Abstract

Rarefied dynamics arising in non-equilibrium flow regimes pose a persistent challenge to computational modeling. Their multiple scales, nonlocal interactions, and high dimensionality require computational strategies that merge physical consistency with computational efficiency. This talk proposes a mulit-fidelity framework that connects structure-preserving moment hierarchies, variational multiscale closures, and computational acceleration with data-driven neural operators. We begin from the method of moments as a hierarchical modeling paradigm and show how invariances and thermodynamic laws guide the development of closure relations consistent with underlying physical symmetries [1]. Building on this, we extend the closure relations using variational multiscale methods to capture unresolved scales while maintaining thermodynamic constraints [2]. Finally, we demonstrate how Neural Green’s Operators can act as data-driven accelerators that can support multi-query problem solving [3].

[1] MRA Abdelmalik, EH van Brummelen. "Moment closure approximations of the Boltzmann equation based on φ-divergences" Journal of Statistical Physics 164.1 (2016): 77-104

[2] FA Baidoo, IM Gamba, TJR Hughes, MRA Abdelmalik. “Extensions to the Navier–Stokes–Fourier equations for rarefied transport: Variational multiscale moment methods for the Boltzmann equation” Mathematical Models and Methods in Applied Sciences 36.01 (2026): 111-172

[3] H Melchers, J Prins, MRA Abdelmalik. "Neural Green's Operators for Parametric Partial Differential Equations." Computer Methods in Applied Mechanics and Engineering 455 (2026): 118893