We propose Geometric Clifford Algebra Networks (GCANs) that are based on
symmetry group transformations using geometric (Clifford) algebras. GCANs are
particularly well-suited for representing and manipulating geometric
transformations, often found in dynamical systems. 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 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.

本文提出了基于几何（Clifford）代数的对称群转换的几何克利福德代数网络（GCAN），并采用组合叠加的方式实现对象变换。通过梯度下降，可以实现这些层作为可调整的几何模板，用于模拟三维刚体变换和大规模流体动力学模拟，相对于传统方法，实现了显著提高的性能。