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

TL;DR

This paper introduces STAResNet, a geometric deep learning architecture in Spacetime Algebra, which significantly improves the accuracy of solving Maxwell's PDEs compared to traditional Clifford ResNet models.

Contribution

The work demonstrates that using Spacetime Algebra in ResNet architectures enhances accuracy and efficiency in solving Maxwell's PDEs, outperforming real-valued and Clifford algebra-based networks.

Findings

STAResNet achieves up to 2.6 times lower MSE than standard Clifford ResNet.

Choosing the correct algebra improves model accuracy, compactness, and generalization.

The approach is effective across various scenarios, including different dimensions and obstacle configurations.

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.