Electromagnetic Neural Networks (EMNNs)

• Electromagnetic Neural Networks (EMNNs) are deep learning models that embed Maxwell’s equations to ensure physics-compliant predictions and improved data efficiency.
• They utilize innovations like level-set encoding, input enrichment, and domain decomposition to accurately model discontinuous media and interface phenomena.
• EMNNs enable accelerated EM simulation and inverse design in applications such as nanophotonics and geophysical inversion, achieving significant speedups over traditional methods.

Electromagnetic Neural Networks (EMNNs) are deep learning models whose core architecture, loss functions, or training procedures explicitly incorporate electromagnetic (EM) physics—primarily via Maxwell’s equations and their boundary/interface conditions. By embedding the mathematical structure and physical constraints of EM phenomena, EMNNs achieve superior generalization, increased data efficiency, and reduced artifact generation compared to traditional black-box neural networks. This synthesis covers EMNN principles, core architectures, domain-specific methods for discontinuous media and scattering, application domains, acceleration strategies, and challenges, drawing on recent developments in physics-informed neural networks (PINNs), quasinormal-mode neural models, convolutional EM inversion, and structured physics-guided frameworks.

1. Core Principles and Formulation

Electromagnetic Neural Networks (EMNNs) are defined by the direct encoding of Maxwell’s equations—either as hard constraints in loss functions or structural priors in network design. A canonical EMNN is a Physics-Informed Neural Network (PINN) outputting electromagnetic fields (E, H) or parameters (ε, μ, σ), trained such that its predictions minimize the residuals of Maxwell's equations and enforce boundary and interface conditions (Abdelraouf et al., 6 May 2025, Nohra et al., 2024). This approach contrasts standard Deep Neural Networks (DNNs), which treat the physical system as a data mapping and require large supervised datasets.

The governing equations typically employed are the first-order Maxwell system in SI units:

$$\begin{cases} \nabla \cdot \mathbf{D} = \rho, \ \nabla \cdot \mathbf{B} = 0, \ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}, \end{cases}$$

with constitutive relations $\mathbf{D} = \epsilon(\mathbf{x}) \mathbf{E}$, $\mathbf{B} = \mu(\mathbf{x}) \mathbf{H}$, and $\mathbf{J} = \sigma(\mathbf{x}) \mathbf{E}$ (Abdelraouf et al., 6 May 2025, Nohra et al., 2024).

The typical PINN loss is constructed as the sum over squared PDE residuals at a set of collocation points, optionally weighted by supervised data fit terms:

$$L(\theta) = L_\mathrm{data}(\theta) + \lambda L_\mathrm{PDE}(\theta)$$

This construction ensures automatic compliance with conservation laws and physical symmetries, yielding physically plausible field predictions and reducing the number of required datapoints versus purely data-driven models.

2. Architectural Innovations for Discontinuous Media

A central challenge in electromagnetic modeling is resolving phenomena with sharp material interfaces and discontinuities. State-of-the-art EMNNs implement the following strategies (Nohra et al., 2024, Nohra et al., 2024):

• Level-set encoding: Discontinuous materials are encoded by a signed-distance level-set function $F(\mathbf{x})$ or $\phi(\mathbf{x})$, which is smoothly approximated via a sharp sigmoid, yielding continuous but rapidly varying parameters, e.g., $\hat H(\mathbf{x}) = S(\alpha\phi(\mathbf{x}))$ with $S(s) = 1/(1 + e^{-s})$. Material properties are then interpolated:

$$\mu(\mathbf{x}) = \mu_1 (1 - \hat{H}(\mathbf{x})) + \mu_2 \hat{H}(\mathbf{x})$$

• Input enrichment: Features such as locally-tuned high-frequency embeddings $S(\beta_k F)$ and interface normals $\nabla S(\gamma F)$ are appended to the network input vector, equipping the network to resolve interface-localized sharp gradients without global spectral bias (Nohra et al., 2024).
• Boundary and initial condition imposition: Strong enforcement is achieved via lift functions $V(\mathbf{x}, t)$ and distance factors $d(\mathbf{x})$, allowing the network to exactly satisfy Dirichlet and initial conditions.
• First-order formulation selection: Neural Tangent Kernel (NTK) analysis demonstrates more favorable convergence properties for first-order (curl-div) Maxwell systems versus second-order PDEs at interfaces, with more uniform eigenvalue spacing and no stiff plateauing (Nohra et al., 2024).
• Domain decomposition: Overlapping subdomain partitioning further accelerates training and enables handling of large or multiply connected domains.

Numerical results on benchmark problems—including 3D spheres-in-cubes, transient and parametric multi-interface scenarios—yield sub-percent relative $L_2$ errors and rapid convergence ($L_\mathrm{PDE} < 10^{-6}$ within 7,000 iterations) (Nohra et al., 2024, Nohra et al., 2024).

