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Bridging Scales: Asymptotic Analysis and AI-Assisted Formalization

  • Xu Yang
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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.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026bridging,
  title = {{Bridging Scales: Asymptotic Analysis and AI-Assisted Formalization}},
  author = {Xu Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.10138},
  eprint = {2606.10138},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.10138},
}
RIS
TY  - UNPB
TI  - Bridging Scales: Asymptotic Analysis and AI-Assisted Formalization
AU  - Xu Yang
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2606.10138
UR  - https://arxiv.org/abs/2606.10138
AB  - 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.
ER  -