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

TL;DR

The paper tackles the high computational cost of transient Maxwell simulations by recasting the discretized update equations into a fixed-weight, two-layer graph neural network (GNN) with predefined edge weights, yielding GEM, a graph-driven solver. By aligning the local message-passing schema with the FDTD update scheme, GEM achieves exact equivalence to standard solvers while exploiting GPU-accelerated GNN implementations for dramatic speedups (up to ~40× over CPU FDTD) and without any training. The authors validate GEM against a conventional FDTD solver across thousands of random scenarios, and demonstrate its applicability to SAR in biological tissue and THz photonic crystals, with perfect fidelity ($R^2$ ~ 0.9999). They also extend the framework to absorbing boundary conditions via a split-field PML formulation and show substantial performance advantages as domain size grows. The work suggests a broader paradigm of graph-driven numerical solvers that can outperform legacy methods and potentially generalize to other PDEs, offering a pathway to fast, accurate simulations in electromagnetics, biology, and photonics.

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

Figures (5)

• Figure 1: Graph-driven solver of Maxwell's Equations; the $E_y$, $H_z$, and $H_x$ components are shown in green, purple, and dark orange color. a, Discretized TE Maxwell equations along with b, a corresponding example 2D TE grid representation of the electromagnetic field component arrangement in space. c, The grid implicitly assumes an equivalent graph topology, and consequently, d, the electromagnetic field update equations can be expressed in a graph-driven manner via mathematical operations between the graph nodes.
• Figure 2: Computational emulation of electromagnetic wave propagation for different use cases; a shows a snapshot of the electric field $E_y$ evolution simulated via GEM, whilst b depicts the snapshot at the same time instance as computed with FDTD. The simulated domain is shown in c, whereas d illustrates the waveform of the $E_y$ component sampled at a random point of the simulated grid.
• Figure 3: Estimation of SAR with GEM for a biological tissue. a, Electric permittivity for the air, skin, fat, and muscle layers of the biological tissue. b, Snapshot of the electric field intensity dissipated throughout the tissue layers. c, Computed SAR with GEM and FDTD along a vertical cross-section passing through the point source.
• Figure 4: Wave propagation in a cavity waveguide. a shows the geometry of the waveguide, while b depicts a snapshot of the electric field intensity across the waveguide simulated with GEM. c Simulated waveform via GEM and FDTD sampled at the end of the waveguide.
• Figure 5: Performance comparison. a, Simulation time for GEM, our in-house FDTD code and Meep Meep. b, Performance gain achieved by GEM and Meep compared to the in-house FDTD code with CPU parallelization.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2