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.

翻译：我们提出几何克利福德代数网络（GCANs）用于建模动力学系统。GCANs基于利用几何（克利福德）代数的对称群变换。我们首先回顾现代（平面基）几何代数的精髓，该代数建立在编码为$\mathrm{Pin}(p,q,r)$群元素的等距变换之上。随后提出群作用层概念，通过预定义的群作用线性组合物体变换。结合新的激活与归一化方案，这些层构成了可通过梯度下降优化的可调$\textit{几何模板}$。理论优势在三维刚体变换建模及大规模流体动力学仿真中得到充分体现，相比传统方法展现出显著提升的性能。