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

TL;DR

The paper examines how symmetry principles in physics, which organize dynamics and constrain degrees of freedom, can be diagnosed from data using representation-learning tools rather than being enforced by architecture. Focusing on variational autoencoders, it argues that the balance between reconstruction and compression reveals latent-space self-organization that mirrors symmetry-induced redundancy, offering a diagnostic for intrinsic dimensionality through a latent-relevance measure. Case studies from simple geometric constraints to particle-physics processes illustrate that exact symmetries produce sharp hierarchies in latent usage, while approximate or emergent symmetries yield graded structures, all without recovering explicit symmetry generators. The work emphasizes the diagnostic nature and practical value of latent-space analysis, while stressing limitations and the need to combine data-driven insights with traditional theory and effective field theory reasoning.

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.

Figures (9)

• Figure 1: Schematic view of the role of symmetries in data-driven representation learning. Physical symmetries induce constraints and redundancies in the data manifold, reducing the effective number of independent degrees of freedom. Representation-learning models trained to balance reconstruction and compression may reflect this structure through self-organization of their latent spaces.
• Figure 2: Illustration of symmetry as information reduction. A high-dimensional space of descriptions (top) contains redundancy due to symmetry transformations, which identify equivalence classes of physically identical configurations. Constraints and conservation laws restrict the data to a lower-dimensional manifold, while symmetry further quotients this manifold into physically inequivalent degrees of freedom (bottom). Figure generated with AI.
• Figure 3: Three paradigms for incorporating symmetry in machine learning. Left: Architectural symmetry, where invariance or equivariance is imposed by construction. Center: Implicit symmetry, introduced through data augmentation or self-supervised objectives. Right: Emergent structure, where symmetry-related organization may arise in latent representations without explicit enforcement, and is interpreted diagnostically. Figure generated with AI.
• Figure 4: Conceptual view of a variational autoencoder as a symmetry probe. Symmetry-induced constraints restrict the data to a lower-dimensional manifold. During training, the VAE balances reconstruction accuracy against compression toward a simple latent prior, leading to suppression of redundant latent directions and preferential use of symmetry- independent degrees of freedom.
• Figure 5: Top plot: Distribution of relevance (as defined in Eq.\ref{['eq:relevance']}) in the latent variables. In orange, the truly two-dimensional dataset $\mathcal{D}^{\text{2D}}$ and in blue the dataset constrained to a circle $\mathcal{D}^{\text{1D}}$. The latent variables are ordered by decreasing relevance. Bottom plot: Illustration of latent-space organization for data in 2D (left) and constrained to a circle (right). The dominant latent variable (denoted by $z_1$) provides a smooth coordinate along the symmetry orbit, while remaining latent directions are suppressed. Figures from Ref. Sanz:2025sld.