Electromagnetic Neural Networks

• Electromagnetic Neural Networks are methods that integrate Maxwell’s equations and electromagnetic laws into neural models to improve inversion, simulation, and hardware co-design.
• They leverage physics-informed loss functions, analytical layers, and tailored architectures to achieve higher accuracy and rapid inference in applications like geophysics and imaging.
• EMNN research spans diverse implementations, including surrogate inversion models, PINNs for discontinuous media, and metasurface-based neural systems for communication tasks.

Searching arXiv for papers on "electromagnetic neural network" and closely related formulations to ground the article. Electromagnetic Neural Network (EMNN) denotes a family of neural-network formulations in which electromagnetic physics is not merely an application domain but an integral part of the model structure, training objective, or hardware substrate. Across the literature, the term has been used in at least three technically distinct senses: as a neural surrogate for electromagnetic forward or inverse operators in geophysics and logging (Bai et al., 2020, Alyaev et al., 2020); as a physics-informed network for solving Maxwell problems in discontinuous media (Nohra et al., 2024); and as a wave-domain architecture in which programmable metasurfaces are abstracted as neural-network layers, so that electromagnetic propagation itself performs parts of sensing or communication computation (Zhang et al., 31 Oct 2025, Huang et al., 2024, Zhang et al., 15 Sep 2025, Li et al., 26 Apr 2026). A broader adjacent literature includes inverse-scattering networks derived from iterative solvers (Li et al., 2018), physics-driven loss-based imaging networks (Du et al., 22 Jul 2025), and knowledge-integrated scattering surrogates based on quasinormal-mode expansions (Lilja et al., 7 Sep 2025). Taken together, these works define EMNN not as a single architecture, but as a research program for coupling electromagnetic operators, constraints, and hardware with learnable representations.

1. Terminological scope and conceptual variants

The label EMNN is applied to multiple constructions that share a common premise: electromagnetic structure is embedded into learning more deeply than in generic black-box regression.

A first usage treats EMNN as a data-driven surrogate or inverse mapper for electromagnetic measurements. In airborne time-domain electromagnetics, a fully connected network is trained to map stacked gate data directly to a 30-layer subsurface model, replacing repeated deterministic inversion with a direct estimate of conductivity or resistivity (Bai et al., 2020). In logging-while-drilling, a residual convolutional network is trained to emulate a proprietary extra-deep electromagnetic forward simulator, accelerating repeated log evaluation in geosteering workflows (Alyaev et al., 2020).

A second usage treats EMNN as a physics-informed solver. In this setting, the network approximates electromagnetic fields while Maxwell residuals appear explicitly in the loss. Nohra and Dufour formulate a single PINN for static and transient three-dimensional problems in discontinuous media, using first-order Maxwell equations, level-set enrichment, and strong boundary or initial condition imposition (Nohra et al., 2024). A plausible implication is that, in this branch of the literature, EMNN refers less to a specific network topology than to a solver class whose inductive bias is supplied by field equations and interface geometry.

A third usage, now prominent in metasurface-assisted communications, identifies EMNN with a physical-computing architecture. Zhang et al. abstract each layer of a stacked intelligent metasurface as a hidden layer composed of fixed free-space propagation matrices and trainable phase-only diagonal matrices, then optimize these jointly with transmitter and receiver deep networks by back-propagation (Zhang et al., 31 Oct 2025). Closely related formulations appear for semantic communications (Huang et al., 2024), end-to-end OFDM (Zhang et al., 15 Sep 2025), and distributed UAV-mounted semantic systems (Li et al., 26 Apr 2026). Here the electromagnetic hardware is the network substrate.

Adjacent but not always explicitly named EMNN are networks that unroll inverse-scattering iterations into learned modules (Li et al., 2018), optimize a CNN solely through physics-based data-consistency and prior losses (Du et al., 22 Jul 2025), or encode scattering physics through a quasinormal-mode layer that guarantees energy conservation and causality (Lilja et al., 7 Sep 2025). This suggests that EMNN is best understood as an umbrella term covering several mechanisms for embedding electromagnetic structure into learning.

