Geometric Clifford Algebra Networks
This work addresses the problem of accurately modeling complex dynamical systems for applications in physics and engineering, representing a novel method rather than an incremental improvement.
The paper tackles modeling dynamical systems by proposing Geometric Clifford Algebra Networks (GCANs), which use symmetry group transformations and geometric algebras to create adjustable geometric templates, resulting in significantly improved performance in 3D rigid body transformations and large-scale fluid dynamics simulations.
We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the $\mathrm{Pin}(p,q,r)$ group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable $\textit{geometric templates}$ that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.