Introduction to Stochastic Differential Equations for Generative Machine Learning: A Variational Perspective

• URL: http://arxiv.org/abs/2606.31576v1
• Date: Tue, 30 Jun 2026 12:34:16 GMT
• Title: Introduction to Stochastic Differential Equations for Generative Machine Learning: A Variational Perspective
• Abstract summary: The use of ordinary and probabilistic differential equations has led to substantial progress in generative machine learning.<n>This paper provides a self-contained and informal introduction to the differential equations and the framework for using them in generative modeling.
• Score: 17.754079500838436
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The use of ordinary and stochastic differential equations has led to substantial progress in generative machine learning with applications to, for example, image, video and biomolecule generation. This paper provides a self-contained and informal introduction to the differential equations, the probabilistic framework for using them in generative modeling and the Fokker--Planck equation that governs the temporal evolution of the marginal distribution of the stochastic variables of the differential equations. The variational lower bound on the log-likelihood (the evidence lower bound, ELBO) is derived and used as a general starting point for a discussion of diffusion models, score matching, and flow matching. All of these approaches may be viewed as specific parameterizations of the most general variational approach. A one-dimensional density modeling problem is used as a simple example to compare different parameterizations.

Related papers

• A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models [15.927114890047035]
  Diffusion models have emerged as a dominant framework for generative modeling.<n>This tutorial provides a unified and self-contained account of these viewpoints from the perspective of differential equations.<n>The tutorial is intended as a bridge between the analytical foundations of diffusion processes and the modern generative algorithms built upon them.
  arXiv  Detail & Related papers  (2026-05-21T14:59:12Z)
• Unifying and extending Diffusion Models through PDEs for solving Inverse Problems [3.1225172236361165]
  Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML)<n>Traditionally, these models have been derived using principles of variational inference, denoising, statistical signal processing, and differential equations.<n>In this study we derive diffusion models using ideas from linear partial differential equations and demonstrate that this approach has several benefits.
  arXiv  Detail & Related papers  (2025-04-10T04:07:36Z)
• Principled model selection for stochastic dynamics [0.0]
  We show that overfitting arises not just from individual model complexity but also from the growth of possible models.<n>We introduce Parsimonious Inference (PASTIS), a principled method combining likelihood-estimation with extreme value theory to suppress superfluous parameters.
  arXiv  Detail & Related papers  (2025-01-17T18:23:16Z)
• On the Trajectory Regularity of ODE-based Diffusion Sampling [79.17334230868693]
  Diffusion-based generative models use differential equations to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models.
  arXiv  Detail & Related papers  (2024-05-18T15:59:41Z)
• Scaling and renormalization in high-dimensional regression [72.59731158970894]
  We present a unifying perspective on recent results on ridge regression.<n>We use the basic tools of random matrix theory and free probability, aimed at readers with backgrounds in physics and deep learning.<n>Our results extend and provide a unifying perspective on earlier models of scaling laws.
  arXiv  Detail & Related papers  (2024-05-01T15:59:00Z)
• Gaussian Mixture Solvers for Diffusion Models [84.83349474361204]
  We introduce a novel class of SDE-based solvers called GMS for diffusion models. Our solver outperforms numerous SDE-based solvers in terms of sample quality in image generation and stroke-based synthesis.
  arXiv  Detail & Related papers  (2023-11-02T02:05:38Z)
• Causal Modeling with Stationary Diffusions [89.94899196106223]
  We learn differential equations whose stationary densities model a system's behavior under interventions. We show that they generalize to unseen interventions on their variables, often better than classical approaches. Our inference method is based on a new theoretical result that expresses a stationarity condition on the diffusion's generator in a reproducing kernel Hilbert space.
  arXiv  Detail & Related papers  (2023-10-26T14:01:17Z)
• A Geometric Perspective on Diffusion Models [57.27857591493788]
  We inspect the ODE-based sampling of a popular variance-exploding SDE. We establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm.
  arXiv  Detail & Related papers  (2023-05-31T15:33:16Z)
• Stochastic Differential Equations with Variational Wishart Diffusions [18.590352916158093]
  We present a non-parametric way of inferring differential equations for both regression tasks and continuous-time dynamical modelling. The work has high emphasis on the part of the differential equation, also known as the diffusion, and modelling it by means of Wishart processes.
  arXiv  Detail & Related papers  (2020-06-26T10:21:35Z)
• Neural Controlled Differential Equations for Irregular Time Series [17.338923885534197]
  An ordinary differential equation is determined by its initial condition, and there is no mechanism for adjusting the trajectory based on subsequent observations. Here we demonstrate how this may be resolved through the well-understood mathematics of emphcontrolled differential equations We show that our model achieves state-of-the-art performance against similar (ODE or RNN based) models in empirical studies on a range of datasets.
  arXiv  Detail & Related papers  (2020-05-18T17:52:21Z)
• Stochastic Normalizing Flows [52.92110730286403]
  We introduce normalizing flows for maximum likelihood estimation and variational inference (VI) using differential equations (SDEs) Using the theory of rough paths, the underlying Brownian motion is treated as a latent variable and approximated, enabling efficient training of neural SDEs. These SDEs can be used for constructing efficient chains to sample from the underlying distribution of a given dataset.
  arXiv  Detail & Related papers  (2020-02-21T20:47:55Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.