2. Physics embedding mechanisms

The unifying technical question is how Maxwellian structure enters the network. The literature exhibits several distinct answers.

In surrogate and inverse models, the embedding may be indirect but still substantial. The airborne TDEM study defines a forward operator $F$ mapping a 30-layer model $m \in \mathbb{R}^n$ to gate data $d \in \mathbb{R}^p$ with $n=30$ and $p=54$, then trains a parameterized inverse map $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$ by minimizing a mean-squared reconstruction loss on synthetic pairs $\{d^{(i)},m^{(i)}\}$ (Bai et al., 2020). The logging study similarly approximates a proprietary frequency-domain electromagnetic forward solver with a network $F_w \approx F$ composed of five residual convolutional blocks and two dense layers (Alyaev et al., 2020). In both cases, the network does not enforce Maxwell equations during inference, but its training set is generated by a full electromagnetic simulator.

In physics-informed EMNNs, the embedding is explicit in the residuals. Nohra and Dufour define

$$R_1(\mathbf{x},t):=\partial_t[\mu H]+\nabla\times E,\quad R_2(\mathbf{x},t):=-\nabla\times H+\sigma E,$$

together with Gauss-law residuals, and minimize a weighted sum of their discrete $L^2$ norms over collocation points (Nohra et al., 2024). Material discontinuities are represented through a level-set function $m \in \mathbb{R}^n$0, and the enriched input

$m \in \mathbb{R}^n$1

injects interface-aware high-frequency features into the network (Nohra et al., 2024). The paper further argues that when the PDE residuals vanish, the four Maxwell-interface jump conditions are automatically satisfied.

In inverse scattering, physics can be enforced through the optimization loop rather than the architecture. The physics-driven neural network of (Du et al., 22 Jul 2025) uses a lightweight CNN, but computes a predicted scattered field from the network output via Method of Moments and penalizes the discrepancy with measured fields. Its loss combines a data term, a lower-bound constraint on $m \in \mathbb{R}^n$2, and a total-variation prior (Du et al., 22 Jul 2025). DeepNIS takes a different route: it exploits the analogy between iterative nonlinear inverse scattering updates and deep residual modules, thereby unrolling solver structure into a cascade of complex-valued CNN blocks (Li et al., 2018).

In scattering surrogates, physics may appear as a differentiable analytical layer. QNM-Net maps device geometry to the components of a quasinormal-mode expansion and assembles the scattering matrix through

$m \in \mathbb{R}^n$3

with energy conservation and causality enforced by construction (Lilja et al., 7 Sep 2025). This is a particularly strong form of inductive bias: rather than learning a free-form spectral response, the network learns resonant parameters with explicit physical meaning.

In metasurface EMNNs, the physics is the network. Each layer comprises a fixed propagation matrix derived from Rayleigh–Sommerfeld diffraction and a trainable diagonal matrix of complex transmission or phase coefficients (Zhang et al., 31 Oct 2025, Huang et al., 2024, Zhang et al., 15 Sep 2025, Li et al., 26 Apr 2026). The forward pass is literal electromagnetic propagation through these layers, while back-propagation adjusts the phase or amplitude settings that correspond one-to-one with hardware controls.

3. Canonical architectural patterns

Although EMNN is not a single model class, several recurrent architectural templates have emerged.

Surrogate and inversion architectures

The airborne TDEM EMNN employs a fully connected architecture with 54 input units, three hidden layers of 100, 500, and 200 neurons, and 30 output units representing the conductivity or resistivity parameters of 30 horizontal layers (Bai et al., 2020). The activations are described only as “continuous and differentiable activation functions,” and the optimizer, learning rate, batch size, and epoch count are not specified (Bai et al., 2020). Each sounding is processed independently after normalization by transmitter waveform and gate widths.

