Title: Latent Mixture of Symmetries for Sample-Efficient Dynamic Learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: dynamic learning; sample efficiency; symmetry preservation; Lie transformations; geometric deep learning
Abstract: Learning dynamics is essential for model-based control and Reinforcement Learning in systems operating in changing environments, such as robotics, autonomous vehicles, and power systems. However, limited system measurements, such as those from low-resolution meters, demand sample-efficient learning. Symmetry provides a powerful inductive bias by characterizing equivalent relations in system behavior to improve sample efficiency. While recent methods attempt to discover symmetries from data, they typically assume a single global symmetry group and treat symmetry discovery and dynamic learning as separate tasks, leading to limited expressiveness and error accumulation. In this paper, we propose the Latent Mixture of Symmetries (Latent MoS), an expressive model that captures symmetry-governed latent factors from complex dynamical measurements. Latent MoS focuses on dynamic learning while locally preserving the underlying symmetric transformations. To further capture long-range temporal equivalence, we introduce a hierarchical architecture that stacks Latent MoS blocks across multiple time scales. Numerical experiments across diverse physical systems demonstrate that Latent MoS significantly outperforms state-of-the-art baselines in interpolation and extrapolation tasks while offering interpretable latent representations suitable for future geometric and safety-critical analysis.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes combining MoEs with Neural ODEs to learn latent dynamics which respect symmetry. The objective is similar to symmetry discovery, but avoids common limitations such as error propagation and limited expressiveness of multiple symmetry groups. The authors use a mixture of latent flows where each flow is governed by a different Lie group and use a encoder-decoder architecture for nonlinear group actions on the data. Experiments on many different domains with ODE dynamics and real-world data show that their method outperforms baselines.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n- The authors provide various theoretical analyses on how Latent MoS can learn several different symmetry groups.\n- The authors experiment on several datasets, including real-world domains such as residential electricity consumption, solar energy, etc.\n\n## Weaknesses\n- I feel like the core flaw of the proposed approach is the linear mixing/weighted average (Equation 8). This seems to go against how symmetries should be composed together (composition/matrix multiplication). It's not clear to me how this method really does learn both symmetries with this mixing technique and it seems to in fact break the group structure, since the mixture isn't a valid group element. Can the authors clarify?\n- Furthermore, the ODE-RNN structure doesn't seem to use the same mixture or symmetry groups for the latent dynamics at all timesteps. Why would a symmetry be present at a certain timestep but not present at all other timesteps? The proposed method doesn't seem to preserve the mixing weights from previous timesteps.\n- LaLiGAN (Reference [38] Yang et al. 2023) and perhaps a downstream dynamics model should be included as baseline. It seems to have the same objective (learning latent symmetries) as this paper.\n\nQUESTIONS:\n- Are the symmetry groups known a priori? Or can the method learn (perhaps through the gating mechanism) subgroups of the general linear/affine group.\n- How are learned symmetries preserved across time steps? Would this method correct learn the Lie point symmetries of the governing equations?\n\nLIMITATIONS:\nSee weaknesses and questions.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper addresses the problem of dynamic learning under data scarcity. The authors introduce the Latent Mixture of Symmetries (MoS) framework based on the Mixture-of-Experts (MoE) approach, efficiently preserving local equivariance through structured transformations. Utilizing a gating mechanism derived from MoE, MoS adaptively selects the most appropriate local Lie group symmetries. Extensive evaluations on a range of synthetic and real-world dynamical systems demonstrate promising improvements compared to several baseline methods.

STRENGTHS AND WEAKNESSES:
### Strengths
1. The concept of modeling different local symmetries within a single trajectory is appealing, clearly motivated, and innovative.
2. The paper is well-structured, presenting detailed theoretical foundations supported by extensive empirical evaluations.
3. The experimental results consistently indicate superior performance of the proposed approach compared to existing baselines.

