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Imperfect score-matching leads to a shift between the training and the sampling distribution of diffusion models. Due to the recursive nature of the generation process, errors in previous steps yield sampling iterates that drift away from…

Machine Learning · Computer Science 2023-02-20 Giannis Daras , Yuval Dagan , Alexandros G. Dimakis , Constantinos Daskalakis

Score-based modeling through stochastic differential equations (SDEs) has provided a new perspective on diffusion models, and demonstrated superior performance on continuous data. However, the gradient of the log-likelihood function, i.e.,…

Machine Learning · Computer Science 2023-03-07 Haoran Sun , Lijun Yu , Bo Dai , Dale Schuurmans , Hanjun Dai

Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time…

Machine Learning · Statistics 2026-04-28 Ludovico T. Giorgini

Denoising diffusion models are a popular class of generative models providing state-of-the-art results in many domains. One adds gradually noise to data using a diffusion to transform the data distribution into a Gaussian distribution.…

Machine Learning · Computer Science 2023-08-21 Francisco Vargas , Will Grathwohl , Arnaud Doucet

We introduce a scheme for deriving an optimally-parametrised Langevin dynamics of few collective variables from data generated in molecular dynamics simulations. The drift and the position-dependent diffusion profiles governing the Langevin…

Statistical Mechanics · Physics 2008-08-22 Cristian Micheletti , Giovanni Bussi , Alessandro Laio

This book presents the core principles that have guided the development of diffusion models, tracing their origins and showing how diverse formulations arise from shared mathematical ideas. Diffusion modeling starts by defining a forward…

Machine Learning · Computer Science 2026-05-28 Chieh-Hsin Lai , Yang Song , Dongjun Kim , Yuki Mitsufuji , Stefano Ermon

Reversing a diffusion process by learning its score forms the heart of diffusion-based generative modeling and for estimating properties of scientific systems. The diffusion processes that are tractable center on linear processes with a…

Machine Learning · Computer Science 2024-07-12 Raghav Singhal , Mark Goldstein , Rajesh Ranganath

Denoising diffusion models (DDMs) have recently attracted increasing attention by showing impressive synthesis quality. DDMs are built on a diffusion process that pushes data to the noise distribution and the models learn to denoise. In…

Machine Learning · Computer Science 2023-05-16 Jaemoo Choi , Yesom Park , Myungjoo Kang

Score-based diffusion models are a powerful class of generative models, but their practical use often depends on training neural networks to approximate the score function. Training-free diffusion models provide an attractive alternative by…

Numerical Analysis · Mathematics 2026-01-28 Pengjun Wang , Zezhong Zhang , Minglei Yang , Feng Bao , Yanzhao Cao , Guannan Zhang

Diffusion models are a popular class of generative models trained to reverse a noising process starting from a target data distribution. Training a diffusion model consists of learning how to denoise noisy samples at different noise levels.…

Machine Learning · Computer Science 2025-10-07 Ameya Daigavane , YuQing Xie , Bodhi P. Vani , Saeed Saremi , Joseph Kleinhenz , Tess Smidt

We present a novel method for training score-based generative models which uses nonlinear noising dynamics to improve learning of structured distributions. Generalizing to a nonlinear drift allows for additional structure to be incorporated…

Machine Learning · Statistics 2025-07-10 Jeremiah Birrell , Markos A. Katsoulakis , Luc Rey-Bellet , Benjamin J. Zhang , Wei Zhu

In this work, we aimed to replicate and extend the results presented in the DiffFluid paper[1]. The DiffFluid model showed that diffusion models combined with Transformers are capable of predicting fluid dynamics. It uses a denoising…

Fluid Dynamics · Physics 2025-07-14 Yannick Gachnang , Vismay Churiwala

Diffusion models, though originally designed for generative tasks, have demonstrated impressive self-supervised representation learning capabilities. A particularly intriguing phenomenon in these models is the emergence of unimodal…

Machine Learning · Computer Science 2026-02-04 Xiao Li , Zekai Zhang , Xiang Li , Siyi Chen , Zhihui Zhu , Peng Wang , Qing Qu

SDE-based methods such as denoising diffusion probabilistic models (DDPMs) have shown remarkable success in real-world sample generation tasks. Prior analyses of DDPMs have been focused on the exponential Euler discretization, showing…

Machine Learning · Computer Science 2025-11-10 Matthew S. Zhang , Stephen Huan , Jerry Huang , Nicholas M. Boffi , Sitan Chen , Sinho Chewi

Denoising diffusion probabilistic models (DDPMs) have achieved high quality image generation without adversarial training, yet they require simulating a Markov chain for many steps to produce a sample. To accelerate sampling, we present…

Machine Learning · Computer Science 2022-10-07 Jiaming Song , Chenlin Meng , Stefano Ermon

Diffusion models (DMs) are a powerful generative framework that have attracted significant attention in recent years. However, the high computational cost of training DMs limits their practical applications. In this paper, we start with a…

Machine Learning · Computer Science 2024-04-12 Tianshuo Xu , Peng Mi , Ruilin Wang , Yingcong Chen

Many stochastic processes in the physical and biological sciences can be modelled as Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the…

Numerical Analysis · Mathematics 2024-04-22 Dominic Phillips , Charles Matthews , Benedict Leimkuhler

Diffusion models have shown incredible capabilities as generative models; indeed, they power the current state-of-the-art models on text-conditioned image generation such as Imagen and DALL-E 2. In this work we review, demystify, and unify…

Machine Learning · Computer Science 2022-08-26 Calvin Luo

Common algorithms for computationally simulating Langevin dynamics must discretize the stochastic differential equations of motion. These resulting finite time step integrators necessarily have several practical issues in common:…

Statistical Mechanics · Physics 2013-01-31 David A. Sivak , John D. Chodera , Gavin E. Crooks

Langevin Dynamics is a Stochastic Differential Equation (SDE) central to sampling and generative modeling and is implemented via time discretization. Langevin Monte Carlo (LMC), based on the Euler-Maruyama discretization, is the simplest…

Machine Learning · Computer Science 2025-10-10 Saravanan Kandasamy , Dheeraj Nagaraj