The extra-deep logging EMNN uses a markedly different topology: five stacked residual blocks based on 1D convolutions with 40 filters and kernel size 3, followed by a fully connected head that maps a flattened 880-dimensional vector to 512 hidden units and finally to 13 outputs (Alyaev et al., 2020). The model has approximately 462 453 trainable parameters, uses ReLU activations throughout, and includes no explicit dropout or $m \in \mathbb{R}^n$4 regularization (Alyaev et al., 2020).

DeepNIS represents an unrolled inverse-scattering architecture. Each complex-valued residual CNN module contains three layers and a residual shortcut, with typical filter choices $m \in \mathbb{R}^n$5, $m \in \mathbb{R}^n$6, and $m \in \mathbb{R}^n$7, $m \in \mathbb{R}^n$8 (Li et al., 2018). The first module takes a back-propagated image as input, and successive modules refine the contrast estimate.

Physics-informed field solvers

The PINN-based EMNN of (Nohra et al., 2024) consists of a fully connected feed-forward network with $m \in \mathbb{R}^n$9 hidden layers, typically 5–8, width $d \in \mathbb{R}^p$0–100, and smooth activations such as $d \in \mathbb{R}^p$1 or $d \in \mathbb{R}^p$2. Boundary and initial conditions are imposed strongly through a trial solution

$d \in \mathbb{R}^p$3

where $d \in \mathbb{R}^p$4 vanishes on the boundary and $d \in \mathbb{R}^p$5 satisfies the prescribed conditions (Nohra et al., 2024).

For EUV mask diffraction, the PINN in (Es'kin et al., 5 Jul 2025) is a single MLP with three hidden layers of 128 neurons each and $d \in \mathbb{R}^p$6 activations, outputting $d \in \mathbb{R}^p$7 and $d \in \mathbb{R}^p$8 at input point $d \in \mathbb{R}^p$9. The same paper also introduces a hybrid Waveguide Neural Operator that replaces the dominant linear solve in a Fourier-Galerkin waveguide method by a small neural operator, while retaining the rest of the waveguide pipeline unchanged (Es'kin et al., 5 Jul 2025).

Metasurface EMNNs

In SIM-based communications, each metasurface layer is represented by two successive operations: a fixed propagation matrix and a trainable diagonal phase matrix (Zhang et al., 31 Oct 2025). For the transmit side, the $n=30$0-th terminal uses the sequence

$n=30$1

and the receive side uses the analogous $n=30$2 and $n=30$3 blocks (Zhang et al., 31 Oct 2025). End-to-end OFDM extends this pattern to per-subcarrier operators $n=30$4 and $n=30$5, embedded between a base-station DNN transmitter and user-equipment DNN receiver (Zhang et al., 15 Sep 2025).

Semantic-communication EMNNs use a similar diffractive stack. In (Huang et al., 2024), layer 0 performs source encoding and layers $n=30$6 perform semantic encoding, each layer being a planar array of $n=30$7 meta-atoms with transmission coefficients $n=30$8. The distributed variant in (Li et al., 26 Apr 2026) duplicates this structure across multiple UAV-mounted SIMs and aggregates their received fields at a ground station.

Comparative summary

EMNN variant	Core architecture	Physics insertion
Airborne TDEM inversion	54–100–500–200–30 fully connected network	Synthetic data from 1D TDEM forward model (Bai et al., 2020)
Extra-deep EM logging	5 residual 1D-conv blocks + dense head	Training targets from proprietary Maxwell solver (Alyaev et al., 2020)
PINN for discontinuous media	Enriched-input MLP, 5–8 layers, width 50–100	First-order Maxwell residuals and level-set interface encoding (Nohra et al., 2024)
SIM-assisted communications	Alternating fixed diffraction matrices and trainable diagonal metasurface layers	Rayleigh–Sommerfeld propagation embedded as hidden layers (Zhang et al., 31 Oct 2025, Zhang et al., 15 Sep 2025, Huang et al., 2024, Li et al., 26 Apr 2026)
Inverse-scattering EMNN	Complex residual CNN cascade or CNN with physics loss	Unrolled solver updates or MoM-based data consistency (Li et al., 2018, Du et al., 22 Jul 2025)
Scattering surrogate with QNM layer	Design encoder + physics parameter heads + QNM layer	Analytical quasinormal-mode expansion (Lilja et al., 7 Sep 2025)

