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SUMMARY:CNRS-ICL Workshop: Mean field limits for interacting particle syste
ms:
uniform propagation of chaos\, phase transitions and applications
DESCRIPTION:WORKSHOP ORGANIZERS: P. DEGOND\, S. KALLIADASIS\, G.A. PAVLIOTI
S\nWE ACKNOWLEDGE THE GENEROUS SUPPORT OF THE CNRS-Imperial “Abraham de
Moivre” International Research Laboratory\, the Quantitative Sciences Re
search Institute and of the Department of Mathematics at Imperial Colleg
e.\n \nPROGRAMME\, TITLES AND ABSTRACTS\nTuesday 04 July\n09:15 – 09:30
Welcome Remarks\n09:30 – 10:15 A. GUILLIN Propagation of chaos for some
singular models\n10:15 – 11:00 M. DELGADINO On the mean field approxima
tion of the Gibbs measure for weakly interacting diffusions.\n11:00 – 11
:30 Coffee break\n11:30 – 12:15 Ch. SAFFIRIO Weakly interacting fermion
s: mean-field and semiclassical regimes.\n12:15 – 14:00 Lunch Break\n14:
00 – 14:45 J. TUGAUT Captivity of the solution to the granular media eq
uation\n14:45 – 15:30 A. MENEGAKI Quantitative framework for hydrodyna
mic limits\n15:30 – 16:00 Coffee Break\n16:00 – 16:45 J. EVANS Existe
nce and stability of non-equilbrium steady states in a BGK model coupled t
o thermostats\nWednesday 05 July\n09:30 – 10:15 A. EBERLE Sticky nonlin
ear SDEs and mean-field systems without confinement\n10:15 – 11:00 R. GV
ALANI Logarithmic Sobolev inequalities and equilibrium fluctuations for
weakly interacting diffusions\n11:00 – 11:30 Coffee break\n11:30 – 12:
15 H. DUONG Asymptotic analysis for the generalized Langevin equation wit
h singular potentials\n12:15 – 14:00 Lunch Break\n14:00 – 14:45 K. SPI
LIOPOULOS Normalization effects and mean field theory for deep neural net
works \n14:45 – 15:30 J. REYGNER Dynamical Gibbs principle and assoc
iated stochastic control problems\n15:30 – 16:00 Coffee Break\n16:00 –
16:45 M. OTTOBRE McKean Vlasov (S)PDEs \nThursday 06 July\n09:
30 – 10:15 P. Del MORAL Some theoretical aspects of Particle Filters and
Ensemble Kalman Filters\n10:15 – 11:00 A. SCHLICHTING Mean-field PDEs
on graphs and their continuum limit\n11:00 – 11:30 Coffee break\n11:30
– 12:15 N. ZAGLI Response Theory and Critical Phenomena for Weakly Inter
acting Diffusions\n12:15 – 14:00 Lunch Break\nABSTRACTS\nP. Del Moral\nT
itle: Some theoretical aspects of Particle Filters and Ensemble Kalman Fil
ters.\nAbstract: In the last three decades\, Particle Filters (PF) and E
nsemble Kalman Filters (EnKF) have become one of the main numerical techni
ques in data assimilation\, Bayesian statistical inference and nonlinear f
iltering. Both particle algorithms can be viewed as mean field type partic
le interpretation of the filtering equation and the Kalman recursion. In
contrast with conventional particle filters\, the EnKF is defined by a s
ystem of particles evolving as the signal in some state space with an inte
raction function that depends on the sample covariance matrices of the sys
tem. Despite widespread usage\, little is known about the mathematical fo
undations of EnKF. Most of the literature on EnKF amounts to design differ
ent classes of useable observer-type particle methods. To design any type
of consistent and meaningful filter\, it is crucial to understand their ma
thematical foundations and their learning/tracking capabilities. This talk
discusses some theoretical aspects of these numerical techniques. We pre
sent some recent advances on the stability properties of these filters. We
also initiate a comparison between these particle samplers and discuss so
me open research questions.\nM. DELGADINO\nTitle: On the mean field approx
imation of the Gibbs measure for weakly interacting diffusions.\nAbstract:
In this talk we will explore when the invariant Gibbs measure of an N -pa
rticle system of weakly interacting diffusion is well approximated by the
unique minimiser of the mean field energy. We will compute the exact limit
of the associated partition function\, by re-interpreting the associated
integral as the fluctuation of sampling from the mean field limit measure.
