Two-Time-Scale Learning Dynamics: A Population View of Neural Network Training

Giacomo Borghi, Hyesung Im, Lorenzo Pareschi (preprint arXiv:2603.19808)

Population-based learning paradigms, including evolutionary strategies, Population-Based Training (PBT), and recent model-merging methods, combine fast within-model optimisation with slower population-level adaptation. Despite their empirical success, a general mathematical description of the resulting collective training dynamics remains incomplete. We introduce a theoretical framework for neural network training based on two-time-scale population dynamics. We model a population of neural networks as an interacting agent system in which network parameters evolve through fast noisy gradient updates of SGD/Langevin type, while hyperparameters evolve through slower selection--mutation dynamics.

Micro-Macro Tensor Neural Surrogates for Uncertainty Quantification in Collisional Plasma

Wei Chen, Giacomo Dimarco, Lorenzo Pareschi (preprint arXiv:2512.24205)

Plasma kinetic equations exhibit pronounced sensitivity to microscopic perturbations in model parameters and data, making reliable and efficient uncertainty quantification (UQ) essential for predictive simulations. However, the cost of uncertainty sampling, the high-dimensional phase space, and multiscale stiffness pose severe challenges to both computational efficiency and error control in traditional numerical methods. These aspects are further emphasized in presence of collisions where the high-dimensional nonlocal collision integrations and conservation properties pose severe constraints. To overcome this, we present a variance-reduced Monte Carlo framework for UQ in the Vlasov--Poisson--Landau (VPL) system, in which neural network surrogates replace the multiple costly evaluations of the Landau collision term. The method couples a high-fidelity, asymptotic-preserving VPL solver with inexpensive, strongly correlated surrogates based on the Vlasov--Poisson--Fokker--Planck (VPFP) and Euler--Poisson (EP) equations.

Asymptotic preserving methods for the low Mach limit in discrete velocity models approximating kinetic equations

Giacomo Dimarco, Axel Klar, Theresa Köfler, Lorenzo Pareschi (preprint arXiv:2512.19847)

We consider a Lattice Boltzmann type discrete velocity model in the low Mach number scaling and develop a corresponding numerical scheme that remains uniformly valid across all regimes of the mean free path, from the kinetic to the hydrodynamic scale. The proposed framework ensures high order temporal accuracy through the use of Implicit Explicit Runge Kutta methods, which provide stability and efficiency in stiff regimes, while spatial resolution is enhanced by combining finite difference WENO reconstructions with high order central difference approximations.

Collective Annealing by Switching Temperatures: a Boltzmann-type description

Frédéric Blondeel, Lorenzo Pareschi, Giovanni Samaey (preprint arXiv:2512.13522)

The design of effective cooling strategies is a crucial component in simulated annealing algorithms based on the Metropolis method. Traditionally, this is achieved through inverse logarithmic decays of the temperature to ensure convergence to global minima. In this work, we propose Collective Annealing by Switching Temperatures (CAST), a novel collective simulated annealing dynamic in which agents interact to learn an adaptive cooling schedule.

High-Order Asymptotic-Preserving IMEX schemes for an ES-BGK model for Gas Mixtures

Domenico Caparello, Lorenzo Pareschi, Thomas Rey (preprint arXiv:2511.22304)

In this work we construct a high-order Asymptotic-Preserving (AP) Implicit-Explicit (IMEX) scheme for the ES-BGK model for gas mixtures introduced in [Brull, Commun. Math. Sci., 2015]. The time discretization is based on the IMEX strategy proposed in [Filbet, Jin, J. Sci. Comput., 2011] for the single-species BGK model and is here extended to the multi-species ES-BGK setting. The resulting method is fully explicit, uniformly stable with respect to the Knudsen number and, in the fluid regime, it reduces to a consistent and high-order accurate solver for the limiting macroscopic equations of the mixture.