ra- dioUNet: Fast radio map estimation with convolu- tional neural networks.IEEE Trans. Wireless Com- mun., 20(6):4001–4015, June 2021

work page 2021
[15]

Zheng, J

Y. Zheng, J. Wang, X. Li, J. Li, and S. Liu. Cell-level RSRP estimation with the image-to-image wireless propagation model based on measured data.IEEE Trans. Cogn. Commun. Netw., 9(6):1412–1425, De- cember 2023

work page 2023
[16]

Raissi, P

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., 378:686–707, 2019

work page 2019
[17]

Approximating Electromagnetic Fields in Discontinuous Media Using a Single Physics-Informed Neural Network.arXiv preprint arXiv:2407.20833, 2024

M. Nohra and S. Dufour. Approximating electro- magnetic fields in discontinuous media using a sin- gle physics-informed neural network.arXiv preprint arXiv:2407.20833, 2024

work page arXiv 2024
[18]

Pinn-fem: A hybrid approach for enforc- ing dirichlet boundary conditions in physics-informed neural networks.arXiv preprint arXiv:2501.07765, 2025

Nahil Sobh, Rini Jasmine Gladstone, and Hadi Mei- dani. Pinn-fem: A hybrid approach for enforc- ing dirichlet boundary conditions in physics-informed neural networks.arXiv preprint arXiv:2501.07765, 2025

work page arXiv 2025
[19]

R. Guo, T. Shan, X. Song, M. Li, F. Yang, S. Xu, and A. Abubakar. Physics embedded deep neural network for solving volume integral equation: 2-D case.IEEE Trans. Antennas Propag., 70(8):6135–6147, August 2022

work page 2022
[20]

Jiang, T

F. Jiang, T. Li, X. Lv, H. Rui, and D. Jin. Physics- informed neural networks for path loss estimation by solving electromagnetic integral equations.IEEE Trans. Wireless Commun., 23(10):15380–15394, Oc- tober 2024

work page 2024
[21]

Huang, M

Z. Huang, M. Wang, X. Wang, et al. Physics-inspired machine learning for radiomap estimation: Integration of radio propagation models and artificial intelligence. IEEE Commun. Mag., 62(8):155–161, August 2024

work page 2024
[22]

H. Jia, W. Chen, Z. Huang, L. Wang, H. Xiao, N. Jia, K. Wu, S. Lai, B. Tian, and Y. Yue. Physics-informed representation alignment for sparse radio-map recon- struction.arXiv preprint arXiv:2503.XXXXX, 2025

work page 2025
[23]

Vaswani, N

A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need.Adv. Neural Inf. Process. Syst. (NeurIPS), 30, 2017

work page 2017
[24]

Q. Ren, Y. Wang, Y. Li, and S. Qi.Sophisticated Electromagnetic Forward Scattering Solver via Deep Learning. Springer, Singapore, 2022

work page 2022
[25]

Highway transformer: Self-gating enhanced self-attentive networks

Yekun Chai et al. Highway transformer: Self-gating enhanced self-attentive networks. InACL, 2020

work page 2020

[1] [1]

S. Faye, R. Camino, G. Rziga, P. A. Sarvari, N. Al-Naffakh, J. C. Estrada-Jim ´enez, E. Pardo, and D. Khadraoui. A survey on EMF-aware mobile net- work planning.IEEE Access, 11:84568–84597, 2023

work page 2023

[2] [2]

Guidelines for limiting exposure to electro- magnetic fields (100 kHz to 300 GHz).Health Phys., 118(5):483–524, 2020

ICNIRP. Guidelines for limiting exposure to electro- magnetic fields (100 kHz to 300 GHz).Health Phys., 118(5):483–524, 2020

work page 2020

[3] [3]

Chan- nel knowledge maps for 6g wireless networks: Con- struction, applications, and future challenges.arXiv preprint arXiv:2505.24151, 2025

Xingchen Liu, Shu Sun, Meixia Tao, et al. Chan- nel knowledge maps for 6g wireless networks: Con- struction, applications, and future challenges.arXiv preprint arXiv:2505.24151, 2025

work page arXiv 2025

[4] [4]

Overview of the first pathloss radio map prediction challenge.IEEE Open Journal of Signal Processing, 2024

Cagkan Yapar et al. Overview of the first pathloss radio map prediction challenge.IEEE Open Journal of Signal Processing, 2024

work page 2024

[5] [5]

Electromagnetic field expo- sure mapping: A taxonomy and survey.Computers, 2025

Yiannis Kiouvrekis et al. Electromagnetic field expo- sure mapping: A taxonomy and survey.Computers, 2025

work page 2025

[6] [6]

Efficient design of electromagnetic field exposure maps with multi-method evolutionary ensembles.Environmen- tal Research, 278:121636, 2025

Jorge Guill ´en-Pina, Jorge P ´erez-Aracil, Ricardo Chocano-del Cerro, Roc ´ıo S´anchez-Montero, Pablo- Luis L´opez-Esp´ı, and Sancho Salcedo-Sanz. Efficient design of electromagnetic field exposure maps with multi-method evolutionary ensembles.Environmen- tal Research, 278:121636, 2025

work page 2025

[7] [7]

Expos- net: A deep learning framework for emf exposure pre- diction in complex urban environments.arXiv preprint arXiv:2503.02966, 2025

