  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.
