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Organizing Committee
University of Oxford
UC Santa Barbara
Technical University of Munich
Johns Hopkins University
Johns Hopkins University
Brown University
Abstract
Systems of interacting particles or agents are studied across many scientific disciplines. They are used as effective models in a wide variety of sciences and applications, to represent the dynamics of particles in physics, cells in biology, people in urban mobility studies, but also, more abstractly in the context of mathematics, as sample particles in Monte Carlo simulations or parameters of neural networks in machine learning.
This workshop aims at bringing together researchers in analysis, computation, inference, control and applications, to facilitate cross-fertilization and collaborations.
This workshop aims at bringing together researchers in analysis, computation, inference, control and applications, to facilitate cross-fertilization and collaborations.
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Workshop Schedule
Monday, May 06, 2024
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
9:00 - 9:40 AM EDT
Interacting particle systems: a journey to kinetic theory and back
Tutorial - 11th Floor Lecture Hall
Speaker
Li Wang, University of Minnesota
Session Chair
Kavita Ramanan, Brown University
9:50 - 10:30 AM EDT
Optimal Control for Mean Field Games and Transition Paths
Tutorial - 11th Floor Lecture Hall
Speaker
Jian-Guo Liu, Duke University
Session Chair
Kavita Ramanan, Brown University
Abstract
Abstract: In this talk, I will present a stochastic optimal control formulation for (i) transition path problems in an infinite time horizon, specifically for Markov jump processes on Polish spaces, and (ii) mean field games in a finite time horizon. Transition paths connecting metastable states in a stochastic model are rare events that appear in many applications in science and engineering. An unbounded terminal cost at a stopping time, along with a controlled transition rate, regulates the transitions between metastable states. To maintain the original bridges after control, the running cost adopts an entropic form for the control velocity. However, the unbounded terminal cost leads to a singular optimal control and presents difficulties in the Girsanov transform. Gamma-convergence techniques and passing the limit in the corresponding Martingale problem allow us to obtain a singular optimally controlled transition rate. We demonstrate that the committor function, which solves a backward equation with specific boundary conditions, provides an explicit formula for the optimal path measure. The optimally controlled process realizes the transition paths almost surely but without altering the bridges of the original process. This stochastic optimal control formulation is also applicable to mean field games.
10:40 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:40 AM EDT
Optimization and sampling by linear kinetic equations
11th Floor Lecture Hall
Virtual Speaker
Lorenzo Pareschi, Heriot Watt University
Session Chair
Kavita Ramanan, Brown University
Abstract
One of the most prominent examples showcasing the close ties between global optimization processes and statistical physics is the simulated annealing method, rooted in the Metropolis-Hasting sampling approach. In this talk, we emphasize the close connections between the underlying stochastic dynamics and linear Boltzmann equations and how convergence to the global minimum can be related to classical entropy inequalities. This allows us to reinterpret classical results, such as the connections between simulated annealing and mean-field Langevin dynamics, and to derive enhanced algorithms based on different strategies to control the annealing process.
11:50 AM - 12:30 PM EDT
Wasserstein gradient flow in an inhomogeneous media: convergence and the effective Wasserstein metric
11th Floor Lecture Hall
Speaker
Yuan Gao, Purdue University
Session Chair
Kavita Ramanan, Brown University
Abstract
The Fokker-Planck equation with fast oscillated coefficients can be regarded as a gradient flow in a Wasserstein space with inhomogeneous dissipation and oscillated free energy. We will use an evolutionary Gamma convergence approach to obtain the homogenized dynamics, which preserves the gradient flow structure in a limiting homogenized Wasserstein space. The comparison between the gradient flow induced limiting Wasserstein distance and the direct Gromov-Hausdorff limiting Wasserstein distance will also be discussed.
12:40 - 2:00 PM EDT
Networking Lunch
Working Lunch
2:00 - 2:40 PM EDT
Quantitative Propagation of Chaos for 2D Viscous Vortex Model on the Whole Space
11th Floor Lecture Hall
Speaker
Zhenfu Wang, Peking University
Session Chair
Jose Carrillo, University of Oxford
Abstract
We derive the quantitative estimates of propagation of chaos for the large interacting particle systems in terms of the relative entropy between the joint law of the particles and the tensorized law of the mean field PDE. We resolve this problem for the first time for the viscous vortex model that approximating 2D Navier-Stokes equation in the vorticity formulation on the whole space. We obtain as key tools the Li-Yau-type estimates and Hamilton-type heat kernel estimates for 2D Navier-Stokes on the whole space. This is based on a joint work with Xuanrui Feng from Peking University.
2:50 - 3:10 PM EDT
Coffee Break
11th Floor Collaborative Space
3:10 - 3:50 PM EDT
Opinion formation in social networks
11th Floor Lecture Hall
Speaker
Oliver Tse, Eindhoven University of Technology
Session Chair
Jose Carrillo, University of Oxford
4:00 - 4:40 PM EDT
Opinion formation in social network
11th Floor Lecture Hall
Speaker
Marie-Therese Wolfram, Emory University
Session Chair
Jose Carrillo, University of Oxford
Abstract
In this talk I will discuss an ODE model for opinion formation in evolving networks. As opposed to existing models, in which the network typically evolves by discretely adding or removing edges, we instead propose a model for opinion formation which is coupled to a network evolving through a system of ordinary differential equations for the edge weights. We interpret each edge weight as the strength of the relationship between a pair of individuals, with edges increasing in weight if pairs continually listen to each others' opinions and decreasing if not. We investigate the impact of various edge dynamics at different timescales on the opinion dynamic itself. Time permitting, I will also discuss a very recent work on how to steer opinions towards a target opinion by controlling the strength of network connections.
4:50 - 6:20 PM EDT
Reception
11th Floor Collaborative Space