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arxiv: 2605.26167 · v1 · pith:6RKLVQXCnew · submitted 2026-05-24 · 💻 cs.LG · cs.AI· math.DS· math.RA

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

Tianwei Wang , Bryan Chen , Qian Zuo , Qiyue Xia , Xin Li , Wei Pang This is my paper

Pith reviewed 2026-06-30 11:33 UTC · model grok-4.3

classification 💻 cs.LG cs.AImath.DSmath.RA
keywords Lie groupdynamical neural networksmanifold learningSE(3)adjoint actionneural ODEroboticscontrol
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The pith

Lie group embedding allows stable and learnable dynamics on manifolds by constraining neural network weights to respect group structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Lie group embedded dynamical neural networks to model systems with continuous symmetries such as rotations and rigid motions. Standard neural networks struggle with Lie groups because they are not closed under addition and their dynamics live on nonlinear manifolds. The approach maps the problem to the Lie algebra using adjoint actions, turning group operations into linear transformations that fit inside perceptron layers as block-wise constraints. Gradient descent with metric projection then trains these constrained weights while preserving stability. This enables applications like controlling telescopic manipulators using SE(3) representations.

Core claim

By parameterizing the adjoint Lie group action as a linear transformation on the Lie algebra, the method induces a block-wise structure on weight matrices that allows addition to operate in vector space while maintaining manifold constraints, enabling gradient-based learning of stable temporal dynamics on general Lie groups such as SE(3).

What carries the argument

The adjoint action of the Lie group on its algebra, which provides a linear mapping that structures the neural network weights into blocks compatible with manifold geometry.

If this is right

  • Stable dynamics can be learned on the manifold for any Lie group.
  • Representation power of groups like SO(3) and SE(3) can be used directly in neural networks for robotics and control.
  • Algorithms based on gradient descent and metric projection guarantee stability of the temporal dynamics.
  • Practical implementation demonstrated on SE(3) for telescopic manipulators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach might extend to other symmetric systems beyond robotics, such as molecular modeling where rotations matter.
  • Integrating with existing neural ODE solvers could become straightforward once the projection step is added.
  • The method suggests that many physical systems could benefit from built-in symmetry preservation rather than learning it from data.

Load-bearing premise

Gradient descent plus metric projection can stably learn the block-wise manifold constraints on the weight matrices without destroying the group structure or causing instability in the dynamics.

What would settle it

Training the network on SE(3) data and checking whether the learned weights satisfy the manifold constraints and whether the simulated trajectories remain on the group manifold without diverging.

Figures

Figures reproduced from arXiv: 2605.26167 by Bryan Chen, Qian Zuo, Qiyue Xia, Tianwei Wang, Wei Pang, Xin Li.

Figure 1
Figure 1. Figure 1: Through conjugate Lie group action ◦ with g and g −1 , we define the adjoint action whereby linearize a Lie group action on Lie algebra. Matrix exponential Exp(·) and logarithm Log(·) allow us to retract Lie algebra g to Lie group manifold G, vice versa. For SO(3) and so(3), adjoint action Ad(·) maps two rigid body rotations which is a linear transformation on so(3) by Thm. 3.1. Theorem 3.1 (Adjoint Action… view at source ↗
Figure 2
Figure 2. Figure 2: (a). Each Lie group neuron is integrated by several [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a). Loss curves of projective manifold learning. Periodic projection will cause fluctuation [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a). Evolution trajectory of trained LieEDNN. Network dynamics converge to the target [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes Lie group embedded dynamical neural networks (LieEDNN) that embed general Lie groups into neural ODEs by using the adjoint action to induce linear mappings on the Lie algebra, which in turn impose block-wise manifold constraints on the weight matrices of the network. These constraints are enforced during learning via gradient descent combined with metric projection on the manifold, with the goal of producing stable temporal dynamics that respect the underlying group structure. The approach is positioned as addressing the incompatibility of Lie groups with Euclidean addition and the nonlinear nature of the dynamics; experiments are reported only for the specific case of SE(3) applied to telescopic manipulator control.

