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arXiv CS.AI·May 27·8 min read
Planning Neural Dynamics with Lie Group Embedding through Supervised Projective Manifold Learning
TL;DR
Neural networks struggle to maintain stable dynamics when modeling continuous transformations like rotations and rigid motions. LieEDNN (Lie group embedded dynamical neural networks) embeds Lie groups—mathematical structures representing continuous symmetries—into neural dynamics using metric projection on smooth manifolds.
✦ Why It Matters
Engineers can now build neural networks for robotics and 3D vision that maintain mathematical stability guarantees while learning from data.
Key Takeaways
- Use LieEDNN when modeling rotations or rigid transformations to guarantee mathematical stability.
- Embed Lie group structure directly into network architecture rather than post-hoc constraints.
- Apply metric projection during training to enforce manifold geometry constraints.
- Leverage SO(3) and SE(3) representations for robotics, computer vision, and physics simulation tasks.
Full Summary
Neural networks often fail to preserve mathematical structure when learning dynamics over continuous symmetries—transformations like 3D rotations (SO(3)) or rigid body motions (SE(3)). LieEDNN addresses this by treating Lie groups (algebraic structures encoding continuous symmetries) as intrinsic representations within neural network dynamics.
The method uses supervised projective manifold learning with gradient descent and metric projection on smooth manifolds to train networks that respect geometric constraints. By embedding Lie group structure directly into the network architecture, the approach achieves both learnable parameters and provably stable dynamics.
This enables practical applications requiring rotation and rigid motion modeling while maintaining mathematical guarantees about system behavior.