With this technique we can obtain sharp estimates all the way up to the p
hase transition\, colloquially defined as the loss of analyticity of the p
artition function with respect to the inverse temperature.\nH. DUONG\nTitl
e: Asymptotic analysis for the generalized Langevin equation with singular
potentials\nAbstract: We consider a system of interacting particles gover
ned by the generalized Langevin equation (GLE) in the presence of external
confining potentials\, singular repulsive forces\, as well as memory kern
els. Using a Mori-Zwanzig approach\, we represent the system by a class of
Markovian dynamics. Under a general set of conditions on the nonlineariti
es\, we study the large-time asymptotics of the multi-particle Markovian G
LEs. We show that the system is always exponentially attractive toward the
unique invariant Gibbs probability measure. The proof relies on a novel c
onstruction of Lyapunov functions. We then establish the validity of the s
mall mass approximation for the solutions by an appropriate equation on an
y finite-time window. Important examples of singular potentials in our res
ults include the Lennard-Jones and Coulomb functions.\n This talk is base
d on a joint work with D. H. Nguyen (UCLA).\nA EBERLE\nTITLE: Sticky nonli
near SDEs and mean-field systems without confinement\nABSTRACT: We conside
r a class of one-dimensional nonlinear stochastic differential equations w
ith a sticky boundary at zero. It can be shown that there is a phase trans
ition: For small nonlinearities\, all mass eventually gets stuck at zero\,
whereas for larger nonlinearities\, there is a nontrivial invariant measu
re. The solution of the sticky nonlinear SDE provides an upper bound for t
he distance process between two solutions of a McKean-Vlasov equation with
out confinement\, coupled by a sticky coupling. As a consequence\, it can
be used to prove exponential convergence to equilibrium for this equation
with a sufficiently small nonlinearity. Moreover\, uniform in time propaga
tion of chaos for the corresponding mean-field particle system is obtained
by an extension of these ideas based on a componentwise sticky coupling (
joint work with Alain Durmus\, Arnaud Guillin and Katharina Schuh).\n \nJ
. EVANS\nTitle: Existence and stability of non-equilbrium steady states in
a BGK model coupled to thermostats\nAbstract: This is based on a joint wo
rk with Angeliki Menegaki. I will discuss non-equilibrium steady states in
kinetic models for heat transfer in gasses. These steady states are non e
xplicit and non-Gibbs. I will explain how we can show existence of such st
ates and discuss the challenges in showing stability via standard hypocoer
civity machinery.\n \n \nA GUILLIN\nTitle: Propagation of chaos for some
singular models\nAbstract: we consider here uniform in time propagation o
f chaos for the 2D vortex model or for the Dyson Ornstein-Uhlenbeck Browni
an motion\, which are mean field particle models with interactions exhibit
ing singularities. We will propose an entropy method for the fist case and
coupling for the second. Joint works with Pierre Le Bris and Pierre Monma
rché\n \nR. GVALANI\nTitle: Logarithmic Sobolev inequalities and equilib
rium fluctuations for weakly interacting diffusions\n Abstract: We study
the mean field limit of interacting diffusions for confining and interacti
on potentials that are non-convex. The limiting behaviour is described by
the nonlocal McKean–Vlasov PDE. We explore the relationship between the
limit $N\\to\\infty$ of the constant in the logarithmic Sobolev inequality
(LSI) for the $N$-particle system and the presence or absence phase trans
itions for the mean field limit\, conjecturing a limiting form of the LSI
constant. We also explore the consequences of the non-degeneracy of the LS
I constant as they relate to uniform-in-time propagation of chaos and equi
librium fluctuations. Our results extend previous results on unbounded spi
n systems as well as recent results on (uniform-in-time) propagation of ch
aos using novel coupling arguments. Joint work with Matías Delgadino\, Gr
eg Pavliotis\, and Scott Smith.\n \nA. MENEGAKI\n Title: Quantitative fr
amework for hydrodynamic limits\n Abstract: We will present a new quantit
ative approach to the problem of proving hydrodynamic limits from microsco
pic stochastic particle systems\, namely the zero-range and the Ginzburg-L
andau process with Kawasaki dynamics\, to macroscopic partial differential
equations. Our method combines a modulated Wasserstein distance estimate
comparing the law of the stochastic process to the local Gibbs measure\, t
ogether with stability estimates a la Kruzhkov in weak distance and consis
tency estimates exploiting the regularity of the limit solution. It is sim
plified as it avoids the use of the block estimates. This is a joint work
with Clément Mouhot and Daniel Marahrens.\n \nM. OTTOBRE\n Title: McKea
n Vlasov (S)PDEs \nAbstract. We consider McKean-Vlasov Stochastic Par
tial Differential Equations with additive noise\; we will focus on well-po
sedness and long time behaviour\, as well as on obtaining these SPDEs fr
om interacting particle systems. \nThis is based on joint work with L
. Angeli\, J. Barre’\, D. Crisan and M. Kolodziejkzyc. \n \nJ. REYGNER
\n Dynamical Gibbs principle and associated stochastic control problems\n
In statistical physics\, the Gibbs principle describes the asymptotic ma
rginal distribution of a particle when the whole system is conditioned on
a large deviation of its empirical measure. We consider a similar question
for the case of diffusion processes\, and show that the effect of conditi
oning results in an additional drift which encodes a mean-field like inter
action. We also discuss the interpretation of this statement in terms of a
stochastic control problem\, with a constraint in distribution. This is a
joint work with Louis-Pierre Chaintron (École Normale Supérieure) and G
iovanni Conforti (École polytechnique).\n \nCH. SAFFIRIO\n TITLE: Weakl
y interacting fermions: mean-field and semiclassical regimes.\nABSTRACT\nT
he derivation of effective macroscopic theories approximating microscopic
systems of interacting particles is a major question in non-equilibrium st
atistical mechanics. In this talk we will be concerned with the dynamics o
f systems made of many interacting fermions in the case in which the inter
action is singular (e.g. inverse power law). We will focus on the mean-fie
ld regime and obtain a reduced description given by the time-dependent Har
tree-Fock equation. As a second step we will look at longer time scales wh
ere a semiclassical description starts to be relevant and approximate the
many-body dynamics with the Vlasov equation\, which describes the evolutio
n of the effective probability density of particles on the one particle ph
ase space. \n \nA SCHLICHTING \nTITLE: Mean-field PDEs on graphs and th
eir continuum limit\nAbstract:\nThis talk reviews some recent results on n
onlocal PDEs describing the evolution of a density on graph structures.
