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

2025-05-12

AI Summary

Statistical, algorithmic, and dynamical complexity measures in complex systems science have long lacked a unified conceptual framework, and their computational realizability remains poorly understood. Method: We propose a three-dimensional conceptual space—“regularity–randomness–complexity”—to systematically integrate multi-paradigm complexity measures; develop a modeling paradigm grounded in a shared latent space, enabling regularity extraction, noise-robust modeling, and structural compression; and rigorously analyze data-driven methods—including autoencoders, latent dynamical models, symbolic regression, and physics-informed neural networks—in terms of their capacity to approximate classical complexity ideals. Contribution/Results: We establish the first cross-paradigm complexity taxonomy, characterize precise computability boundaries for these methods, and provide both a theoretical foundation and a scalable, latent-space-based modeling framework for physics-informed AI and AI-driven scientific discovery.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Metareasoning and Metaheuristics
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Organizing statistical, algorithmic, and dynamical complexity measures
• Addressing uncomputability challenges in complexity science
• Bridging theoretical complexity with data-driven modeling methods

Innovation

• Unified framework for statistical, algorithmic, dynamical measures
• Latent spaces for regularity extraction and noise management
• Data-driven methods as approximations to complexity ideals