Wei Zhang's homepage

Summer semester 2025

Course at FUB: Mathematical strategies for complex stochastic dynamics

General information

Lectures

Location: T9/053 Seminarraum (Takustr. 9)

Schedule: Wednesday from 14:00-16:00

Class starts on: 2025-04-23

FU link: 19234501
Practice courses

Location: T9/046 Seminarraum (Takustr. 9)

Schedule: Friday from 10:00-12:00

Class starts on: 2025-04-25

FU link: 19234502

Contents

Basic of stochastic processes
- Langevin dynamics, Markov chains, generators, Fokker-Planck equation,
  convergence to equilibrium, Ito's formula
Model reduction for stochastic dynamics
- averaging technique, effective dynamics, Markov state modeling
Machine learning techniques for stochastic dynamics
- stochastic gradient descent, autoencoders, PDE eigenvalue problems by deep learning,
  diffusion models, continuous normalizing flow, flow-matching

Course materials

Date	Notes & Slides	Topics	Further reading
April 23, 2025	lecture_01, introduction slides	ODEs and Gaussian variables
April 30, 2025	lecture_02, slides_02	Ito integral, Ito's lemma, and Fokker-Planck equation	Ref. 1 and 2
May 07, 2025	1. lecture_03, slides_03 2. slides_03-DL	1. Feynman-Kac, invariant distribution, Brownian and Langevin dynamics 2. short introduction to ML/DL	Ref. 1 and 2 Ref. 4
May 14, 2025	lecture_04	eigenvalues, convergence to equilibrium, and Markov chains	Ref. 2
May 21, 2025	1. lecture_05 2. slides_05	1. Markov state models and variational principle 2. PyTorch tutorial	Ref. 5
May 28, 2025	lecture_06, slides_06	score-based diffusion models	Ref. 6
June 11, 2025	lecture_07, slides_07	Dirac delta function, conditional expectation, flow-based generative models	Ref. 7 and 8
June 18, 2025	lecture_08, slides_08	denoising diffusion probabilistic models, normalizing flows	Ref. 9 and 10
June 25, 2025	lecture_09, slides_09	k-means clustering, PCA, and autoencoders	Ref. 4
July 02, 2025	lecture_10, slides_10	transition path, minimal energy path, string method	Ref. 11

Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications. 5th. Springer, 2000. book
G.A. Pavliotis. Stochastic Processes and Applications: Diffusion Processes, the Fokker--Planck and Langevin Equations, Springer, 2014. book
G.A. Pavliotis and A.M. Stuart. Multiscale Methods: Averaging and Homogenization, Springer, 2008. book
Kevin P. Murphy. Probabilistic Machine Learning: An introduction. MIT Press, 2022. url: probml.ai. book
J.-H. Prinz et al. Markov models of molecular kinetics: Generation and validation. J. Chem. Phys. 134.17, 174105 (2011), p. 174105 doi
Yang Song et al. Score-Based Generative Modeling through Stochastic Differential Equations, ICLR 2021. arxiv Song's blog
Xingchao Liu, Chengyue Gong and Qiang Liu. Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow, ICLR 2023. arxiv
Michael S. Albergo and Eric Vanden-Eijnden. Building Normalizing Flows with Stochastic Interpolants, ICLR 2023. arxiv
Jonathan Ho, Ajay Jain and Pieter Abbeel. Denoising Diffusion Probabilistic Models, NeurIPS 2020. link
George Papamakarios et al. Normalizing Flows for Probabilistic Modeling and Inference, Journal of Machine Learning Research 22 (2021) 1-64. pdf
Weinan E, Weiqing Ren and Eric Vanden-Eijnden. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events, J. Chem. Phys. 126, 164103 (2007), doi