A fast spectral particle method for the Landau equation

 

Giacomo Borghi, Lorenzo Pareschi (preprint arXiv:2609.23086)

We propose a fast deterministic particle method for the spatially homogeneous Landau equation. The method combines a particle representation of the solution with a Fourier approximation of the nonlinear collision flux. Nonuniform fast Fourier transforms are used to reconstruct the density from the particles and to evaluate the flux and density at the particle locations, while the convolutional structure of the flux enables its efficient computation by FFTs. For a fixed transform tolerance, the cost per time step is O(N+M^d logM), where N is the number of particles and M the number of Fourier modes per velocity dimension. We establish a consistency estimate for the reconstructed velocity field on regions where the reference density is bounded away from zero. The estimate separates the spectral truncation error from the particle-density reconstruction error, showing how the latter can dominate for sufficiently smooth densities.

Control variates with neural surrogates for uncertainty quantification in kinetic equations

Wei Chen, Giacomo Dimarco, Lorenzo Pareschi (preprint arXiv:2608.01360)

Efficient uncertainty quantification for kinetic equations with random inputs is challenging because it requires repeated simulations of high-dimensional models, such as the Boltzmann, Landau, and related collisional equations, whose computational cost can quickly become prohibitive. Multifidelity control variates address this difficulty by coupling a small number of high-fidelity simulations with many evaluations of lower-complexity reduced models. In this work, we analyze the case in which the reduced model is replaced by a neural surrogate rather than evaluated through a classical numerical scheme.

High-Order Asymptotic-Preserving Schemes for Kinetic Equations from Rarefied to Incompressible Regimes

Giacomo Dimarco, Axel Klar, Theresa Köfler, Lorenzo Pareschi, Tiwari Sudarshan (preprint arXiv:2606.28040)

This work introduces a novel high-order numerical framework for solving kinetic equations, designed to remain uniformly valid across all regimes of the mean free path, spanning from the rarefied kinetic scale to the incompressible hydrodynamic limit. The method is built upon a micro-macro decomposition, which reformulates the underlying kinetic equation into a coupled system consisting of a macroscopic part, representing the fluid-dynamic evolution, and a microscopic part, describing the non-equilibrium deviations. 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.

How opinions shape epidemics: a graphon-based kinetic approach

Abu Safyan Ali, Elisa Calzola, Giacomo Dimarco, Lorenzo Pareschi, Thomas Rey (preprint arXiv:2605.14139)

Understanding the mutual influence between social behavior and physical health is crucial for designing effective epidemic mitigation strategies. Individual interactions drive the evolution of opinions, which in turn shape how infectious diseases are perceived and consequently how they spread within a population, for instance through the adoption or rejection of preventive measures. At the same time, the distribution and dynamics of physical contacts play a fundamental role in determining transmission patterns.

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.