# Critically-Damped Higher-Order Langevin Dynamics for Generative Modeling

arXiv: https://arxiv.org/abs/2506.21741 · HTML: https://mlsanity.com/paper/2506.21741/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 15 of its 22 references that are in the map.
- First posted on arXiv 26 Jun 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 15 of 15)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Wavelet Score-Based Generative Modeling](https://mlsanity.com/paper/2208.05003/) (2022)
- [Normalizing flow neural networks by JKO scheme](https://mlsanity.com/paper/2212.14424/) (2022)
- [Where to Diffuse, How to Diffuse, and How to Get Back: Automated Learning for Multivariate Diffusions](https://mlsanity.com/paper/2302.07261/) (2023)
- [Underdamped Diffusion Bridges with Applications to Sampling](https://mlsanity.com/paper/2503.01006/) (2025)
- [Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups](https://mlsanity.com/paper/2405.16381/) (2024)
- [What's the score? Automated Denoising Score Matching for Nonlinear Diffusions](https://mlsanity.com/paper/2407.07998/) (2024)
- [Generative Modelling with High-Order Langevin Dynamics](https://mlsanity.com/paper/2404.12814/) (2024)
- [Critically Damped Third-Order Langevin Dynamics](https://mlsanity.com/paper/2409.07697/) (2024)

## Built on by (most cited first, 3 of 3)

- [Defending Diffusion Models Against Membership Inference Attacks via Higher-Order Langevin Dynamics](https://mlsanity.com/paper/2509.14225/) (2025)
- [Contact Wasserstein Geodesics for Non-Conservative Schr"odinger Bridges](https://mlsanity.com/paper/2511.06856/) (2025)
- [Reducing Diffusion Model Memorization with Higher Order Langevin Dynamics](https://mlsanity.com/paper/2605.19170/) (2026)

## Abstract (arXiv)

Denoising diffusion probabilistic models (DDPMs) represent an entirely new class of generative AI methods that have yet to be fully explored. They use Langevin dynamics, represented as stochastic differential equations, to describe a process that transforms data into noise, the forward process, and a process that transforms noise into generated data, the reverse process. Many of these methods utilize auxiliary variables that formulate the data as a ``position" variable, and the auxiliary variables are referred to as ``velocity", ``acceleration", etc. In this sense, it is possible to ``critically damp" the dynamics. Critical damping has been successfully introduced in Critically-Damped Langevin Dynamics (CLD) and Critically-Damped Third-Order Langevin Dynamics (TOLD++), but has not yet been applied to dynamics of arbitrary order. The proposed methodology generalizes Higher-Order Langevin Dynamics (HOLD), a recent state-of-the-art diffusion method, by introducing the concept of critical damping from systems analysis. Similarly to TOLD++, this work proposes an optimal set of hyperparameters in the n-dimensional case, where HOLD leaves these to be user defined. Additionally, this work provides closed-form solutions for the mean and covariance of the forward process that greatly simplify its implementation. Experiments are performed on the CIFAR-10 and CelebA-HQ 256 × 256 datasets, and validated against the FID metric.