# Latent Mixture of Symmetries for Sample-Efficient Dynamic Learning

arXiv: https://arxiv.org/abs/2510.03578 · HTML: https://mlsanity.com/paper/2510.03578/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in RL for robot locomotion and manipulation, one of 15 directions in Deep reinforcement learning methods.
- Builds on 44 of its 82 references that are in the map.
- First posted on arXiv 4 Oct 2025.

## Placed in

- Deep reinforcement learning methods / [RL for robot locomotion and manipulation](https://mlsanity.com/deeprl/rl-for-robot-locomotion-and-manipulation/)

## Builds on (most cited first, 30 of 44)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Neural Network Dynamics for Model-Based Deep Reinforcement Learning with Model-Free Fine-Tuning](https://mlsanity.com/paper/1708.02596/) (2017)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [SO(2)-Equivariant Reinforcement Learning](https://mlsanity.com/paper/2203.04439/) (2022)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [ContiFormer: Continuous-Time Transformer for Irregular Time Series Modeling](https://mlsanity.com/paper/2402.10635/) (2024)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Relaxing Equivariance Constraints with Non-stationary Continuous Filters](https://mlsanity.com/paper/2204.07178/) (2022)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [Lie Point Symmetry and Physics Informed Networks](https://mlsanity.com/paper/2311.04293/) (2023)
- [A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning](https://mlsanity.com/paper/2311.00212/) (2023)
- [Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution](https://mlsanity.com/paper/2310.02299/) (2023)
- [Time-Reversal Symmetric ODE Network](https://mlsanity.com/paper/2007.11362/) (2020)

## Built on by (most cited first, 5 of 5)

- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)
- [Predicting Power-System Dynamic Trajectories with Foundation Models](https://mlsanity.com/paper/2604.14991/) (2026)
- [LASS-ODE: Scaling ODE Computations to Connect Foundation Models with Dynamical Physical Systems](https://mlsanity.com/paper/2602.01009/) (2026)
- [Cross-Domain Generalization of Multimodal LLMs for Global Photovoltaic Assessment](https://mlsanity.com/paper/2511.19537/) (2025)

## Abstract (arXiv)

Learning dynamics is essential for model-based control and Reinforcement Learning in engineering systems, such as robotics and power systems. However, limited system measurements, such as those from low-resolution sensors, demand sample-efficient learning. Symmetry provides a powerful inductive bias by characterizing equivariant relations in system states 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 a mixture of symmetry-governed latent factors from complex dynamical measurements. Latent MoS focuses on dynamic learning while locally and provably preserving the underlying symmetric transformations. To further capture long-term equivariance, we introduce a hierarchical architecture that stacks MoS blocks. Numerical experiments in diverse physical systems demonstrate that Latent MoS outperforms state-of-the-art baselines in interpolation and extrapolation tasks while offering interpretable latent representations suitable for future geometric and safety-critical analyses.