Mathematical strategies for complex stochastic dynamics
Wei Zhang
Additional information / Pre-requisites
Prerequisites: Basic knowledge of stochastics, and numerical methods
Comments
Content:
Stochastic dynamics are widely studied in scientific fields such as physics, biology, and climate. Understanding these dynamics is often challenging due to their high dimensionality and multiscale characteristics. This lecture provides an introduction to theoretical and numerical techniques, including machine learning techniques, for studying such complex stochastic dynamics. The following topics will be covered:
- Basic of stochastic processes:
Langevin dynamics, overdamped Langevin dynamics, Markov chains, generators and Fokker-Planck equation, convergence to equilibrium, Ito’s formula
- Model reduction techniques for stochastic dynamics:
averaging technique, effective dynamics, Markov state modeling
- Machine learning techniques using/for stochastic dynamics:
dynamics of stochastic gradient descent, autoencoders, solving eigenvalue problems by deep learning, generative modeling using diffusion models, continuous normalizing flow, or flow-matching
closeSuggested reading
1) Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications. 5th. Springer, 2000
2) Kevin P. Murphy. Probabilistic Machine Learning: An introduction. MIT Press, 2022. url: probml.ai
3) J.-H. Prinz et al. “Markov models of molecular kinetics: Generation and validation”. In: J. Chem. Phys. 134.17, 174105 (2011), p. 174105
4) W. Zhang, C. Hartmann, and C. Schütte. “Effective dynamics along given reaction coordinates and reaction rate theory”. In: Faraday Discuss. 195 (2016), pp. 365–394
5) Mardt, A., Pasquali, L., Wu, H. et al. VAMPnets for deep learning of molecular kinetics. Nat Commun 9, 5 (2018).
6) Score-Based Generative Modeling through Stochastic Differential Equations, Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, Ben Poole, ICLR 2021.
close13 Class schedule
Regular appointments