4. Training regimes and optimization strategies

EMNN training differs substantially across subfields because the source of supervision differs.

For supervised surrogates and inverse maps, synthetic data generation is central. The airborne TDEM study samples 30-layer 1D models to form a laterally coherent “pseudo-2D” section informed by prior geological knowledge, solves the forward problem for each model, and uses approximately $n=30$9 pairs for training (Bai et al., 2020). No field data are used in training or fine-tuning; the VTEM dataset from Sakatti, Finland, is held out entirely for testing and validation (Bai et al., 2020). Multi-start training is used to avoid poor local minima, and early stopping is based on a held-out validation subset (Bai et al., 2020).

The logging EMNN is trained on a dataset of 78 877 pairs after trimming, with an 80/10/10 train/validation/test split (Alyaev et al., 2020). Optimization uses Adam with $p=54$0, $p=54$1, default learning rate, batch size 512, and early stopping with patience 800 epochs (Alyaev et al., 2020). Training on an NVIDIA RTX 2080 Ti takes approximately 13 hours and converges in approximately 11 745 epochs (Alyaev et al., 2020).

PINN-based EMNNs typically use two-stage optimization. Nohra and Dufour employ Adam for 1 000–3 000 iterations with batch sampling of collocation points, then full-batch L-BFGS for 2 000–5 000 iterations to drive residuals to $p=54$2–$p=54$3 (Nohra et al., 2024). A curriculum on the interface sharpness parameter $p=54$4 in $p=54$5 avoids vanishing-gradient traps when modeling discontinuities (Nohra et al., 2024). The EUV diffraction PINN similarly uses Adam at learning rate $p=54$6 followed by L-BFGS, whereas WGNO is trained in two Adam stages with learning rates $p=54$7 and $p=54$8 (Es'kin et al., 5 Jul 2025).

Metasurface EMNNs are trained end to end over both digital and wave-domain parameters. In SIM-assisted full-duplex communications, base training uses statistical CSI, random bit vectors, and binary cross-entropy loss, updating transmitter DNN weights, metasurface phases, and receiver DNN weights through AdamW (Zhang et al., 31 Oct 2025). Fine-tuning then replaces the statistical channel with instantaneous CSI and adapts the model using a small number of epochs (Zhang et al., 31 Oct 2025). The OFDM EMNN uses BCE loss, back-propagates through BS-DNN, TX-SIM, channel layer, RX-SIM, and UE-DNN, and employs transfer learning from statistical to instantaneous CSI (Zhang et al., 15 Sep 2025). The semantic SIM letter uses mini-batch gradient descent via Adam with batch size $p=54$9, initial learning rate $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$0 decayed by 0.8 if loss plateaus, and total epochs $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$1 (Huang et al., 2024). The distributed semantic system introduces a trainable softmax temperature and a temperature-adaptive gradient optimization algorithm to mitigate gradient vanishing under wide channel attenuations (Li et al., 26 Apr 2026).

Physics-driven inverse-scattering EMNNs may be trained per case rather than over a population. In (Du et al., 22 Jul 2025), the network weights are initialized randomly and updated for a given measured dataset by minimizing the physics-driven loss; no supervised pretraining dataset is required. This differentiates PDNN from population-trained surrogates and suggests a continuum between learned solvers and optimization-based inversion.

