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Speed-Accuracy Relations for Diffusion Models: Wisdom from Nonequilibrium Thermodynamics and Optimal Transport

Kotaro Ikeda1,*, Tomoya Uda2, Daisuke Okanohara3, and Sosuke Ito4,5,†

  • 1Department of Mathematical Engineering and Information Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
  • 2Department of Earth and Planetary Physics, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
  • 3Preferred Networks Inc., 1-6-1, Otemachi, Chiyoda-ku, Tokyo 100-0004, Japan
  • 4Universal Biology Institute, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
  • 5Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-00033, Japan

  • *Contact author: kotaro-ikeda@g.ecc.u-tokyo.ac.jp
  • †Contact author: sosuke.ito@ubi.s.u-tokyo.ac.jp
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Phys. Rev. X 15, 031031 – Published 30 July, 2025

DOI: https://doi.org/10.1103/x5vj-8jq9

Abstract

We discuss a connection between a generative model, called the diffusion model, and nonequilibrium thermodynamics for the Fokker-Planck equation, called stochastic thermodynamics. Using techniques from stochastic thermodynamics, we derive the speed-accuracy relations for diffusion models, which are inequalities that relate the accuracy of data generation to the entropy production rate. These relations can be interpreted as the relations between accuracy and the speed of the diffusion dynamics in the absence of the nonconservative force. From a stochastic thermodynamic perspective, our results provide quantitative insight into how best to generate data in diffusion models. The optimal learning protocol is introduced by the geodesic of space of the 2-Wasserstein distance in optimal transport theory. We numerically illustrate the validity of the speed-accuracy relations for diffusion models with different noise schedules and different data. We numerically discuss our results for optimal and suboptimal learning protocols. We also demonstrate the applicability of our results to data generation from the real-world image datasets.

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Physics Subject Headings (PhySH)

Popular Summary

Diffusion models are a type of generative model that uses diffusion dynamics to generate data. These models are known for producing high-quality images, audio, and video. Although initially inspired by nonequilibrium thermodynamics, this analogy has not been fully exploited. In this study, we demonstrate that thermodynamic dissipation can limit the quality of data generated by diffusion models.

While optimal transport, closely related to nonequilibrium thermodynamics, has been discussed in the context of diffusion models, previous studies have overlooked the significance of thermodynamic concepts. We address this gap by organizing the relationship between diffusion models and nonequilibrium thermodynamics and by considering an analogy to thermodynamic trade-off relations. This approach reveals how dissipation constrains the quality of generated data and shows that optimal transport dynamics yields the most accurate data generation, while empirical methods of diffusion models remain suboptimal.

Our findings provide a theoretical foundation for selecting learning methods for diffusion models from the perspective of nonequilibrium thermodynamics. Thus, these results highlight the value of the analogy between nonequilibrium thermodynamics and diffusion models, paving the way for new research directions that apply nonequilibrium thermodynamics to the study of diffusion models.

Article Text

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Phys. Rev. X 15, 031031– Published 30 July, 2025
  • Received 21 July 2024
  • Accepted 4 June 2025
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