Stochastic Thermodynamics of Score Matching in Diffusion Models

Abstract

Score-based diffusion models are a powerful class of generative AI systems capable of sampling from complex, high-dimensional probability distributions. Their dynamics consist of a forward diffusion process that transforms data into noise and a learned reverse process that reconstructs data by reversing the probability flow. Here, we develop a stochastic thermodynamic framework for diffusion models and their score-matching objective. We introduce a trajectory-dependent quantity, time-asymmetry entropy production (TAEP), defined from the forward and reverse diffusion dynamics, and show that it obeys exact fluctuation theorems. Remarkably, Hyv\"{a}rinen's implicit score-matching kernel emerges naturally as a fluctuating component of TAEP, while the average TAEP is exactly proportional to the score-matching objective. We further show that fluctuations of TAEP quantify sampling unevenness and provide a thermodynamic measure of data-manifold coverage. These results yield a quantitative explanation for the superior sampling diversity of diffusion models and reveal a thermodynamic mechanism by which stochastic gradient descent favors flatter, more generalizable solutions. By uncovering the entropic nature of score matching, our work establishes fundamental statistical-mechanical principles underlying diffusion-based generative AI.