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：我们提议采用几何(Cliford)代数进行对称组变换,以对称组变换为基础。 GCAN特别适合代表并操纵往往是在动态系统中发现的几何变换,我们首先审查现代(基于飞机的)几何代数(代数)的精度,这些变数以以美元/mathrm{Pin}(p,q,r)美元组数编码成元素的等离子体变数为基础。然后,我们提出分组行动层的概念,即使用预先指定的组数动作线性地将物体变形合并起来。与新的激活和正常化计划一起,这些层可用作可以通过梯度下降加以改进的可调整的几何结构。理论优势在三维硬体变形模型和大型流体动态模拟中得到了有力的反映,表明相对于传统方法的性能显著改善。