2026-05-27原文

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

The paper proposes Lie group embedded dynamical neural networks (LieEDNN) that leverage adjoint action on Lie algebra to overcome incompatibility with addition and non-Euclidean dynamics, enabling stable learning on manifolds. Experiments on SE(3) for telescopic manipulators validate the approach.

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Key points

Introduces LieEDNN with Lie group as intrinsic representation of manifold symmetry
Uses adjoint action to enable addition on Lie algebra
Parameterizes algebra and adjoint as linear transformations to integrate with neural networks
Demonstrates on SE(3) for telescopic manipulator control

Why it matters

This matters because introduces LieEDNN with Lie group as intrinsic representation of manifold symmetry.

Technical impact

May affect research directions, evaluation methods, open-source reproduction, and productization paths.

[2605.26167] Planning Neural Dynamics with Lie Group Embedding through Supervised Projective Manifold Learning

[Submitted on 24 May 2026]

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

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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.

Comments: Preprint. Under review

Subjects:

Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Dynamical Systems (math.DS); Rings and Algebras (math.RA)

Cite as: arXiv:2605.26167 [cs.LG]

(or arXiv:2605.26167v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2605.26167

arXiv-issued DOI via DataCite

Submission history

From: Tianwei Wang [view email] [v1] Sun, 24 May 2026 22:47:25 UTC (2,170 KB)

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