Compression, Regularity, Randomness and Emergent Structure: Rethinking Physical Complexity in the Data-Driven Era

TL;DR

This paper proposes a unified framework for understanding various measures of complexity, integrating statistical, algorithmic, and dynamical approaches, and highlights modern data-driven methods as practical tools for complexity analysis in high-dimensional systems.

Contribution

It introduces a conceptual space organizing complexity measures and discusses how modern data-driven techniques approximate classical complexity concepts.

Findings

Unified framework for complexity measures along regularity, randomness, and complexity axes

Mapping of statistical, algorithmic, and dynamical measures into a common space

Modern data-driven methods serve as pragmatic approximations to classical complexity ideals

Abstract

Complexity science offers a wide range of measures for quantifying unpredictability, structure, and information. Yet, a systematic conceptual organization of these measures is still missing. We present a unified framework that locates statistical, algorithmic, and dynamical measures along three axes (regularity, randomness, and complexity) and situates them in a common conceptual space. We map statistical, algorithmic, and dynamical measures into this conceptual space, discussing their computational accessibility and approximability. This taxonomy reveals the deep challenges posed by uncomputability and highlights the emergence of modern data-driven methods (including autoencoders, latent dynamical models, symbolic regression, and physics-informed neural networks) as pragmatic approximations to classical complexity ideals. Latent spaces emerge as operational arenas where regularity extraction, noise management, and structured compression converge, bridging theoretical foundations with practical modeling in high-dimensional systems. We close by outlining implications for physics-informed AI and AI-guided discovery in complex physical systems, arguing that classical questions of complexity remain central to next-generation scientific modeling.