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

Alberto Pepe		
Cambridge University, UK

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 allows for a mean square error (MSE)  between ground truth (GT) and estimated fields up to 2.6 times lower as opposed to Clifford ResNet with 6 times fewer trainable parameters, consistently lower MSE and higher correlation regardless of the sampling period of the dataset, the presence of obstacles with either seen or unseen configurations, the number of channels in the ResNet architecture, the number of rollout steps and whether the field is in 2D or in 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.