# A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning

arXiv: https://arxiv.org/abs/2311.00212 · HTML: https://mlsanity.com/paper/2311.00212/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Nov 2023.

## Built on by (most cited first, 22 of 22)

- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [EdgeBench: Unveiling Scaling Laws of Learning from Real-World Environments](https://mlsanity.com/paper/2607.05155/) (2026)
- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [Learning (Approximately) Equivariant Networks via Constrained Optimization](https://mlsanity.com/paper/2505.13631/) (2025)
- [Latent Mixture of Symmetries for Sample-Efficient Dynamic Learning](https://mlsanity.com/paper/2510.03578/) (2025)
- [Explainable AI: Learning from the Learners](https://mlsanity.com/paper/2601.05525/) (2026)
- [Decoding complexity: how machine learning is redefining scientific discovery](https://mlsanity.com/paper/2405.04161/) (2024)
- [Discovering Symmetry Groups with Flow Matching](https://mlsanity.com/paper/2512.20043/) (2025)
- [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)
- [Learning Equivariant Functions via Quadratic Forms](https://mlsanity.com/paper/2509.22184/) (2025)
- [Approximate Equivariance via Projection-based Regularisation](https://mlsanity.com/paper/2601.05028/) (2026)
- [Least-Action-Guided Diffusion for Physical Extrapolation](https://mlsanity.com/paper/2606.11277/) (2026)
- [Data-driven discovery of self-similarity using neural networks](https://mlsanity.com/paper/2406.03896/) (2024)
- [Discover physical concepts and equations with machine learning](https://mlsanity.com/paper/2412.12161/) (2024)
- [RECON: Robust symmetry discovery via Explicit Canonical Orientation Normalization](https://mlsanity.com/paper/2505.13289/) (2025)
- [Think, Act, Learn: A Framework for Autonomous Robotic Agents using Closed-Loop Large Language Models](https://mlsanity.com/paper/2507.19854/) (2025)
- [Can AI Follow In Einstein's Footsteps?](https://mlsanity.com/paper/2607.27794/) (2026)
- [Cartan Networks: Group theoretical Hyperbolic Deep Learning](https://mlsanity.com/paper/2505.24353/) (2025)
- [QCNN with Rough Path Signature Kernels](https://mlsanity.com/paper/2607.07634/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)

## Abstract (arXiv)

Symmetry is present throughout nature and continues to play an increasingly central role in physics and machine learning. Fundamental symmetries, such as Poincar'e invariance, allow physical laws discovered in laboratories on Earth to be extrapolated to the farthest reaches of the universe. Symmetry is essential to achieving this extrapolatory power in machine learning applications. For example, translation invariance in image classification allows models with fewer parameters, such as convolutional neural networks, to be trained on smaller data sets and achieve state-of-the-art performance. In this paper, we provide a unifying theoretical and methodological framework for incorporating symmetry into machine learning models in three ways: 1. enforcing known symmetry when training a model; 2. discovering unknown symmetries of a given model or data set; and 3. promoting symmetry during training by learning a model that breaks symmetries within a user-specified group of candidates when there is sufficient evidence in the data. We show that these tasks can be cast within a common mathematical framework whose central object is the Lie derivative associated with fiber-linear Lie group actions on vector bundles. We extend and unify several existing results by showing that enforcing and discovering symmetry are linear-algebraic tasks that are dual with respect to the bilinear structure of the Lie derivative. We also propose a novel way to promote symmetry by introducing a class of convex regularization functions based on the Lie derivative and nuclear norm relaxation to penalize symmetry breaking during training of machine learning models. We explain how these ideas can be applied to a wide range of machine learning models including basis function regression, dynamical systems discovery, neural networks, and neural operators acting on fields.