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.

翻译：我们提出基于几何（克利福德）代数的对称群变换的几何克利福德代数网络（GCANs）。GCANs特别适合表示和操作动态系统中常见的几何变换。我们首先回顾现代（基于平面的）几何代数的精髓，该代数建立在由$\mathrm{Pin}(p,q,r)$群编码的等距变换之上。随后提出群作用层的概念，该层利用预先指定的群作用线性组合对象变换。结合新的激活与归一化方案，这些层可作为可通过梯度下降优化的可调几何模板。理论优势在三维刚体变换建模及大规模流体动力学模拟中得到充分体现，相较于传统方法展现出显著性能提升。