Nonequilbrium physics of generative diffusion models

TL;DR

This paper reframes generative diffusion models (GDMs) as a nonequilibrium physics problem, treating forward diffusion as an Ornstein–Uhlenbeck Langevin process and the reverse generative step as a statistical-inference-driven dynamics with the score function guiding denoising. Using a path-integral perspective, it derives fluctuation theorems and entropy production for both forward and reverse processes, and introduces a potential/free-energy framework to characterize phase transitions in the reverse dynamics, including a speciation transition and a glass-like fragmentation analyzed via the Franz–Parisi potential. The work provides analytic results in a Gaussian-mixture data setting, revealing that reverse diffusion can be viewed as minimizing a generalized free energy and that the dynamic-state variable acts as quenched disorder akin to spin-glass systems. By linking stochastic thermodynamics, statistical inference, and geometry-based methods, the paper offers a coherent theoretical picture of GDMs with implications for understanding sampling dynamics and guiding the design of diffusion-based generative models.

Abstract

Generative diffusion models apply the concept of Langevin dynamics in physics to machine leaning, attracting a lot of interests from engineering, statistics and physics, but a complete picture about inherent mechanisms is still lacking. In this paper, we provide a transparent physics analysis of diffusion models, formulating the fluctuation theorem, entropy production, equilibrium measure, and Franz-Parisi potential to understand the dynamic process and intrinsic phase transitions. Our analysis is rooted in a path integral representation of both forward and backward dynamics, and in treating the reverse diffusion generative process as a statistical inference, where the time-dependent state variables serve as quenched disorder akin to that in spin glass theory. Our study thus links stochastic thermodynamics, statistical inference and geometry based analysis together to yield a coherent picture about how the generative diffusion models work.

Figures (11)

• Figure 1: A schematic illustration of the generative diffusion process of two-dimensional Gaussian mixture data. The forward process from time $t = 0$ to $t = 3$ is shown together with the reverse process from $t = 3$ back to $t = 0$. The gradient of log-state-likelihood can be analytically estimated as the state probability is given by $p({X_t},t) = \frac{1}{2}\mathcal{N}\left( {\boldsymbol{\mu}{e^{ - t}},\mathbf{I}_d} \right) + \frac{1}{2}\mathcal{N}\left( { - \boldsymbol{\mu}{e^{ - t}},\mathbf{I}_d} \right)$ .
• Figure 2: Ensemble average of $e^{ - \Delta {S_{tot}}}$ as a function of the number of trajectories used to do the average, simulated by solving the forward OU process. $\mu = 1$, and other parameters are detailed in Appendix \ref{['app-a']}.
• Figure 3: Normalized histogram of entropy changes estimated from $10\,000$ trajectories for the forward OU process. $\mu = 1$, and other parameters are detailed in Appendix \ref{['app-a']}. (a) Statistics of system entropy change. (b) Statistics of environment entropy change. (c) Statistics of the total entropy change. (d) Statistics of $e^{-\Delta S_{\text{tot }}}$.
• Figure 4: Evolution of potential energy ($\mu = 1$) in one dimension. The speciation transition time is given by $t_S \approx 0.35$.
• Figure 5: Phase diagram of symmetry breaking in one-dimensional example of arbitrary data mean $\mu$ and variance $\sigma^2$. The concrete examples of three phases are shown in Fig. \ref{['Typ']}.