5. Application domains and reported performance

The EMNN literature spans geophysical inversion, well logging, inverse scattering, computational electromagnetics, sensing, and communication systems. Because evaluation protocols differ across domains, reported results are best read in their application-specific context.

Airborne and borehole electromagnetics

For airborne TDEM inversion, the reconstructed synthetic models yield forward responses differing from the originals by $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$2 per sounding, while the ratio $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$3 stays within roughly $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$4 over most of the section (Bai et al., 2020). On the Sakatti VTEM field survey, the ANN inversion fits real data to $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$5 per sounding, compared with $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$6 for the deterministic 1D inversion, and the ratio of ANN to deterministic inversion conductivities lies between 0.3 and 3.0 in most places (Bai et al., 2020). Computationally, 14 346 soundings with 54 gates are inverted in approximately 24 s on a standard laptop, about 1.7 ms per sounding, whereas the deterministic inversion takes on the order of $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$7 s on a 64-CPU server (Bai et al., 2020).

For extra-deep electromagnetic logs, the DNN surrogate achieves $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$8 on every one of the 13 log channels on the held-out test set (Alyaev et al., 2020). In a synthetic geosteering run of 901 logging positions, the network computes all 13 channels in 0.13 s total, approximately 0.15 ms per position, exceeding 6 500 forward calls per second on a 10-core Xeon (Alyaev et al., 2020). The largest discrepancies occur near layer boundaries and in some semi-degenerate shale scenarios, but are reported to remain within engineering tolerances for real-time inversion (Alyaev et al., 2020).

Electromagnetic field solvers and diffraction

For static and transient discontinuous-media problems, the single-PINN EMNN achieves a maximum pointwise error $g_\theta : \mathbb{R}^p \to \mathbb{R}^n$9 after 1 000 Adam and 2 000 L-BFGS iterations in a steady-state sphere-in-cube benchmark with smooth interface $\{d^{(i)},m^{(i)}\}$0 (Nohra et al., 2024). With curriculum to $\{d^{(i)},m^{(i)}\}$1, interface error is $\{d^{(i)},m^{(i)}\}$2 (Nohra et al., 2024). Multiple interfaces and non-trivial shapes are handled with comparable relative errors below 3%, and transient tests show excellent agreement with FEM while avoiding spurious divergence currents because $\{d^{(i)},m^{(i)}\}$3 is enforced (Nohra et al., 2024).

In EUV mask diffraction, the WGNO attains relative $\{d^{(i)},m^{(i)}\}$4 errors from $\{d^{(i)},m^{(i)}\}$5 to $\{d^{(i)},m^{(i)}\}$6 on 2D analytical tests, with inference around $\{d^{(i)},m^{(i)}\}$7 s (Es'kin et al., 5 Jul 2025). For realistic 2D and 3D masks, reported relative $\{d^{(i)},m^{(i)}\}$8 errors are as low as $\{d^{(i)},m^{(i)}\}$9 and $F_w \approx F$0, with over 200× inference speedup versus rigorous waveguide computation (Es'kin et al., 5 Jul 2025). The PINN baseline is substantially less accurate and much slower to train in these tests (Es'kin et al., 5 Jul 2025).

Inverse scattering and sensing

DeepNIS reports mean SSIM around 0.9 and MSE around 0.03 on 2 000 unseen MNIST-like shapes, compared with SSIM below 0.1 and MSE around 0.8 for back-propagation, and SSIM around 0.2 and MSE around 0.6 for contrast source inversion (Li et al., 2018). Runtime per image is under 1 s on GPU, compared with about 8 s for back-propagation and about 10 min for CSI on CPU (Li et al., 2018). On the FoamDielExt experimental dataset, DeepNIS attains SSIM 0.83, MSE 0.09, and about 1 s runtime (Li et al., 2018).