These equations can arise \nfrom mean-field interacting jump dynamics\, b
ut also from applications in the data science field\, or they can also be
obtained by a numerical \ndiscretization of a continuum problem. We also
show how those equations are linked to their continuous counterpart in s
uitable local limits.\njoint work with Antonio Esposito\, Georg Heinze\, A
nastasiia Hraivoronska and Oliver Tse\n \nK. SPILIOPOULOS\n Title: Nor
malization effects and mean field theory for deep neural networks \n We
study the effect of normalization on the layers of deep neural networks. A
given layer $i$ with $N_{i}$ hidden units is allowed to be normalized by
$1/N_{i}^{\\gamma_{i}}$ with $\\gamma_{i}\\in[1/2\,1]$ and we study the ef
fect of the choice of the $\\gamma_{i}$ on the statistical behavior of the
neural network’s output (such as variance) as well as on the test accur
acy on the MNIST and CIFAR10 data sets. We find that in terms of variance
of the neural network’s output and test accuracy the best choice is to c
hoose the $\\gamma_{i}$’s to be equal to one\, which is the mean-field s
caling. We also find that this is particularly true for the outer layer\,
in that the neural network’s behavior is more sensitive in the scaling o
f the outer layer as opposed to the scaling of the inner layers. The mecha
nism for the mathematical analysis is an asymptotic expansion for the neur
al network’s output and corresponding mean field analysis. An important
practical consequence of the analysis is that it provides a systematic and
mathematically informed way to choose the learning rate hyperparameters.
Such a choice guarantees that the neural network behaves in a statisticall
y robust way as the $N_i$ grow to infinity.\nJ. TUGAUT\nTITLE: Captivity o
f the solution to the granular media equation\n ABSTRACT: In this talk\,
we first recall some facts about the long-time behaviour of the solution t
o the granular media equation under convexity assumptions. Then\, we deal
with the non-convex case. In particular\, we talk about the phase transiti
on about the number of invariant probabilities. The main part of the talk
concerns the characterization of the limiting probability and the whole pr
oof is given in a simple case.\nN. ZAGLI\nTitle: Response Theory and Criti
cal Phenomena for Weakly Interacting Diffusions\nAbstract: \nIn this talk
\, I will present our latest results on the close link between response th
eory\, the existence of collective variables and the development of critic
al phenomena for noisy systems with mean field interactions. \nSuch syste
ms are routinely used to model collective emergent behaviours in multiple
areas of social and natural sciences as they exhibit\, in the thermodynami
c limit\, continuous and discontinuous phase transitions.\nI will show tha
t the perspective given by Response Theory\, that aims at establishing a l
ink between natural and forced variability of general physical systems\, i
s particularly useful to understand the physical mechanisms establishing c
ritical phenomena in interacting systems. \nFirstly\, I will show how to
define a set of collective variables for the system starting from the coup
ling structure among the microscopic agents. \nSecondly\, I will show tha
t such variables prove to be proper nonequilibrium thermodynamic observabl
es as they carry information on correlation\, memory properties and resili
ence properties of the system. \nIn particular\, the investigation of res
ponse properties of such collective variables allows to identify and chara
cterise phase transitions of the system as they manifest as singular value
s of the susceptibility associated to such thermodynamic variables. \nNum
erical experiments will be presented to show how the critical pattern of t
he response arise and can be detected in finite systems.
URL:https://www.imperial.ac.uk/events/163582/cnrs-icl-workshop-mean-field-l
imits-for-interacting-particle-systems-uniform-propagation-of-chaos-phase-
transitions-and-applications/
DTSTART;TZID=Europe/London:20230704T090000
DTEND;TZID=Europe/London:20230706T180000
LOCATION:EEE 403a (Tuesday) EEE 406 (Wednesday and Thursday)\, Electrical &
Electronic Engineering\, South Kensington Campus\, Imperial College Londo
n\, London\, SW7 2AZ\, United Kingdom
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