Solving Maxwell's equations with Non-Trainable Graph Neural Network Message Passing

TL;DR

This paper introduces a novel graph neural network approach to solve Maxwell's equations efficiently, achieving high accuracy with reduced computational time, and suggests potential applications to other PDE-based scientific problems.

Contribution

It demonstrates that Maxwell's equations can be solved using a simple two-layer GNN with static edge weights, offering a new, efficient computational method.

Findings

Achieves comparable accuracy to traditional methods

Reduces computational time significantly

Provides a framework adaptable to other PDEs

Abstract

Computational electromagnetics (CEM) is employed to numerically solve Maxwell's equations, and it has very important and practical applications across a broad range of disciplines, including biomedical engineering, nanophotonics, wireless communications, and electrodynamics. The main limitation of existing CEM methods is that they are computationally demanding. Our work introduces a leap forward in scientific computing and CEM by proposing an original solution of Maxwell's equations that is grounded on graph neural networks (GNNs) and enables the high-performance numerical resolution of these fundamental mathematical expressions. Specifically, we demonstrate that the update equations derived by discretizing Maxwell's partial differential equations can be innately expressed as a two-layer GNN with static and pre-determined edge weights. Given this intuition, a straightforward way to numerically solve Maxwell's equations entails simple message passing between such a GNN's nodes, yielding a significant computational time gain, while preserving the same accuracy as conventional transient CEM methods. Ultimately, our work supports the efficient and precise emulation of electromagnetic wave propagation with GNNs, and more importantly, we anticipate that applying a similar treatment to systems of partial differential equations arising in other scientific disciplines, e.g., computational fluid dynamics, can benefit computational sciences