### Weaknesses
See Questions below.

QUESTIONS:
1. How sensitive is Latent MoS to the accurate identification of underlying symmetry groups? As indicated by ablation studies, including unnecessary symmetries can negatively impact performance. Could the authors comment on any conceptual connections between this scenario and overcomplete dictionary learning, considering symmetries analogous to dictionary items?

2. Could the authors clarify how the interpolation and extrapolation tasks are precisely defined?

3. In Figure 3 (Lotka System - Drop rate 30%), the blue bar appears to be missing. Could the authors provide clarification or correct this oversight?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Latent Mixture of Symmetries (MoS), a continuous-time dynamics model that keeps physical equivariance by treating each latent time-step as a learnable Lie-group action and then blending several such actions with a Mixture-of-Experts gate. The theoretical analysis gives a simple commutation criterion that guarantees any selected action remains equivariant with respect to the known symmetry in the observation space. Experiments on both synthetic ODEs and real-world datasets show improved accuracy and qualitatively symmetry-consistent latents compared with baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrength\n1. Using MoS to learn a data-driven mixture of symmetry experts is a neat way to model symmetry in dynamical systems with unknown and/or shifting symmetries.\n2. The commutation-based design yields interpretable latent operators while still providing provable equivariance guarantees, which helps both sample efficiency and interpretability.\n\nWeakness\n1. The paper does not compare with symmetry-discovery baselines that explicitly discover symmetries from data. \n2. The paper does not compare itself against the symmetry-breaking literature (methods that gradually relax or modulate equivariance). Can one view the proposed method like breaking the symmetry from the affine group? A discussion here would be helpful.\n\nQUESTIONS:\n1. It would be helpful to state explicitly which symmetry (or combination) exists in each experimental dataset and provide quantitative evidence that MoS captures it (the PCA in Fig. 2 is a good start, but more discussion would be beneficial).\n2. Why is Corollary 1 limited to planar rotation? Did any experiment involve true 3D rotational symmetry, and if so how was it handled?\n3. Could the authors clarify how MoS might extend beyond rotation/translation/scaling?\n4. How sensitive is performance to the size of the symmetry candidate library? Will more irrelevant experts hinder the performance?\n\nLIMITATIONS:\nAlthough the authors list some limitations, the paper would benefit from a failure-case analysis, e.g., examples where the gating network consistently picks the wrong expert and how that might be resolved.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose the Latent Mixture of Symmetries (Latent MoS), a model for learning dynamics by leveraging symmetry as an inductive bias. Unlike traditional approaches, which either impose a priori available symmetries or discover global ones, Latent MoS learns a mixture of local latent symmetries through Lie group transformations. As a result, it preserves local equivariance and allows for time-dependent adaptations. To enable long-term equivariance, the authors propose hierarchically stacking MoS modules. The authors provide a theoretical justification for the model's ability to preserve equivariance. Empirical experiments in physical systems demonstrate improved performance over baselines in terms of interpolation, extrapolation, and sample efficiency.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Quality:\n  - The authors propose a mathematically rigorous model with formal guarantees of equivariance preservation.\n  - Experimental results, with thoroughly studied ablations, on a diverse set of physical systems demonstrate strong performance in low-data settings.\n- Clarity:\n  - The flow from single-group equivariance to mixtures and temporal hierarchy clarifies the fundamental ideas.\n  - In general, the paper is well-written.\n- Significance:\n  - The MoS formulation provides structured latent dynamics in various settings, including model-based RL and scientific computing.\n- Originality:\n  - Time-varying mixture of experts for modeling latent symmetry transformations seems like a novel idea.\n\nWeaknesses:\n- Quality:\n  - The ablation study does not address imperfect gating.\n- Clarity:\n  - Derivations in section 3, especially 3.2, may benefit from a more intuitive explanation to non-expert readers.\n- Significance:\n  - Applicability to closed-loop control, planning, RL, etc, has not been empirically tested.\n- Originality:\n  - The novelty mainly lies in the combination of well-known components.\n\nQUESTIONS:\n- Can you provide qualitative evidence to support that latent symmetries are interpretable?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}