Significance. If the stability guarantees and the preservation of group structure under the proposed projection step hold beyond the demonstrated case, the work would offer a concrete mechanism for incorporating continuous symmetries into neural dynamical models, which could be valuable for robotics, graphics, and control applications where SO(3) and SE(3) representations are already standard. The reduction of adjoint actions to linear transformations on the algebra that align with perceptron layers is a technically interesting device that could bridge geometric and standard neural architectures.

major comments (2)
  1. [Abstract] Abstract: the claim that 'algorithms with stability guarantees of the temporal neural network dynamics' are developed for general Lie groups is load-bearing for the central contribution, yet the only reported experiments use SE(3) on telescopic manipulators; no results, ablation, or analysis are supplied for other groups (e.g., SO(3) or non-compact cases), so the generality of the metric-projection step remains unverified.
  2. [The section on the learning algorithm and stability guarantees] The section describing the adjoint-action parameterization and the induced block-wise manifold constraints: the assumption that gradient descent plus metric projection on these constraints will preserve the original Lie-algebra embedding and produce stable dynamics is stated but not accompanied by a derivation, Lyapunov-style argument, or counter-example check that would apply outside the SE(3) setting; if the projection step introduces drift for other groups, the claim that the method works for general Lie groups does not hold.
minor comments (1)
  1. [Abstract] The abstract and introduction would benefit from an explicit statement of the precise form of the metric projection operator and the manifold on which it acts (e.g., an equation defining the projection).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments highlighting the importance of verifying generality and providing explicit stability arguments. We address each major point below, indicating where revisions will be made to clarify the scope and strengthen the presentation without altering the core claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'algorithms with stability guarantees of the temporal neural network dynamics' are developed for general Lie groups is load-bearing for the central contribution, yet the only reported experiments use SE(3) on telescopic manipulators; no results, ablation, or analysis are supplied for other groups (e.g., SO(3) or non-compact cases), so the generality of the metric-projection step remains unverified.

    Authors: The theoretical construction using adjoint actions to induce block-wise manifold constraints on weights and metric projection to enforce them is derived without reference to a specific group, relying only on the vector space structure of the Lie algebra and the existence of a suitable metric. SE(3) serves as a representative non-compact example with practical relevance; the same parameterization applies directly to SO(3) and other compact groups. We will revise the abstract to state that the method is developed for general Lie groups with empirical demonstration on SE(3), and add a short paragraph in the experiments section discussing extension to SO(3). revision: partial

  2. Referee: [The section on the learning algorithm and stability guarantees] The section describing the adjoint-action parameterization and the induced block-wise manifold constraints: the assumption that gradient descent plus metric projection on these constraints will preserve the original Lie-algebra embedding and produce stable dynamics is stated but not accompanied by a derivation, Lyapunov-style argument, or counter-example check that would apply outside the SE(3) setting; if the projection step introduces drift for other groups, the claim that the method works for general Lie groups does not hold.

    Authors: The projection step is defined to be the nearest-point projection onto the submanifold of weight matrices satisfying the block-wise constraints induced by the adjoint representation; by construction this restores exact satisfaction of the constraints after each gradient step, thereby preserving the Lie-algebra embedding. Stability follows from the fact that the resulting vector field on the algebra is linear and the manifold is invariant under the group action. We will expand the learning-algorithm section with an explicit invariance argument showing that the projected dynamics remain on the manifold for any Lie group admitting a bi-invariant metric, and note that full Lyapunov analysis for arbitrary groups is left for future work. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins from standard Lie group properties (adjoint action inducing linear maps on the algebra), parameterizes the embedding to produce block-wise manifold constraints on weights, and proposes GD + metric projection algorithms claimed to preserve structure and yield stable dynamics. No equation or step reduces the claimed stability guarantees or general-Lie-group applicability to a fitted parameter, self-citation chain, or input by construction. The method is presented as an architectural choice aligned with neural ODEs, with experiments on SE(3) serving as validation rather than the sole source of the result. This is a self-contained proposal without the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the method appears to rest on the standard Lie-algebra adjoint representation and the existence of a Riemannian metric allowing projection, both treated as background rather than derived.

axioms (2)
  • domain assumption Adjoint action of a Lie group on its algebra induces a linear map compatible with neural-network addition
    Invoked to solve the incompatibility of Lie groups with vector addition
  • domain assumption Metric projection onto the manifold after each gradient step preserves stability of the learned dynamics
    Central to the learning algorithm but not justified in abstract

pith-pipeline@v0.9.1-grok · 5811 in / 1389 out tokens · 29713 ms · 2026-06-30T11:33:20.760339+00:00 · methodology

discussion (0)

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