This seminar reviews the main concepts of stochastic differential equations (SDEs) in view of better understanding diffusion models, a state of the art family of deep generative models. This seminar takes place on the first half of the winter term. Participants can (voluntarily) attend to the follow-up seminar Score-based Generative Models for Machine Learning (Master Seminar), which take part on the second half of the winter term.

The topics covered in this seminar are broad. We intend not to dive into small details, but to handle instead the most important aspects in view on the aforementioned generative models. We start this seminar reviewing the most important results on differential equations, putting a special focus on the initial value problem. We review existence and uniqueness results as well as the numerical analysis for integration. We continue the seminar revisiting the basic mathematical notations and statistical concepts needed to introduce the Ito integrals. We derive the most important properties for the Ito calculus, as it is the Ito isometry. Having this setup, we derive the Ito formula and handle some examples and applications. We conclude this seminar studying the statistics of SDEs. We derive the Fokker-Planck-Kolmogorov equation, review the Markov and Martigale Properties of SDEs, and derive general equations for the moments of SDEs.

• Prerequisites: Basic knowledge in probability theory and statistics
• Registration: Via Müsli. Link
• First (organizational) meeting: Kalenderwoche 42. Specific day and time will be announced soon.
• Time and Location: Time and location will be announced soon.

Further information on the seminar will be announced in the first organizational meeting. For any specific question you can contact Daniel Gonzalez.

• Stochastic differential equations: an introduction with applications, Oksendal, Bernt, Springer Science & Business Media (2013)
• Neural ordinary differential equations, Chen, Ricky TQ and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David K NeurIPS (2018)
• Ffjord: Free-form continuous dynamics for scalable reversible generative models, Grathwohl, Will and Chen, Ricky TQ and Bettencourt, Jesse and Sutskever, Ilya and Duvenaud, David arXiv preprint (2018)
• Ffjord: Free-form continuous dynamics for scalable reversible generative models, Grathwohl, Will and Chen, Ricky TQ and Bettencourt, Jesse and Sutskever, Ilya and Duvenaud, David arXiv preprint (2018)
• Applied stochastic differential equations, Särkkä, Simo and Solin, Arno, Cambridge University Press (2019)