Structure and asymptotic preserving deep neural surrogates for uncertainty quantification in multiscale kinetic equations

Wei Chen, Giacomo Dimarco, Lorenzo Pareschi (preprint arXiv:2506.10636)

The high dimensionality of kinetic equations with stochastic parameters poses major computational challenges for uncertainty quantification (UQ). Traditional Monte Carlo (MC) sampling methods, while widely used, suffer from slow convergence and high variance, which become increasingly severe as the dimensionality of the parameter space grows. To accelerate MC sampling, we adopt a multiscale control variates strategy that leverages low-fidelity solutions from simplified kinetic models to reduce variance.

Superlinear Drift in Consensus-Based Optimization with Condensation Phenomena

Jonathan Franceschi, Lorenzo Pareschi, Mattia Zanella (preprint arXiv:2506.09001)

Consensus-based optimization (CBO) is a class of metaheuristic algorithms designed for global optimization problems. In the many-particle limit, classical CBO dynamics can be rigorously connected to mean-field equations that ensure convergence toward global minimizers under suitable conditions. In this work, we draw inspiration from recent extensions of the Kaniadakis--Quarati model for indistinguishable bosons to develop a novel CBO method governed by a system of SDEs with superlinear drift and nonconstant diffusion. The resulting mean-field formulation in one dimension exhibits condensation-like phenomena, including finite-time blow-up and loss of L2-regularity.

Robust feedback control of collisional plasma dynamics in presence of uncertainties

Giacomo Albi, Giacomo Dimarco, Federica Ferrarese, Lorenzo Pareschi (preprint arXiv:2505.19992)

Magnetic fusion aims to confine high-temperature plasma within a device, enabling the fusion of deuterium and tritium nuclei to release energy. Due to the very large temperatures involved, it is essential to isolate the plasma from the device walls to prevent structural damage and the external magnetic fields play a fundamental role in achieving this confinement. In realistic settings, the physical mechanisms governing plasma behavior are highly complex, involving numerous uncertain parameters and intricate particle interactions, such as collisions, that significantly affect both confinement efficiency and overall stability.

Hierarchical dynamic domain decomposition for the multiscale Boltzmann equation

Domenico Caparello, Lorenzo Pareschi, Thomas Rey (preprint arXiv:2505.03360)

In this work, we present a hierarchical domain decomposition method for the multi-scale Boltzmann equation based on moment realizability matrices, a concept introduced by Levermore, Morokoff, and Nadiga. This criterion is used to dynamically partition the two-dimensional spatial domain into three regimes: the Euler regime, an intermediate kinetic regime governed by the ES-BGK model, and the full Boltzmann regime. The key advantage of this approach lies in the use of Euler equations in regions where the flow is near hydrodynamic equilibrium, the ES-BGK model in moderately non-equilibrium regions where a fluid description is insufficient but full kinetic resolution is not yet necessary, and the full Boltzmann solver where strong non-equilibrium effects dominate, such as near shocks and boundary layers. This allows for both high accuracy and significant computational savings, as the Euler solver and the ES-BGK models are considerably cheaper than the full kinetic Boltzmann model.

Wasserstein convergence rates for stochastic particle approximation of Boltzmann models

 G. Borghi, L. Pareschi (preprint arXiv: 2504.10091). 

We establish quantitative convergence rates for stochastic particle approximation based on Nanbu-type Monte Carlo schemes applied to a broad class of collisional kinetic models. Using coupling techniques and stability estimates in the Wasserstein-1 (Kantorovich-Rubinstein) metric, we derive sharp error bounds that reflect the nonlinear interaction structure of the models. Our framework includes classical Nanbu Monte Carlo method and more recent developments as Time Relaxed Monte Carlo methods. The results bridge the gap between probabilistic particle approximations and deterministic numerical error analysis, and provide a unified perspective for the convergence theory of Monte Carlo methods for Boltzmann-type equations. As a by-product, we also obtain existence and uniqueness of solutions to a large class of Boltzmann-type equations.