Planning Neural Dynamics with Lie Group Embedding through Supervised Projective Manifold Learning

2026-05-24

AI Summary

This work proposes a neural dynamical modeling approach that embeds Lie group structures to address the incompatibility between Lie group operations and standard neural network additive arithmetic, as well as the failure of Neural Ordinary Differential Equations (Neural ODEs) caused by dynamical evolution in the nonlinear Lie algebra space. By introducing a Lie group action–induced linear mapping mechanism, the method parameterizes Lie algebra elements as linear transformations and embeds network weights under block-wise manifold constraints. It integrates gradient descent on smooth manifolds with supervised projection learning, thereby preserving continuous symmetries while ensuring dynamical stability and learnability. The approach is successfully demonstrated on SE(3) for telescope manipulator control, validating its effectiveness in constructing stable, differentiable neural dynamical systems for engineering applications such as robotics.

Abstract

We propose Lie group embedded dynamical neural networks (LieEDNN) and the corresponding learning algorithms based on gradient descent and metric projection on smooth manifold, where we treat Lie group as an intrinsic representation for continuous symmetry of manifold geometry. Thereby we achieve learnable and stable dynamics on the underlying manifold for general Lie group, and we are able to utilize the powerful representation capability of Lie group such as SO(3) and SE(3) to solve real world engineering problems in areas such as robotics, graphics, and control. Two core challenges are: (i) General Lie groups are incompatible with addition arithmetic, which is necessary for neural network interactions. (ii) The dynamics evolve in the nonlinear representation space of special algebra rather than the normal Euclidean space, which violates the paradigm of common neural ODEs. To address these two challenges, we firstly introduce adjoint Lie group action on the Lie algebra, which induces a linear mapping and transfer to the block-wise structure of weight matrices, such that addition could operate on the Lie algebra as a vector space. Then we parameterize the Lie algebra and the adjoint action as linear transformation so that the architecture is aligned with neural network perceptrons. Explicitly, this embedding appears as block-wise manifold constraints on weights, and we develop algorithms to learn the equilibrium with stability guarantees of the temporal neural network dynamics. Experiments are implemented on a specific Lie group SE(3), with the application scenario of telescopic manipulators.

Problem

• Lie group
• neural dynamics
• manifold learning
• non-Euclidean space
• continuous symmetry

Innovation

• Lie group embedding
• manifold learning
• neural dynamics
• adjoint action
• SE(3)