Graph neural networks solve Maxwell’s equations numerically

Solving Maxwell’s equations, a set of four fundamental equations that define electromagnetism and optics, is fundamental to every task in computational photonics. Solving these equations for modern problems is often too complex for traditional numerical methods, so researchers have increasingly turned to artificial neural networks. However, these networks are limited by a fixed scale and resolution.

Kuhn et al. developed a graph neural network to solve Maxwell’s equations numerically. Graph neural networks are useful for processing unstructured and non-uniform data and work independently of graph size or structure.

“With these graph neural networks, the same limitations do not apply,” said author Lina Kuhn. “We can train the network on relatively small domains and extrapolate afterwards to larger domains in the inference process.”

The authors used their network to calculate a single time step using the finite-difference time-domain method, one of the most common numerical simulations for understanding how light interacts with a structure. The method works by advancing the field propagation by a discrete time step instead of focusing on a steady-state solution for the field.

“We are excited by the simplicity with which we could solve the problem,” Kuhn said.

The team’s neural network is a proof of concept but could be used to describe more complex systems, such as simulations in higher dimensions or with dispersive materials. However, they say more work is needed to accelerate the simulations, such as by adding sparse matrices, and look forward to additional contributions from the machine learning community.

“This is rewarding work with many opportunities,” Kuhn said.

Source: “Exploiting graph neural networks to perform finite-difference time-domain based optical simulations,” by L. Kuhn, T. Repän, and C. Rockstuhl, APL Photonics (2023). The article can be accessed at https://doi.org/10.1063/5.0139004 .