Lie Group Embedding Enables Stable Neural Dynamics on Manifolds

Researchers have introduced Lie group embedded dynamical neural networks (LieEDNN), utilizing learning techniques rooted in gradient descent and metric projection on smooth manifolds. This innovative method employs Lie groups as fundamental representations of continuous symmetry within manifold geometry, facilitating stable and learnable dynamics across various general Lie groups. It capitalizes on the representation strengths of groups such as SO(3) and SE(3), which are essential in robotics, graphics, and control systems. The study tackles two primary issues: the incompatibility of Lie groups with addition arithmetic for neural network functions and the evolution of dynamics in a nonlinear representation space of special algebra, diverging from traditional neural ODE frameworks. To address this, the approach incorporates adjoint operators for linear operations on Lie algebras.