Interacting particle systems are used in many contexts, for modeling purposes (collective behaviour, mean field games, statistical physics) or in numerical methods (optimization, sampling methods, simulation of rare events). The objective of this workshop is to gather mathematicians working on different aspects of these subjects: algorithms, models and mathematical analysis.
Monday, 09 December
10:30 – 11:00 Welcome and Coffee/Tea/Pastries
11:00 – 12:00 Dr Anastasia Borovykh
12:00 – 13:30 Lunch
13:30 – 14:30 Dr Geneviève Dusson
14:30 – 15:30 Dr Andrew Duncan
15:30 – 16:00 Coffee/Tea break
16:00 – 17:00 Professor Johannes Zimmer
Workshop dinner from 19:30 onwards (by invitation only)
Tuesday, 10 December
09:30 – 10:00 Coffee/Tea//Pastries
10:00 – 11:00 Dr Noé Cuneo
11:00 – 12:00 Professor Julien Barré
12:00 – 13:30 Working lunch
12:15 – 13:15 Dr Charles-Edouard Bréhier
13:30 – 14:30 Dr Pierre Monmarché
14:30 – 15:30 Professor Pierre Degond
15:30 – 16:00 Coffee/Tea and Close
Book of Abstracts
Title: Understanding the output evolution of a deep neural network during training
Abstract: In this talk we will gain insight into the effects of the optimization method and its hyperparameters on the neural network output evolution during training. We will start with stochastic gradient Langevin dynamics (SGLD). We first discuss a previously obtained result which shows that under specific assumptions a deep neural network becomes equivalent to a linear model. In this case one can explicitly solve for the network output during training, and the effects of the training hyperparameters can be studied. For general deep networks the linear approximation is no longer sufficient and higher order approximations are required. Obtaining explicit expressions in this case if however no longer trivial. We present a Taylor-expansion based method to solve for higher-order approximations of the network output during training, and also in this case study the effects of the hyperparameters of the optimization algorithm on the network output and generalization capabilities. We then extend the standard SGLD algorithm to a system of interacting SGLD particles. We will discuss how the interaction term can aid in the optimization of the neural network.
Title: Interatomic Potentials from Symmetry-Adapted Polynomial Fits
Abstract: For large molecules and materials, ab initio computations are too computationally demanding, so that Interatomic Potentials, cheap to compute but less accurate, are in use. Originally derived from empirical models, these potentials have over the past few years mainly been developed from data-driven methods. In this talk, I will present the construction of interatomic potentials based on polynomials satisfying the symmetries of the “exact” potential energy surface, i.e. invariant under rotations and permutations of identical particles. These potentials are practically computed in a data-driven fashion using linear fits. I will then explain on a toy model how these invariances can be exploited to improve the cost vs error ratio in a polynomial approximation of the energy. Finally, I will report simulation results on materials systems and molecules, illustrating the accuracy, the low computational cost, and the systematic improvability of the potentials. Joint work with Alice Allen (Cambridge), Markus Bachmayr (Mainz), Gabor Csanyi (Cambridge), Cas van der Oord (Cambridge), and Christoph Ortner (Warwick).
Title: On the geometry of Stein variational gradient descent
Abstract: Bayesian inference problems require sampling or approximating high-dimensional probability distributions. The focus of this talk is on the recently introduced Stein variational gradient descent methodology, a class of algorithms that rely on iterated steepest descent steps with respect to a reproducing kernel Hilbert space norm. This construction leads to interacting particle systems, the mean-field limit of which is a gradient flow on the space of probability distributions equipped with a certain geometrical structure. We leverage this viewpoint to shed some light on the convergence properties of the algorithm, in particular addressing the problem of choosing a suitable positive definite kernel function. Our analysis leads us to considering certain singular kernels with adjusted tails. This is joint work with N. Nusken (U. of Pottsdam).
Title: On fluctuations in particle system s and their links to macroscopic models
Abstract: We study particle systems and analyse their fluctuations. These fluctuations can be described by stochastic differential equations or variation al fo rmulations related to large deviations. The link between finite systems and their m any -particle limit will be an alysed in these f ormulations. This leads to an Onsager-type functional describing the evolution of finite state Markov chains. Another one result is that fluctuations determine in the purely diffusive case the associated macroscopic operator, of Wasserstein type.
Title: Nonequilibrium steady states for chains of oscillators and rotors: an overview
Abstract: I will talk about chains of oscillators and rotors interacting with stochastic heat baths at different temperatures. I will introduce these very simple models in the framework of the (yet unsolved!) problem of heat conduction. Then, we will focus on a much more elementary question: the existence of an invariant measure (called non-equilibrium steady state), which has been proved only in some specific cases over the past 20 years. I will explain how distinct models lead to distinct difficulties, and sketch some of the ideas used to overcome them.
Title: Hydrodynamic limits, finite size fluctuations, large deviations and Gamma-convergence
Abstract: Deriving deterministic fluid partial differential equations from a system of interacting particles is an old problem. It often involves two limiting procedures: a large number N of particles, which leads to a kinetic equation, for instance Boltzmann; and a time scale separation between local equilibration and large scale dynamics, materialized for instance by a small Knudsen number. How can one deal with situations where N is large, but finite? Motivated by models of active particles, we will describe an approach based on large deviation theory and gamma-convergence. Collaboration with Cédric Bernardin, Raphaël Chétrite, Yash Chopra, Mauro Mariani (+others)
Title: A coupling proof for the convergence of the Fleming-Viot algorithm
Abstract: The Fleming-Viot algorithm is based on a system of interacting particles and designed to target the quasi-stationary distribution (QSD) of a killed Markov process. In the case of a diffusion in a compact set with smooth killing, simple coupling arguments yield quantitative error estimates (in the number of particle, the simulation time and the timestep of the Euler scheme) between the empirical measure of the particles and the QSD.
Title: Models of emergent networks
Abstract: In this talk, we will present a modelling framework for the emergence of networks and their evolution. We will provide various examples of such emergent networks: ant trails, extracellular fibers and blood capillaries. We believe this framework can apply to other types of networks in which the topology and topography of nodes and links is fuzzy and evolutive.
Title: Analysis of Adaptive Biasing methods based on self-interacting diffusions
Abstract: An efficient strategy to compute equilibrium averages, for multimodal distributions, consists in biasing the associated ergodic diffusion process, using the free energy function relative to a given rea ction coordinate. In practice, one computes an approximation of the free energy on-the-fly. I will present two methods, the Adaptive Biasing Potential and the Adaptive Biasing Force algorithms, based on self-interacting dynamics: the bias depends on the empirical distribution of the process, thus the process interacts with its past. We prove the convergence of the bias to the free energy function, which justifies the efficiency of the algorithm. Joint work with Michel Benaïm and Pierre Monmarché.
For general enquiries: Anuj Sood