Artificial Intelligence and Symmetries: Learning, Encoding, and Discovering Structure in Physical Data

TL;DR

This paper explores how machine learning, especially variational autoencoders, can identify and encode symmetries in physical data, revealing underlying structures and conservation laws without explicit symmetry enforcement.

Contribution

It reviews recent advances in data-driven symmetry discovery in physical systems using representation learning, highlighting theoretical and practical challenges.

Findings

Symmetries reduce the intrinsic dimensionality of physical datasets.

Latent spaces in generative models can self-organize to reflect symmetries.

Limitations exist in inferring symmetry structures without explicit biases.

Abstract

Symmetries play a central role in physics, organizing dynamics, constraining interactions, and determining the effective number of physical degrees of freedom. In parallel, modern artificial intelligence methods have demonstrated a remarkable ability to extract low-dimensional structure from high-dimensional data through representation learning. This review examines the interplay between these two perspectives, focusing on the extent to which symmetry-induced constraints can be identified, encoded, or diagnosed using machine learning techniques. Rather than emphasizing architectures that enforce known symmetries by construction, we concentrate on data-driven approaches and latent representation learning, with particular attention to variational autoencoders. We discuss how symmetries and conservation laws reduce the intrinsic dimensionality of physical datasets, and how this reduction may manifest itself through self-organization of latent spaces in generative models trained to balance reconstruction and compression. We review recent results, including case studies from simple geometric systems and particle physics processes, and analyze the theoretical and practical limitations of inferring symmetry structure without explicit inductive bias.