STAResNet: a Network in Spacetime Algebra to solve Maxwell's PDEs

TL;DR

The paper addresses solving Maxwell's PDEs with neural networks by exploiting Geometric Algebra embeddings, comparing standard Clifford networks with a Spacetime Algebra (STA) formulation. It introduces STAResNet, a ResNet-like architecture operating on spacetime bivectors in STA, and evaluates it on 2D and 3D Maxwell problems with varying sampling, obstacles, and rollout horizons. STAResNet achieves up to about 2.6× lower mean squared error and improved correlation using roughly the same or fewer parameters, demonstrating that the choice of algebra critically impacts accuracy and generalization. The findings suggest that embedding physical fields in the appropriate geometric algebra yields more descriptive, compact, and robust PDE solvers, with potential applications across physics-informed machine learning.

Abstract

We introduce STAResNet, a ResNet architecture in Spacetime Algebra (STA) to solve Maxwell's partial differential equations (PDEs). Recently, networks in Geometric Algebra (GA) have been demonstrated to be an asset for truly geometric machine learning. In \cite{brandstetter2022clifford}, GA networks have been employed for the first time to solve partial differential equations (PDEs), demonstrating an increased accuracy over real-valued networks. In this work we solve Maxwell's PDEs both in GA and STA employing the same ResNet architecture and dataset, to discuss the impact that the choice of the right algebra has on the accuracy of GA networks. Our study on STAResNet shows how the correct geometric embedding in Clifford Networks gives a mean square error (MSE), between ground truth and estimated fields, up to 2.6 times lower than than obtained with a standard Clifford ResNet with 6 times fewer trainable parameters. STAREsNet demonstrates consistently lower MSE and higher correlation regardless of scenario. The scenarios tested are: sampling period of the dataset; presence of obstacles with either seen or unseen configurations; the number of channels in the ResNet architecture; the number of rollout steps; whether the field is in 2D or 3D space. This demonstrates how choosing the right algebra in Clifford networks is a crucial factor for more compact, accurate, descriptive and better generalising pipelines.

Figures (20)

• Figure 1: 3D GA approach: solving Maxwell's PDEs through Clifford ResNet, a ResNet-inspired network working in G(3,0,0).
• Figure 2: STA approach: solving Maxwell's PDEs through STAResNet, our ResNet-inspired network working in G(1,3,0).
• Figure 3: Training and validation losses versus number of epochs for 2D Maxwell's PDEs for instances sampled at (a) $25$s, (b) $50$s, (c) $75$s, (d) $100$s.
• Figure 4: (a) Mean squared error and (b) correlation between estimated and ground truth EM fields in the test set for varying $\Delta t$.
• Figure 5: Ground truth and estimated EM field with Clifford ResNet and STAResNet for datasets sampled at (a) $\Delta t = 25$s and (b) $\Delta t = 50$s.