3. Scattering, Surrogates, and Physics-Based Priors

EMNNs have recently been extended to surrogate electromagnetic scattering models via direct encoding of physical modal expansions. The quasinormal-mode (QNM) neural framework ("QNM-Net") employs an analytic resonant scattering matrix decomposition,

$$S(\omega) = e^{i\omega \tau} \left\{ C(\omega) + D \left[ i\omega - i\tilde{\Omega} \right]^{-1} M^{-1} D^\dagger C(\omega) \right\} e^{i\omega \tau}$$

where the NN learns the background $C(\omega)$, port delays $\tau$, mode couplings $D$, eigenfrequencies $\tilde{\Omega}$, and overlap matrix $M$ (Lilja et al., 7 Sep 2025). This architecture guarantees energy conservation (exact unitarity for any truncation) and causality (poles in the lower half-plane), resulting in order-of-magnitude data efficiency improvements over standard supervised NNs.

QNM-Net achieves sub-percent spectral prediction errors for photonic crystal slabs and all-dielectric metasurface scattering with 2–10× fewer samples than traditional networks, and applies equally to arbitrary N-port, multi-resonant EM devices. A plausible implication is that modal physics priors enable leaner, more interpretable surrogates across virtually all linear wave-scattering applications.

4. EMNNs in Inverse Design and High-Fidelity Simulation

Physics-informed architectures facilitate both forward EM simulation and inverse design for nanophotonic devices, antennas, and metamaterials. Fast surrogates (e.g., WaveY-Net) amortize full-wave simulation costs—achieving field prediction and optimization convergence speeds up to 10³× faster than FDTD/FEM solvers (Abdelraouf et al., 6 May 2025, Dove et al., 2024).

Hybrid models like the Waveguide Neural Operator (WGNO) embed the physics of classical modal methods but replace the linear-solve bottleneck with a compact neural operator acting in Fourier space, achieving $10^{-6}$–$10^{-7}$ prediction error and $\gtrsim 200$× speedups for 3D mask simulations (Es'kin et al., 5 Jul 2025).

In foundation-simulator frameworks (e.g., UCMax), multi-conditioned convolutional U-Nets can generalize to arbitrary wavelength, incidence, time step, and material profile, supporting both forward and inverse tasks via backpropagation. Attentional conditioning and non-recurrent supervision enable rigorous, provable inference-time error bounds (energy-weighted MSE), fostering confidence in substitution for classical solvers (Dove et al., 2024).

5. Structured Physics-Guided Learning for Device Modeling

In mechatronics, EMNNs with embedded physics structures have been leveraged to model static position-dependent force-current mappings in electromechanical actuators such as linear motors. Physics-Guided Neural Networks (PGNNs) retain analytical invertibility by incorporating known harmonic/sinusoidal backbone models, with additional small NNs learning parasitic or manufacturing-induced residual effects (Bolderman et al., 2024).

In the PGNN framework, the predictive force law takes the form:

$$\hat{F}(t) = \hat{K}^l(\theta^l, y(t))\,i^l(t) + \hat{F}_{cog}(\phi^l, y(t))$$

allowing for real-time commutation law inversion and robust correction of position-dependent errors. Real-world experiments demonstrate 10× reductions in commutation error and 4× improvement in position tracking error compared to classical commutation (Bolderman et al., 2024).

6. Convolutional Architectures and Geophysical EM Inversion

Fully convolutional EMNNs have been successfully deployed in high-dimensional EM imaging and inversion tasks, notably geophysical applications such as CO₂ plume monitoring via CSEM (Puzyrev, 2018). By framing the inversion as an image-to-image regression and using pixel-wise loss functions (IoU for binary delineation, RMSE for continuous resistivity), these architectures provide real-time subsurface reconstructions, robust to noise and anomaly geometry.

Training on a few thousand synthetic scenarios is sufficient when augmented, and real-time inference (<1 s) makes EMNNs attractive for online data analysis and as initial models for traditional, iterative inversion schemes. A plausible implication is that black-box convolutional EMNNs, while less interpretable, provide practical acceleration and are extensible to anisotropic and time-dependent modeling.

7. Remaining Challenges and Outlook

EMNNs combine rigorous physics compliance with deep-learning flexibility, but outstanding challenges remain. These include interpretability of latent representations, coverage of unconventional modes, and scalability to multi-physics coupling (e.g., combining Maxwell and thermal equations). Multi-fidelity and transfer learning strategies, spectral normalization, and domain-decomposition approaches are active areas of development (Abdelraouf et al., 6 May 2025).

Recent models demonstrate robust generalization, accelerated simulation, and modular adaptation across nanophotonics, metamaterials, lithography mask design, and electromagnetic actuator control. The inclusion of physics-informed priors is central to data efficiency, physical fidelity, and model interpretability.

A plausible implication is that, as EMNN frameworks mature, one can expect broader use as universal surrogates for EM simulation, optimization, and control, enabling real-time design pipelines, adaptive device functionality, and physically trustworthy inverse problem solutions across science and engineering.