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ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients — arXiv:2608.29499

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.29499
License (as archived): CC0-1.0
Archived: 2026-10-17T09:28:26.631Z
SHA-256 of the archived text: cb51a676599fedbe3717286b36f48ea64b029ebfc6416891d2add5552526295f
Stable URL of this page: https://metraipolis.com/documento/dc_efecf522a195

How to cite

ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients — arXiv:2608.29499 — arXiv (en), https://arxiv.org/abs/2608.29499 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_efecf522a195 (Metraipolis archive dc_efecf522a195, sha256 cb51a676599fedbe3717286b36f48ea64b029ebfc6416891d2add5552526295f)

BibTeX

@misc{dcefecf522a195,
  title = {ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients — arXiv:2608.29499},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.29499},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_efecf522a195; sha256 cb51a676599fedbe3717286b36f48ea64b029ebfc6416891d2add5552526295f},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.29499. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Physics-informed neural networks (PINNs) have emerged as a meshless alternative to conventional numerical methods for solving partial differential equations (PDEs). However, their limited ability to capture sharp gradients can lead to substantial errors when resolving boundary and interior layers. Here, we introduce an energy-norm-enhanced PINN (ENPINN) that incorporates gradient information and variational structure into the loss function to improve the resolution of layer-dominated solutions. We first examine two related formulations: weak-loss PINNs (WLPINNs), which incorporate test functions into the conventional PINN residual, and gradient-enhanced PINNs (gPINNs), which augment the loss with spatial derivatives of the PDE residual. By analyzing these formulations, we identify their limitations in resolving steep solution gradients and motivate the systematic construction of ENPINN. We establish theoretically how the energy-norm error depends on the ENPINN loss and show that a suitably modified residual-derivative term is essential for accurately capturing boundary layers. We further establish the existence of neural-network approximations with arbitrarily small energy error and derive corresponding derivative bounds, providing a theoretical foundation for the proposed framework. The performance of ENPINN is assessed through systematic comparisons with existing PINN variants for convection-diffusion-reaction problems exhibiting steep gradients. Numerical experiments include a combustion model, a coupled multi-scale system, a two-dimensional Burgers equation with an interior layer, and a three-dimensional time-dependent problem.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-01 in math.NA).
Categorie arXiv: Numerical Analysis (math.NA) ; Mathematical Physics (math-ph).
Pagina del lavoro: https://arxiv.org/abs/2608.29499 — 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 2026-10-17T08:59:54.352Z dalla pagina https://arxiv.org/catchup/math.NA/2026-09-01?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).