The PDNN inverse-scattering scheme reports average relative errors of 2.1%, 3.0%, 2.4%, and 3.5% on square, circles, ring, and “Austria” profiles, improved further marginally by subregion reduction (Du et al., 22 Jul 2025). Runtime drops from about 360 s to about 50–80 s on an RTX 4090 after domain cropping (Du et al., 22 Jul 2025). At SNR $F_w \approx F$1 dB, PDNN remains within 5% error; at SNR 10 dB, error is about 8% (Du et al., 22 Jul 2025).

In learned integrated sensing with a physical dynamic metasurface layer, the learned EMNN reaches 92.5% classification accuracy with $F_w \approx F$2 measurements and saturates at approximately 95% with $F_w \approx F$3 for MNIST-like microwave object classification using $F_w \approx F$4 metamaterial elements (Hougne et al., 2019). Accuracy variance is reported below 1% for $F_w \approx F$5, and polarizability perturbations up to $F_w \approx F$6 degrade performance by less than 1% (Hougne et al., 2019).

Communications and semantic systems

In SIM-assisted co-frequency co-time full-duplex communications, increasing metasurface layers from 1 to 3 yields 1–2 dB additional self-interference suppression and reduces BER by an order of magnitude at mid-range SNR; a system with far fewer RF chains and 3 SIM layers of $F_w \approx F$7 EM units can match or outperform a conventional CCFD configuration with much larger antenna counts (Zhang et al., 31 Oct 2025). Transfer learning reduces convergence time by over 50% relative to training from scratch (Zhang et al., 31 Oct 2025).

In task-oriented semantic communications, a 10-layer-plus-input SIM with $F_w \approx F$8 meta-atoms per layer, trained on MNIST at approximately 28 GHz, achieves more than 90% test recognition accuracy purely through wave-domain computation, with only a simple energy-comparison rule at the receiver (Huang et al., 2024). The distributed UAV-enabled EMNN reports an average 8% accuracy improvement over the single-SIM baseline across multiple datasets, and on MNIST specifically reports 93.06% for the distributed architecture versus about 85% for centralized EMNN and about 75% for a standard DNN in the parameter-matched comparison (Li et al., 26 Apr 2026).

For end-to-end OFDM, simulation results show that randomized transmit-power schedules improve robustness across 0–30 dBm, fine-tuning from a base model converges approximately 10× faster than training from scratch, DPSIM consistently outperforms SIM by 5–10 dB in BER versus SNR, and a SIM-E2E system with only 16T × 9R can match a conventional massive-MIMO 256T × 49R digital-only system (Zhang et al., 15 Sep 2025).

6. Relations to neighboring research areas and recurrent misconceptions

EMNN overlaps with several broader research programs but is not reducible to any single one.

It is not synonymous with PINNs. PINNs are one important EMNN instantiation when Maxwell residuals define the loss, as in discontinuous-media field modeling (Nohra et al., 2024) or the EUV diffraction PINN (Es'kin et al., 5 Jul 2025). However, the term EMNN is also used for fully supervised surrogates without PDE residuals (Alyaev et al., 2020), direct inverse mappers (Bai et al., 2020), hardware-in-the-loop metasurface networks (Zhang et al., 31 Oct 2025), and solver-unrolled inverse-scattering networks (Li et al., 2018). Conversely, many electromagnetic PINNs are not labeled EMNN.

It is also not limited to neural surrogates for forward simulation. In some literature, EMNN performs inversion directly from measurements to subsurface parameters (Bai et al., 2020), whereas in metasurface communications it represents a differentiable transceiver in which conventional tasks such as modulation, precoding, combining, demodulation, and decoding are carried out jointly during electromagnetic forward propagation (Zhang et al., 31 Oct 2025, Zhang et al., 15 Sep 2025).

A further misconception is that EMNN necessarily implies black-box learning. Several of the strongest examples are explicitly knowledge-integrated. QNM-Net constrains the learned scattering response through a quasinormal-mode expansion that guarantees energy conservation and causality (Lilja et al., 7 Sep 2025). The PDNN inverse-scattering method is trained only from the measured fields and forward computations corresponding to the predicted solution, rather than from a supervised dataset of labeled scatterers (Du et al., 22 Jul 2025). These examples indicate that EMNN often serves precisely to reduce black-box behavior.

