Tuesday, November 1 |
07:30 - 09:00 |
Breakfast (Restaurant Hotel Hacienda Los Laureles) |
09:00 - 09:25 |
Shi Jin: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networks ↓ Neural message passing is a basic feature extraction unit for graph-structured data considering neighboring node features in network propagation from one layer to the next. We model such process by an interacting particle system with attractive and repulsive forces and the Allen-Cahn force arising in the modeling of phase transition. The dynamics of the system is a reaction-diffusion process which can separate particles without blowing up. This induces an Allen-Cahn message passing (ACMP) for graph neural networks where the numerical iteration for the particle system solution constitutes the message passing propagation. ACMP which has a simple implementation with a neural ODE solver can propel the network depth up to one hundred of layers with theoretically proven strictly positive lower bound of the Dirichlet energy. It thus provides a deep model of GNNs circumventing the common GNN problem of oversmoothing. GNNs with ACMP achieve state of the art performance for real-world node classification tasks on both homophilic and heterophilic datasets. (Zoom) |
09:30 - 09:55 |
Weizhu Bao: Uniform error bounds on numerical methods for long-time dynamics of dispersive PDEs ↓ In this talk, I report our recent work of error estimates on different numerical methods for the long-time dynamics of dispersive PDEs with small potential or weak nonlinearity, such as the Schroedinger equation with small potential, the nonlinear Schroedinger equation with weak
nonlinearity, the nonlinear Klein-Gordon equation with weak nonlinearity, the Dirac equation with small electromagnetic potential, and the nonlinear Dirac equation with weak nonlinearity, etc. By introducing a new technique of regularity compensation oscillatory (RCO), we can establish improved uniform error bounds on time-splitting methods for dispersive PDEs with small potentials and/or weak nonlinearity. This talk is based on joint works with Yongyong Cai and Yue Feng. (Zoom) |
10:00 - 10:30 |
Coffee Break (Conference Room San Felipe) |
10:30 - 10:55 |
Mattia Zanella: Uncertainty quantification for kinetic equations of emergent phenomena ↓ Kinetic equations play a leading role in the modelling of large systems of interacting particles/agents with a recognized effectiveness in describing real world phenomena ranging from plasma physics to multi-agent dynamics. The derivation of these models has often to deal with physical, or even social, forces that are deduced empirically and of which we have limited information. Hence, to produce realistic descriptions of the underlying systems it is of paramount importance to consider the effects of uncertain quantities as a structural feature in the modelling process.
In this talk, we focus on a class of numerical methods that guarantee the preservation of main physical properties of kinetic models with uncertainties. In contrast to a direct application of classical uncertainty quantification methods typically leading to the loss of positivity of the numerical solution of the problem, we discuss the construction of novel particle Galerkin schemes that are capable of achieving high accuracy in the random space without losing nonnegativity of the solution [1,2,3]. Applications of the developed methods are presented for the Boltzmann equation and for kinetic equations of plasmas with uncertainties.
Bibliography:
[1] J. A. Carrillo, L. Pareschi, M. Zanella. Particle based gPC methods for mean-field models of swarming with uncertainty. Commun. Comput. Phys., 25(2): 508-531, 2019.
[2] L. Pareschi, M. Zanella. Monte Carlo stochastic Galerkin methods for the Boltzmann equation with uncertainties: space-homogeneous case. J. Comput. Phys. 423:109822, 2020.
