Statistical Physics of Learning and AI: Disorder, Free Energy, Phase Transitions, and Neural Scaling (Mathematics for Artificial Intelligence)

Why do modern learning systems exhibit sharp transitions, multiple solution regimes, scaling laws, and collective behavior that cannot be understood from parameter-by-parameter analysis alone? Statistical Physics of Learning and AI develops a unified framework for answering these questions through the language of disorder, free energy, phase transitions, mean-field theory, and high-dimensional dynamics.

Written as a graduate-level text for students and researchers in artificial intelligence, applied mathematics, statistics, and statistical physics, the book builds the essential ideas from first principles and connects them to contemporary learning systems. It explains how quenched disorder, energy and loss landscapes, replica and cavity reasoning, message passing, state evolution, finite-size effects, and dynamical mean-field methods illuminate high-dimensional inference and learning. Classical models such as perceptrons, teacher-student systems, spin glasses, and associative memories are used not as historical endpoints, but as conceptual laboratories for understanding modern phenomena including double descent, feature learning, stochastic-gradient dynamics, neural scaling laws, attention, and in-context learning.

A central theme is the passage from microscopic mechanisms to macroscopic behavior. Throughout, mathematical structure is tied to the actual AI development pipeline—from initialization and pretraining to optimization, representation formation, scaling, and emergent behavior—so that statistical-physics concepts become practical tools for reasoning about large learning systems. The result is both a rigorous introduction to the field and a bridge between classical statistical mechanics and the mathematics of modern AI.