In metasurface settings, EMNN should not be confused with a metaphorical analogy alone. In the SIM papers, there is a literal correspondence between trainable network parameters and hardware control variables: per-element phases or transmission coefficients (Zhang et al., 31 Oct 2025, Huang et al., 2024, Zhang et al., 15 Sep 2025, Li et al., 26 Apr 2026). This physical realizability distinguishes these architectures from purely software analogies to wave propagation.

Finally, fast inference should not be conflated with universal generalization. Several papers explicitly rely on representative synthetic training distributions (Bai et al., 2020, Alyaev et al., 2020). The geophysical studies note that training sections are informed by prior geological knowledge (Bai et al., 2020), and the logging surrogate is built from datasets designed to embrace geological rules and geosteering specifics supported by the forward model (Alyaev et al., 2020). This suggests that EMNN performance can be highly contingent on the fidelity and coverage of the generative model.

7. Open problems and likely directions

Several themes recur across the literature as unresolved technical issues.

One is generalization across operating conditions, geometries, and materials. Supervised EMNN surrogates can be extremely fast, but their validity depends on training support. The QNM-based approach addresses this by encoding resonance structure directly and reports one to two orders of magnitude better data efficiency than black-box feed-forward networks in a photonic-crystal slab benchmark (Lilja et al., 7 Sep 2025). A plausible implication is that future EMNN development will increasingly emphasize structured latent physics rather than larger unconstrained networks.

A second theme is representation of discontinuities, resonances, and multiscale structure. The discontinuous-media PINN shows that first-order Maxwell formulations, level-set enrichment, and curriculum on interface sharpness are materially beneficial (Nohra et al., 2024). The EUV WGNO likewise suggests that hybridization with an established solver can outperform both generic PINNs and purely black-box approximators (Es'kin et al., 5 Jul 2025). This suggests continued movement toward solver-informed and decomposition-based EMNNs.

A third theme is co-design of hardware and learning. In metasurface EMNNs, analog wave-domain processing and digital processing are already optimized jointly (Zhang et al., 31 Oct 2025, Zhang et al., 15 Sep 2025). Distributed variants introduce architectural questions absent from centralized digital networks, such as phase coherence across UAV-mounted SIMs, pre- versus post-detection combining, and the role of softmax temperature under channel attenuation (Li et al., 26 Apr 2026). These systems raise broader issues of calibration, quantization, control-channel overhead, and hardware nonidealities that are only partly addressed in current simulations.

A fourth theme is scalability to full-vector, broadband, and three-dimensional Maxwell problems. DeepNIS notes that extension to truly large 3D geometries and full-vector scattering remains under investigation (Li et al., 2018). PDNN identifies the Method of Moments inside the loss as the major bottleneck, with scaling concerns for high-resolution domains (Du et al., 22 Jul 2025). Even the successful discontinuous-media PINN reports training cost of about 5 min on a single GPU versus about 0.5 s for one FEM run in a benchmark, with amortization justified mainly over parametric sweeps (Nohra et al., 2024). This indicates that EMNN methods are not uniformly superior to classical solvers; their advantage is often conditional on repeated evaluations, inverse loops, or hardware co-design.

A final direction is interpretability and physical guarantees. QNM-Net is notable because it outputs resonant frequencies, port couplings, and phase delays with direct physical meaning, while enforcing causality and energy conservation by construction (Lilja et al., 7 Sep 2025). Comparable guarantees are less mature in other EMNN branches. A plausible implication is that future encyclopedia definitions of EMNN may increasingly center on this transition: from networks merely trained on electromagnetic data to networks whose internal organization is aligned with electromagnetic law, device physics, or physical hardware actuation.