Yarui Zhang, Shanshan Wang, and Joe Wiart. Expos- net: A deep learning framework for emf exposure pre- diction in complex urban environments.arXiv preprint arXiv:2503.02966, 2025

work page arXiv 2025

[8] [8]

A recent survey on radio map estimation methods for wireless networks.Electronics, 14(8):1564, 2025

Bin Feng, Meng Zheng, Wei Liang, and Lei Zhang. A recent survey on radio map estimation methods for wireless networks.Electronics, 14(8):1564, 2025

work page 2025

[9] [9]

Design of an integrated platform for mapping residential exposure to rf-emf sources.In- ternational Journal of Environmental Research and Public Health, 17(18):6775, 2020

Claire Regrain, Roger Lobjois, Abdelhamid Hadjem, and Joe Wiart. Design of an integrated platform for mapping residential exposure to rf-emf sources.In- ternational Journal of Environmental Research and Public Health, 17(18):6775, 2020

work page 2020

[10] [10]

Z. Yi, Z. Liu, R. Huang, J. Wang, W. Xie, and S. Liu. Feature extraction in reference signal received PREPRINT 12 power prediction based on convolution neural net- works.IEEE Commun. Lett., 25(6):1751–1755, June 2021

work page 2021

[11] [11]

Lee and A

J.-H. Lee and A. F. Molisch. A scalable and general- izable pathloss map prediction.IEEE Trans. Wireless Commun., 2024

work page 2024

[12] [12]

Saeizadeh, M

A. Saeizadeh, M. Tehrani-Moayyed, D. Villa, J. G. Beattie, P. Johari, S. Basagni, and T. Melodia. AIRMap: AI-generated radio maps for wireless digital twins.arXiv preprint, 2025

work page 2025

[13] [13]

Overview of the first pathloss radio map prediction challenge

C ¸ a˘gkan Yapar, Kehai Qiu, Tao Cheng, et al. Overview of the first pathloss radio map prediction challenge. IEEE Open Journal of Signal Processing, 5:588–605, 2024

work page 2024

[14] [14]

Levie, C

R. Levie, C. Yapar, G. Kutyniok, and G. Caire. Ra- dioUNet: Fast radio map estimation with convolu- tional neural networks.IEEE Trans. Wireless Com- mun., 20(6):4001–4015, June 2021

work page 2021

[15] [15]

Zheng, J

Y. Zheng, J. Wang, X. Li, J. Li, and S. Liu. Cell-level RSRP estimation with the image-to-image wireless propagation model based on measured data.IEEE Trans. Cogn. Commun. Netw., 9(6):1412–1425, De- cember 2023

work page 2023

[16] [16]

Raissi, P

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., 378:686–707, 2019

work page 2019

[17] [17]

Approximating Electromagnetic Fields in Discontinuous Media Using a Single Physics-Informed Neural Network.arXiv preprint arXiv:2407.20833, 2024

M. Nohra and S. Dufour. Approximating electro- magnetic fields in discontinuous media using a sin- gle physics-informed neural network.arXiv preprint arXiv:2407.20833, 2024

work page arXiv 2024

[18] [18]

Pinn-fem: A hybrid approach for enforc- ing dirichlet boundary conditions in physics-informed neural networks.arXiv preprint arXiv:2501.07765, 2025

Nahil Sobh, Rini Jasmine Gladstone, and Hadi Mei- dani. Pinn-fem: A hybrid approach for enforc- ing dirichlet boundary conditions in physics-informed neural networks.arXiv preprint arXiv:2501.07765, 2025

work page arXiv 2025

[19] [19]

R. Guo, T. Shan, X. Song, M. Li, F. Yang, S. Xu, and A. Abubakar. Physics embedded deep neural network for solving volume integral equation: 2-D case.IEEE Trans. Antennas Propag., 70(8):6135–6147, August 2022

work page 2022

[20] [20]

Jiang, T

F. Jiang, T. Li, X. Lv, H. Rui, and D. Jin. Physics- informed neural networks for path loss estimation by solving electromagnetic integral equations.IEEE Trans. Wireless Commun., 23(10):15380–15394, Oc- tober 2024

work page 2024

[21] [21]

Huang, M

Z. Huang, M. Wang, X. Wang, et al. Physics-inspired machine learning for radiomap estimation: Integration of radio propagation models and artificial intelligence. IEEE Commun. Mag., 62(8):155–161, August 2024

work page 2024

[22] [22]

H. Jia, W. Chen, Z. Huang, L. Wang, H. Xiao, N. Jia, K. Wu, S. Lai, B. Tian, and Y. Yue. Physics-informed representation alignment for sparse radio-map recon- struction.arXiv preprint arXiv:2503.XXXXX, 2025

work page 2025

[23] [23]

Vaswani, N

A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need.Adv. Neural Inf. Process. Syst. (NeurIPS), 30, 2017

work page 2017

[24] [24]

Q. Ren, Y. Wang, Y. Li, and S. Qi.Sophisticated Electromagnetic Forward Scattering Solver via Deep Learning. Springer, Singapore, 2022

work page 2022

[25] [25]

Highway transformer: Self-gating enhanced self-attentive networks

Yekun Chai et al. Highway transformer: Self-gating enhanced self-attentive networks. InACL, 2020

work page 2020