[3] A. Medaglia, L. Pareschi, M. Zanella. Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties. Preprint arXiv:2208.00692. (Zoom) |
11:00 - 11:25 |
Francis Filbet: On a discrete framework of hypocoercivity for kinetic equations ↓ We study a class of spatial discretizations for the Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials.Then we propose a specific finite volume discretization for which we can prove hypocoercive estimates. This allows us to explore the long time behavior of the numerical solution and the asymptotitc limit (diffusive limit). (Zoom) |
11:30 - 12:00 |
Mária Lukácová: Hybrid multiscale methods for polymeric fluids ↓ I present our recent results on hybrid multiscale methods for polymeric fluids. I will concentrate on a class of kinetic models for polymeric fluids motivated by the Peterlin dumbbell theories for dilute polymer solutions with a nonlinear spring law. The polymer molecules are suspended in an incompressible viscous Newtonian fluid confined to a bounded domain in two
or three space dimensions. The unsteady motion of the solvent is described by
the incompressible Navier–Stokes equations with the elastic extra stress tensor
appearing as a forcing term in the momentum equation. The elastic stress tensor is defined by Kramer’s expression through the probability density function that satisfies the corresponding Fokker–Planck equation. In this case a coefficient depending on the average length of polymer molecules appears in the latter equation. I present the main steps of the proof of the existence of global-in-time weak solutions to the kinetic Peterlin model.
Numerical simulations are obtained by the conservative scheme for a high-dimensional Fokker–Planck equation that models polymer molecules. The hybrid kinetic-continuum scheme combines the Lagrange–Galerkin method for the solvent and the Hermite spectral method for the Fokker-Planck equation together with a space splitting approach. Several numerical experiments will be presented to illustrate the performance of the scheme, and to confirm the conservation of mass at the discrete level.
If time permits I can mention recent results of the hybrid molecular dynamics-continuum methods. (Zoom) |
12:00 - 12:10 |
Group Photo (Hotel Hacienda Los Laureles) |
12:30 - 14:00 |
Lunch (Restaurant Hotel Hacienda Los Laureles) |
14:00 - 14:40 |
Domenec Ruiz I Balet: The interplay between control and deep learning ↓ This talk will be about Neural Ordinary Differential Equations from a control theoretical perspective. We will see how they have strong simultaneous control properties that allow interpolation and approximation results. The simultaneous control problem of the differential equations can be interpreted as a bilinear-type control of the associated continuity equation, from which controllability results will be presented. Finally, we will also introduce the so-called Momentum ResNet, a variation that leads to a kinetic equation. (Conference Room San Felipe) |
14:45 - 15:15 |
Coffee Break (Conference Room San Felipe) |
15:15 - 15:55 |
Matias Delgadino: Phase transitions and log Sobolev inequalities ↓ In this talk, we will study the mean field limit of weakly interacting diffusions for confining and interaction potentials that are not necessarily convex. We explore the relationship between the large N limit of the constant in the logarithmic Sobolev inequality (LSI) for the N-particle system and the presence or absence of phase transitions for the mean field limit. The non-degeneracy of the LSI constant will be shown to have far reaching consequences, especially in the context of uniform-in-time propagation of chaos and the behaviour of equilibrium fluctuations. This will be done by employing techniques from the theory of gradient flows in the 2-Wasserstein distance, specifically the Riemannian calculus on the space of probability measures. (Conference Room San Felipe) |
16:00 - 16:40 |
Sui Tang: Bridging the interacting particle models and data science via Gaussian process ↓ System of interacting particles that display a wide variety of collective behaviors are ubiquitous in science and engineering, such as self-propelled particles, flocking of birds, milling of fish. Modeling interacting particle systems by a system of differential equations plays an essential role in exploring how individual behavior engenders collective behaviors, which is one of the most fundamental and important problems in various disciplines. Although the recent theoretical and numerical study bring a flood of models that can reproduce many macroscopical qualitative collective patterns of the observed dynamics, the quantitative study towards matching the well-developed models to observational data is scarce.
We consider the data-driven discovery of macroscopic particle models with latent interactions. We propose a scalable learning approach that models the latent interactions as Gaussian processes, which provides an uncertainty-aware modeling of interacting particle systems. We introduce an operator-theoretic framework to provide a detailed analysis of recoverability conditions, and establish statistical optimality of the proposed approach. Numerical results on prototype systems and real data demonstrate the effectiveness of the proposed approach. (Conference Room San Felipe) |
18:00 - 20:00 |
Dinner (Restaurant Hotel Hacienda Los Laureles) |