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A simple shallow neural network for emulating the solution to singularly perturbed problems — arXiv:2609.05180

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.05180
License (as archived): CC0-1.0
Archived: 2026-10-16T23:19:35.636Z
SHA-256 of the archived text: a7246ab1200dabb85683e992071cf376467e111de504a384e0b44454d9223290
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How to cite

A simple shallow neural network for emulating the solution to singularly perturbed problems — arXiv:2609.05180 — arXiv (en), https://arxiv.org/abs/2609.05180 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7fd4a2bfa03b (Metraipolis archive dc_7fd4a2bfa03b, sha256 a7246ab1200dabb85683e992071cf376467e111de504a384e0b44454d9223290)

BibTeX

@misc{dc7fd4a2bfa03b,
  title = {A simple shallow neural network for emulating the solution to singularly perturbed problems — arXiv:2609.05180},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.05180},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7fd4a2bfa03b; sha256 a7246ab1200dabb85683e992071cf376467e111de504a384e0b44454d9223290},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.05180. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We consider (feed-forward) Neural Networks (NNs) for the emulation of the solution to singularly perturbed second order boundary value problems, with two small parameters. We describe a shallow NN which exploits available asymptotic expansions for the solution. These additive decompositions into smooth and layer components, allow for derivative estimates which are explicit in the order of differentiation as well as the singular perturbation parameter(s) \cite{melenk, Irene, SX}. Utilizing such decompositions, we propose a simple NN for emulating the solution to such problems using the $\tanh$ activation function together with different training objectives, such as residual or energy minimization. The key idea is to augment the approximation space with suitable exponential functions, similar to enriched spaces in finite element methods, e.g.~\cite{Kellogg}. Numerical examples in one and two dimensions, including a smooth non-tensor-product domain, illustrate the resulting parameter-robust behavior over the tested perturbation ranges.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-07 in math.NA).
Categorie arXiv: Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2609.05180 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T23:02:57.309Z dalla pagina https://arxiv.org/catchup/math.NA/2026-09-07?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).