Math757: a topic course in stochastic dynamical systems

Interacting particle systems
dynamics, inference and applications in machine learning

Instructor:     Fei Lu
Class meets:    TTh, 10:30-11:45, Shriver Hall 104
Office Hours: TTh 9:30--10:30,    Krieger 301 
Email:             feilu##   ( ## =

This topic course will run in a reading/discussion/projects fashion. References are attached to each topic. In addition, the following books may be helpful (To be updated).

Course plan (tentative): This topic course will explore systems of interacting particles or agents: their dynamics, inference, and applications in machine learning. For the study of the dynamics, topics include stability and ergodicity of the systems, including first-order gradient systems, 2nd-order Hamiltonian systems and related mean field equations and propagation of chaos. For the applications, topics include optimization algorithms such as stochastic gradient decent (SGD) and particle SGD, and sampling methods using particle systems such as Stein variational gradient decent. For the inference, we will consider the estimation of the interaction kernels as well as state estimation using data assimilation techniques. If time permits applications in network and control with be explored. This topic course will run in a reading/discussion/projects fashion.

Prerequisite: probability; differential equations (preferably stochastic differential equations).

Grading: Grade will be based on project assignments and presentations. There is no exam.
Tentative schedule (will be updated weekly):
week Topics References Other
1/28, 1/30 Course plan overview
Review ODE (Caratheodory DE)
non-Lipschitz / discontinuous RHS; existence, uniqueness, global solution
Homework 1   tex file
Due on 2/11; solution
2/4, 2/6 Consensus and flocking Dynamics
Motsch+Tadmor14, Heterophilious dynamics enhances concensus
Cucker+Smale07, Emergent behavior in flocks

2/11, 2/13 Consensus and flocking Dynamics
2/18, 2/20 Mean field
Bertozzi et al: Blow-up in multidimensional aggregation equations with mildly singular interaction kernels
Reading: Simione et al: Existence of ground state of nonlocal-interaction energies
Ambrosio+Bernard: Uniqueness of signed measures solving the continuity equation for Osgood vector fields
2/25, 2/27 stochastic systems: 1st order
basics LectureNotes4
3/3, 3/5 stochastic systems: 1st order
Malrieu03: Convergence to equilibrium for granular media equations and their Euler schemes
Toscani+Villani00: On the Trend to Equilibrium for Some Dissipative Systems with Slowly Increasing a Priori Bounds
Jabin+Wang17: mean field limit for stochastic particle systems
Homework2   tex file
Due on 3/24 Extended to 3/31
3/10, 3/12 stochastic systems: 1st order
Implicit Euler LectureNotes6
3/17, 3/19 spring break
3/24, 3/26 stochastic systems: 1st order
  2nd order
Implicit Euler (Malrieu03) LectureNotes6
Further reading: Guionnet+Zegarlinski02: lectures on LSI
Bolley+Gentil+Guillin13: Uniform convergence to Equilibrium for Granular media
Bakry04 Functional Inequalities for Markov semigroups
Linear 2nd-order models LectureNote Record

3/31, 4/2 stochastic systems: 2nd order stochastic Cucker-Smale model: LectureNote0331    LectureNote0402
Pedeches18: Asymptotic properties of various stochastic Cucker-Smale dynamics
Cattiaux+Delebecque+Pedeches18: stochastic Cucker-Smale models: old and new

4/7, 4/9 Exponential Ergodicity of SDEs Mattingly+Stuart+Higham02SPA
4/14, 4/16 Inference of the interaction function basics of non-parametric regression:
(CS01) Cucker-Smale01: On the mathematical foundations of learning
Chp10 of (GKKW02) Gyorfi+Kohler+Krzyzak+Walk: A distribution-free theory of nonparametric regression

4/21, 4/23 Inference of the interaction function
deterministic and stochastic systems LecNote
4/28, 4/30 Inference of the interaction function:
stochastic systems

Likelihood ratio: Chp1.1.1.4 Kutoyants04 Statistic inference for ergodic diffusion processes ;
Chapter 3.5 of Karatzas+Shreve: Brownian motion and stochastic calculus
Concentration for Martingales:
Gene+Nickl: Math Foundation of infinite-dimensional statistical models;
Lemma 1 of Nickl+Ray1810