Bridging Scales: Asymptotic Analysis and AI-Assisted Formalization

Abstract

Asymptotic analysis is one of the classical tools for bridging models across scales. Behind many such derivations lies a common symbolic structure: an ansatz, a substitution, an order-by-order matching procedure, and the extraction of effective equations or interface conditions. This article revisits that structure through two representative bridges: the kinetic-to-fluid limit, illustrated by radiative transfer with interface layers and by neural-network approximations of Boltzmann equations, and the quantum-to-classical limit, illustrated by the Frozen Gaussian Approximation and its Dirac extension. We then explain why such derivations are natural candidates for AI-assisted formalization: their recurring symbolic structures can be organized, verified, and reused. In this sense, a carefully structured expository paper may serve not only as a review, but also as a mathematical seed for future AI-assisted environments.