Kevin Cooper

Alexander Panchenko, Lyudmyla L. Barannyk, Kevin Cooper

The paper introduces a general framework for derivation of continuum equations governing meso-scale dynamics of large particle systems. The balance equations for spatial averages such as density, linear momentum, and energy were previously derived by a number of authors. These equations are not in closed form because the stress and the heat flux cannot be evaluated without the knowledge of particle positions and velocities. We propose a closure method for approximating fluxes in terms of other meso-scale averages. The main idea is to rewrite the non-linear averages as linear convolutions that relate micro- and meso-scale dynamical functions. The convolutions can be approximately inverted using regularization methods developed for solving ill-posed problems. This yields closed form constitutive equations that can be evaluated without solving the underlying ODEs. We test the method numerically on Fermi-Pasta-Ulam chains with two different potentials: the classical Lennard-Jones, and the purely repulsive potential used in granular materials modeling. The initial conditions incorporate velocity fluctuations on scales that are smaller than the size of the averaging window. The results show very good agreement between the exact stress and its closed form approximation.

Kevin Cooper, Michael Geller

In the age of data-driven decision making, preserving privacy while providing personalized experiences has become paramount. Personalized Federated Learning (PFL) offers a promising framework by decentralizing the learning process, thus ensuring data privacy and reducing reliance on centralized data repositories. However, the integration of advanced Artificial Intelligence (AI) techniques within PFL remains underexplored. This paper proposes a novel approach that enhances PFL with cutting-edge AI methodologies including adaptive optimization, transfer learning, and differential privacy. We present a model that not only boosts the performance of individual client models but also ensures robust privacy-preserving mechanisms and efficient resource utilization across heterogeneous networks. Empirical results demonstrate significant improvements in model accuracy and personalization, along with stringent privacy adherence, as compared to conventional federated learning models. This work paves the way for a new era of truly personalized and privacy-conscious AI systems, offering significant implications for industries requiring compliance with stringent data protection regulations.