# Nonlinear reduced-order modeling for time-domain electromagnetics

Carlotta Filippin, Maria Strazzullo, Stéphane Lanteri, Federico Pichi

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Carlotta Filippin, Maria Strazzullo, Stéphane Lanteri, Federico Pichi. Nonlinear reduced-order modeling for time-domain electromagnetics. EMS-TAG-SciML 2025 - Annual Meeting of EMS activity group on Scientific Machine Learning, Mar 2025, Milan, Italy. hal-05070275

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https://inria.hal.science/hal-05070275v1

Submitted on 16 May 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Nonlinear reduced-order modeling for time-domain electromagnetics

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Carlotta Filippin¹ Maria Strazzullo², Stéphane Lanteri¹, Federico Pichi³,⁴

¹Université Côte d'Azur, Inria, CNRS, LJAD, France. ²Politecnico di Torino, DISMA, Italy. ³mathLab, Mathematics Area, SISSA, Italy. ⁴Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland.
carlotta.filippin@inria.fr

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## 1. Introduction

This work is concerned with the design of fully data-driven reduced-order models in the field of time-domain electromagnetics and nanophotonics. Numerical modeling plays a crucial role in revealing the behavior of light and matter interactions at the nanoscale, exploiting computational schemes, such as the Finite-Differences (FD) Time-Domain method and, as in our case, the Discontinuous Galerkin (DG) Time-Domain method. In this context, we study reduced-order modeling (ROM) as a consequence of the pressing need for fast surrogate model capable of handling physically and geometrically parameterized electromagnetic problems. Traditional ROM techniques like the Proper Orthogonal Decomposition (POD) and the Greedy algorithm have already been investigated in the literature, along with their inherent limitations in effectively capturing nonlinear phenomena. Here, we exploit a deep learning-based ROM strategy, the Graph Convolutional Autoencoder (GCA) method [1], serving as a nonlinear extension of POD compression, harnessing the power of Graph Neural Networks (GNNs) to retain geometric structures within unstructured meshes. We present some preliminary results for simple 2D benchmarks showing promising directions toward 3D problems involving complex geometries.

## 2. Nanophotonics

The numerical modeling of **light-matter interactions** at the nanoscale emerges as a powerful tool to enhance technological innovation:

1. harnessing solar energy with nanostructured **solar cells**,
2. enhancing digital imaging sensitivity through nanostructured **CMOS** sensors,
3. optimizing light extraction and emission in **microLED** displays.

## 3. PEC cavity model

As a first example, let us consider the two-dimensional Maxwell's equations in the transverse magnetic form (TM),

$$ 
\left\{ 
\begin{aligned}
\bar{\mu} \frac{\partial \tilde{H}^x}{\partial \tilde{t}} &= -\frac{\partial \tilde{E}^z}{\partial \tilde{y}}, \\
\bar{\mu} \frac{\partial \tilde{H}^y}{\partial \tilde{t}} &= \frac{\partial \tilde{E}^z}{\partial \tilde{x}}, \\
\bar{\varepsilon} \frac{\partial \tilde{E}^z}{\partial \tilde{t}} &= \frac{\partial \tilde{H}^y}{\partial \tilde{x}} - \frac{\partial \tilde{H}^x}{\partial \tilde{y}},
\end{aligned}
\right. 
$$

where $\bar{\mathbf{E}}$ and $\bar{\mathbf{H}}$ are, respectively, the electric and the magnetic field. As a second example, we will examine the inclusion of a lossless dielectric medium with a relative permittivity $\varepsilon_1$ within a cavity, as illustrated in the figure below and described in [2].

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## 7. References

[1] F. Pichi, B. Moya, and J. Hesthaven. A graph convolutional autoencoder approach to model order reduction for parametrized PDEs. *Journal of Computational Physics*, 2024.

[2] Hassan Fahs. *High-order discontinuous Galerkin methods for solving the time-domain Maxwell equations on non-conforming simplicial meshes*. Theses, Université Nice Sophia Antipolis, 2008.

[3] J. Hesthaven and T. Warburton. *Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications*. Springer Verlag, 2007.

## 4. Simulation techniques and GCA

### Discontinuous Galerkin Methods

* Ability of dealing with **interface discontinuities** and complex geometries
* Local mass matrix: less expensive inversion and explicit semidiscrete scheme
* Numerical flux design suited for wave-dominates problems
* High-order accuracy on general grids
* It increases the total degrees of freedom through decoupling of elements

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Time-domain Maxwell's equations are key candidates with problems that are well suited to the DG approach [3]. Representation of a classical Galerkin method (a) and a DG method (b) is shown in the figure above.

**GCA-ROM:** handles graph-based input with nodal values from classical FEM simulations.
**Adaptation to DG context:** requires a new graph definition to retain all degrees of freedom.

1. **multiple values** exist at each mesh node,
2. **edge** values are also considered to better capture discontinuities in physical quantities.

Two approaches are considered:

1. **AVG-DG-GCA:** uses the same architecture as FEM, averaging all degrees of freedom at each node.
2. **DG-GCA:** preserves all degrees of freedom using a newly defined graph structure, where the nodes of the graph represent the elements of the mesh.

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## 5. AVG-DG-GCA vs. DG-GCA

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Reconstructed output is compared to input using a loss function for **unsupervised** learning.

A term for **supervised** learning compares the dataset and latent vector.

Increasing the DG method's order does not significantly impact training time.

Higher-order DG methods create a more accurate dataset of snapshots.

## 6. Conclusions

1. **Results:** AVG-DG-GCA produces good results. DG-GCA provides more uniform accuracy but longer training times, especially with higher-order elements ($P_2$).
2. **Challenges:** The presence of different media is challenging due to discontinuities in electromagnetic fields, and adjusting hyperparameters has not fully resolved these issues.