LieEDNN embeds Lie groups into neural nets for stable robot dynamics

A team of researchers led by Tianwei Wang has published a new method on arXiv that embeds Lie groups directly into neural network dynamics, addressing two major challenges that have limited the use of geometric structures in deep learning. The paper, "Planning Neural Dynamics with Lie Group Embedding through Supervised Projective Manifold Learning," introduces Lie group embedded dynamical neural networks (LieEDNN). Traditional neural networks operate in Euclidean space and rely on addition, which is incompatible with general Lie groups such as SO(3) and SE(3). Additionally, the nonlinear manifold geometry violates the standard neural ODE paradigm.

The researchers solve both issues by leveraging the adjoint action of a Lie group on its Lie algebra. This induces a linear mapping that transfers into block-wise structure in the weight matrices, allowing addition to operate on the Lie algebra as a vector space. The architecture then imposes manifold constraints as block-wise projections during training, with stability guarantees for the temporal dynamics. Experiments are conducted on SE(3), the special Euclidean group used in robotics, with a telescopic manipulator scenario demonstrating stable and learnable dynamics on the manifold.

The method opens up powerful representation capabilities for engineering problems. By treating Lie groups as intrinsic representations of continuous symmetry, LieEDNN can be applied to robotics, computer graphics, and control systems that require precise geometric reasoning. The paper is currently under review and offers a rigorous mathematical framework that bridges differential geometry and modern deep learning.