{"ID":5438851,"CreatedAt":"2026-07-01T01:17:58.482524686Z","UpdatedAt":"2026-07-03T12:27:39.719939621Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.31576","arxiv_id":"2606.31576","title":"Introduction to Stochastic Differential Equations for Generative Machine Learning: A Variational Perspective","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.","short_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 f...","url_abs":"https://arxiv.org/abs/2606.31576","url_pdf":"https://arxiv.org/pdf/2606.31576v1","authors":"[\"Ole Winther\",\"Paul Jeha\",\"Sander Dieleman\",\"Andriy Mnih\",\"Manfred Opper\",\"Andrea Dittadi\"]","published":"2026-06-30T12:34:16Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[\"Diffusion Model\"]","has_code":false}