# Adaptive Symmetry Discovery for Dynamical System Identification

arXiv: https://arxiv.org/abs/2608.08091 · HTML: https://mlsanity.com/paper/2608.08091/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 59 references that are in the map.
- First posted on arXiv 8 Aug 2026.

## Builds on (most cited first, 30 of 41)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [The Exact Sample Complexity Gain from Invariances for Kernel Regression](https://mlsanity.com/paper/2303.14269/) (2023)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [Learning from many trajectories](https://mlsanity.com/paper/2203.17193/) (2022)
- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [Learning Layer-wise Equivariances Automatically using Gradients](https://mlsanity.com/paper/2310.06131/) (2023)
- [A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning](https://mlsanity.com/paper/2311.00212/) (2023)
- [Frame Averaging for Equivariant Shape Space Learning](https://mlsanity.com/paper/2112.01741/) (2021)
- [Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency](https://mlsanity.com/paper/2406.07598/) (2024)
- [Learning Infinitesimal Generators of Continuous Symmetries from Data](https://mlsanity.com/paper/2410.21853/) (2024)
- [Symmetry-Informed Governing Equation Discovery](https://mlsanity.com/paper/2405.16756/) (2024)
- [Symmetry Discovery Beyond Affine Transformations](https://mlsanity.com/paper/2406.03619/) (2024)
- [Sample Complexity Bounds for Estimating Probability Divergences under Invariances](https://mlsanity.com/paper/2311.02868/) (2023)
- [On the Ability of Deep Networks to Learn Symmetries from Data: A Neural Kernel Theory](https://mlsanity.com/paper/2412.11521/) (2024)
- [Symmetry Discovery for Different Data Types](https://mlsanity.com/paper/2410.09841/) (2024)
- [Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetry](https://mlsanity.com/paper/2512.11855/) (2025)
- [Learning with Exact Invariances in Polynomial Time](https://mlsanity.com/paper/2502.19758/) (2025)
- [Discovering Symbolic Differential Equations with Symmetry Invariants](https://mlsanity.com/paper/2505.12083/) (2025)
- [Explicit Discovery of Nonlinear Symmetries from Dynamic Data](https://mlsanity.com/paper/2510.01855/) (2025)
- [AtlasD: Automatic Local Symmetry Discovery](https://mlsanity.com/paper/2504.10777/) (2025)

## Built on by (most cited first, 3 of 3)

- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)
- [Sparse Data Augmentation for Optimization with Provable Guarantees](https://mlsanity.com/paper/2609.08133/) (2026)
- [Dimension-Free Scaling Laws for Size Generalization: An Exponential Separation](https://mlsanity.com/paper/2610.19279/) (2026)

## Abstract (arXiv)

Dynamical systems model trajectory data generated by fixed underlying dynamics, with applications ranging from biology to physics. Especially in scientific settings, dynamical systems are not generic but often exhibit symmetries imposed by physical laws, formalized through equivariance with respect to group actions. The identification problem concerns recovering the parameters of a system from observed trajectories. In this work, we study adaptive symmetry discovery for dynamical system identification and address how a system can be identified from a single trajectory when it is equivariant with respect to an unknown symmetry group. To this end, we first show that for known symmetries, the system can be identified from a significantly shorter single trajectory than in the generic setting, and we precisely characterize this improvement. We then consider the automatic symmetry discovery setting, proposing a method to learn the symmetry group directly from a single trajectory and incorporate it into the identification procedure, achieving the same optimal trajectory length as in the known-symmetry case. Our analysis relies on tools from group representation theory and the expander properties of Cayley graphs, and may be of independent interest for the study of symmetries in dynamical systems.