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

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.